Reliability analysis of freight elevator systems based on FFMEC, FTA and BN

Abstract

Reliability design is critical to the safe operation of complex systems, and the large-tonnage freight elevator (LTFE) is a typical complex system. In order to improve the reliability of LTFE in the textile dyeing and printing industry, a comprehensive analysis method is proposed, which is based on Fuzzy Failure Mode, Effect and Criticality Analysis (FFMECA), Fault Tree Analysis (FTA) and Bayesian Networks (BN). First, FFMECA is analyzed by fuzzy analytic hierarchy Process (FAHP) and entropy-weight method. The fuzzy function was designed according to the characteristics of parameters Severity (S), Occurrence (O) and Detection (D), and the Fuzzy Risk Priority Number (FRPN) are calculated and ranked to identify high-risk failure modes. Next, the key causes of failure are identified by the analysis of FTA and BN. FTA model is constructed by the high-risk failure modes, then the FTA model is mapped into a BN model, and an inferential analysis is executed. Finally, a case study is conducted, and six high-risk failure modes are found, including gate structure fault, tractor failure, electromagnetic braking fault. Improvement measures were proposed and simulation verified that the system reliability is more than 94%, meeting the system design requirements.

Introduction

In recent years, the textile printing and dyeing industry developed rapidly, and the large-tonnage freight elevator plays a crucial role in the production of vertical transportation. The production environment of textile printing and dyeing industry is characterized by high temperature, humidity, strong corrosion and high load ¹, and involves a large number of personnel, vehicles and materials. Under the influence of multiple factors, the main failure modes of common freight elevators are corrosion, overheating, aging and impact damage, which greatly reduce the service life, safety and reliability of the freight elevator system ². In the textile printing and dyeing industry, the tonnage of most freight elevator is generally more than 3,000, and can be classified into LTFEs. According to business statistics, in the past ten years, the average failure rate of LTFEs reaches 63.9% by special production conditions, and the average failure rate is 32.7% higher than that of common freight elevators. High-frequent failures of elevator have caused the serious problems of economic losses and personnel safety. In order to ensure the operation of LTFE in textile printing and dyeing industry, the design and development costs must be taken into account.

In order to improve the reliability of the freight elevator system, scholars have carried out various studies to assess the risks of the freight elevator system. Wang Yi et al.³ combined mechanical principles and fault diagnosis technology, and proposed fault prediction and preventive maintenance strategies to ensure the operation of the freight elevator system from the starting point of service maintenance. Guo Hui et al.^4–7 studied the establishment of fault tree models for a series of elevator failures such as roof stroke, bottom squat and rolling, and made a qualitative and quantitative analysis of the freight elevator system through static reliability analysis methods such as FMEA and FTA, and put forward reasonable suggestions for fault prevention. By combining design failure analysis method with process failure analysis method (DFMEA-PFMEA), Shi Jiasong et al.⁸ carried out reliability analysis and optimization design of elevator tractor, and effectively identified tractor faults. Li Zhongxing et al.⁹ proposed an assessment method combining pre-risk analyzation-failure mode and impact analysis (PHA-FMEA) to establish a comprehensive risk assessment model for elevator systems, which reduces the influence of factors considered in traditional methods on elevator safety assessment results and makes the analysis more systematic and comprehensive. Qiang Wang et al.¹⁰ proposed a reliability analysis model based on the combination of orthogonal defect classification (ODC) and Markov analysis (MA), which can be used to evaluate the reliability of in-service traction elevators, and gave prevention strategies. Feng Wang et al.¹¹ established the FTA model by sorting out the reasons of elevator risks, and introduced the triangular fuzzy number (TFN) to reduce the influence of human factors and improve the assessment accuracy.

Although there are many literatures on the FMECA and FTA analysis of elevator systems, most of them are limited to specific subsystems, and use a single static reliability analysis method that lack the systematic, objective and dynamic analysis ability. In order to overcome the limitations of traditional FMECA and FTA analysis, scholars introduce enhancement methods to improve the accuracy of analysis. Sharma Arpit et al.¹² introduced the Fuzzy analytic hierarchy process (FAHP) to comprehensively evaluate the risk of failure mode and reduce the uncertainty of practice. Qingji Zhou et al.¹³ proposed a Fuzzy FMECA method to obtain the critical failure modes of Fuzzy Risk Priority Numbers (FPRNs) based on fuzzy theory and Alpha-level set theory. Esma Uflaz et al.¹⁴ applied Dempster-Shafe theory and Bayesian network model combining with expert opinions to optimize the defects during the FMECA analysis process and further reduce subjective influencing factors and enhance dynamic reliability assessment. Feng Zhou et al.¹⁵ put forward a comprehensive reliability analysis method based on FMEA-FTA that can identify the system failure mode and expand it into a fault tree model, and the weak links in the assembly process can be obtained, and the organic combination of FMEA and FTA can be realized. Longlong He et al.¹⁶ proposed an accident analysis method of coal mining and transportation system (CMST) based on fault tree analysis (FTA) combining with Bayesian network (BN) and pre-hazard analysis (PHA) to provide strong support for system safety and reliability. Shuai Zong et al.¹⁷ combined FTA with Bayesian network, and introduced mixture theory and fuzzy theory to simplify the fault tree model, so that the project in the early stage can obtain effective risk assessment. In addition, scholars also try to combine Bayesian networks^18,19 with fuzzy theory and dynamic analysis to reduce the influence of human and achieve more accurate diagnosis and prediction. Based on the above analysis, the improved reliability analysis method still has some problems, such as complicated analysis process, insufficient dynamic analysis ability and systematic analysis.

Aiming at the above problems, an improved FAHP and entropy-based methods to calculate index weights according to expert scores individually, the FRPNs are obtained and the high-risk failure modes are ranked. For the high-risk failure modes, an FTA model is constructed to perform root-cause analysis, which can also been mapped into BN for bidirectional reasoning. The results of FTA and BN are compared and analyzed to identify the primary causes of the high-risk failure modes, then give the improvement and optimization measures to enhance the reliability of the freight elevator system. The final verification of the design optimization effect through Monte Carlo simulation for system reliability. The highlights of this study are as follows:

1. The fuzzy theory-based FMECA method is introduced into the reliability analysis of LTFEs. Characteristic-failure mapping rules are constructed to supplement fault modes under special operating conditions. Expert experience and objective fault data are combined to provide quantitative indicators for system evaluation.

2. The FFMECA-FTA-BN system evaluation framework is developed, which links FMECA failure modes with FTA events and simplifies the reliability evaluation process. This framework integrates expert evaluation, objective data, and quantitative indicators, it improves design efficiency and enhances systematization in reliability analysis.

3. Bayesian networks are introduced to optimize fault tree analysis, and an FTA-BN-based method is proposed to identify key influencing factors. Monte Carlo simulation is used to verify the reliability of the optimized design, this approach enables dynamic analysis, supports iterative optimization, and ensures reliability validation.

This study aims to develop a compressive reliability analysis method integrating FFMECA, FTA and BN to improve the reliability design of LTFEs , and provide technical reference for the reliability design of other complex systems under special operating conditions.

Overall method

The overall analysis process of LTFE is shown in Fig. 1. Step 1: Failure modes, causes, and impacts are determined through expert experience and freight elevator failure data. A fuzzy semantic rule is developed based on the three critical parameters—Severity (S), Occurrence (O), and Detection (D), with each failure mode being scored individually by experts. The weights of these three parameters are independently calculated using FAHP and entropy-based methods, resulting in a comprehensive weight used to calculate the Failure Probability Risk Network (FPRN) and rank the high-risk failure modes. For.

Fig. 1.

FFMECA-FTA-BN general framework.

the identified high-risk failure modes, a Fault Tree Analysis (FTA) model is constructed for root cause analysis. Through quantitative analysis, the importance of the basic event structure is determined. The FTA model is then mapped to a BN for bidirectional reasoning, and a comparative analysis of the FTA and BN results is conducted to identify the primary causes of the high-risk failure modes. Based on the identified high-risk failure modes and their primary causes, targeted improvement and optimization measures are proposed to enhance the reliability of the freight elevator system in the textile dyeing and printing industry. Finally, Monte Carlo simulation is used to analyze the effectiveness of the optimized design, and the system reliability is verified to determine whether it meets the requirements or if further iterative design is needed.

Fuzzy FMECA method

Fault mode identification and fuzzy semantic specification development

FFMECA is an enhanced reliability analysis method that combines fuzzy theory with FMECA, which overcomes the limitations of traditional FMECA in dealing with the ambiguity and uncertainty of complex systems. According to the "Elevator Manufacturing and Installation Safety Code^20,21, the LTFE system is divided into eight subsystems: traction system, guide system, weight balance system, electrical control system, electrical drive system, safety protection system, door system, and cabin system. The textile printing and dyeing industry is characterized by high temperature, high humidity, and severe corrosion. By combining expert experience with elevator fault data, a feature-based failure mapping rule is developed. This rule supplements the characteristic failure modes of general condition system faults at three levels: system, subsystem, and physical components, as shown in Fig. 2. Based on operational data, 21 key failure modes are identified, as listed in Table 1. This rule can be expanded and modified according to the application of complex systems in different operating conditions to adapt to the required evaluation objects and their characteristics.

Fig. 2.

Industry characteristics and fault mapping rules.

Table 1.

LTFE System critical fault mode.

NO	Failure mode	NO	Failure mode	NO	Failure mode
FM 1	Car structure failure	FM 8	Sensor failure	FM 15	Floor anomaly
FM 2	Traction rope wear	FM 9	Lighting system anomaly	FM 16	Gate structure fault
FM 3	Guide mechanism failure	FM 10	Buffer failure	FM 17	Scheduling control fault
FM 4	Electromagnetic braking fault	FM 11	Weight balance anomaly	FM 18	Tractor failure
FM 5	Lubrication failure	FM 12	Control board fault	FM 19	The rectifier module is damaged
FM 6	Mechanical corrosion aging	FM 13	Abnormal power supply	FM 20	Electrical driving anomaly
FM 7	Control panel fault	FM 14	Electromagnetic compatibility fault	FM 21	Electrical control fault

The results of traditional FMECA analysis are often affected by subjectivity, leading to discrepancies between the results and actual situations. To address this issue, fuzzy theory is introduced to reduce the influence of subjective factors on the final results. In order to improve the objectivity of experts’ evaluations in the FRPN criteria, precise fuzzy semantic standards are defined for the expert’s capabilities, the three critical parameters in FRPN are: Severity (S), Occurrence (O), and Detection (D), and different fuzzy numbers are selected based on the characteristics of each parameter to enhance the accuracy of the evaluation, as shown in Table 2. The levels of S and O show different significance and exhibit relatively low fuzziness, here, a triangular fuzzy number is taken to denote it. For D, due to the variant effectiveness of detection methods, and considerable uncertainty, so trapezoidal fuzzy numbers are chosen. Experts

Table 2.

Failure severity, occurrence and detection fuzzy scoring criteria.

No	Severity(S)	Occurrence(O)	S&O fuzzy number	No	Detection(D)	D fuzzy number
1	None(N)	None (N)	(1,1,2)	1	Almost impossible (AI)	(1,1,2,3)
2	Negligible (NE)	Remote (RE)	(1,2,3)
3	Minor (MI)	Very Unlikely (VL)	(2,3,4)	2	Low probability (l)	(1,2,3,4)
4	Limited (LE)	Unlikely (L)	(3,4,5)
5	Moderate (M)	Infrequent (I)	(4,5,6)	3	Moderate probability (M)	(3,4,6,7)
6	Significant (SI)	Moderate (M)	(5,6,7)
7	Serious (S)	Occasional (O)	(6,7,8)	4	High probability (H)	(6,7,8,9)
8	Very Serious (VS)	Frequent (F)	(7,8,9)
9	Hazardous (H)	Very Frequent (VF)	(8,9,10)	5	Almost certain (AC)	(8,9,10,10)
10	Catastrophic (C)	Almost Certain (AC)	(9,10,10)

.

score the failure modes using the fuzzy semantic criteria from Table 2. Considering the potential biases in evaluations due to variations in experts’ experience, domain knowledge, and educational background, develop criteria for assessing expert competence, as shown in Table 3. Then the ability of each expert is assessed according to the standards of Table 3, and the scores are then normalized, and the weights W_l are assigned to each expert to make the evaluation results more reflective to the actual situations.

1

Table 3.

Evaluation criteria for comprehensive competence of experts.

Project	Evaluation index item	Score
Professional title a l (1)	Senior	4
	Deputy senior	3
	Intermediate	2
	Novice	1
Years of work in the field a l (2)	≤ 20 years	4
	≤ 15 years	3
	≤ 10 years	2
	≤ 5 years	1
Educational level a l (3)	PH.D	3
	Master	2
	Undergraduate course	1
Job post a l (4)	Research Institute	4
	Manufacturing enterprise	3
	Special Inspection Institute	2
	University	1

where, W_l is the normalized weight for the l-th expert; is the capability score of the k-th item for the l-th expert.

Fuzzy risk priority numbers (frpns) calculation

The Risk Priority Number (RPN) is a key index used in FMECA to assess the impact of failure modes. It is primarily calculated by multiplying three parameters: Severity (S), Occurrence (O), and Detection (D). To reduce the influence of human factors in the RPN calculation, a combined weighting method is introduced, which integrates the subjective Analytical Hierarchy Process (FAHP) and the objective Entropy method. The weights for S, O, and D are calculated separately using these two methods. The normalized weight is derived by calculating the weighted average of the FAHP weight and the Entropy method weight by Eq. (2). The use of both weighting methods effectively combines their advantages, balancing expert experience with objective data to achieve a unified approach, thus improving the objectivity and accuracy of the calculation results.

2

where, is the comprehensive normalized weight; is the FAHP weight; is the entropy weight; is corresponding to S, O, and D in sequence.

FAHP subjective empowerment method

First, the complementary judgment matrix is constructed according to the index parameters shown in Table 4, the fuzzy complementary judgment matrix is combined with expert scores, and the elements in the judgment matrix meet (3). Secondly, the eigenvalues and eigenvectors of the matrix are calculated, and the fuzzy consistency matrix is constructed by the normalization of the eigenvectors. Finally, the weight vector is determined according to the maximum membership degree method, the Consistency index (CI)is calculated using Eq. (4), and the Consistency Ratio (CR) is tested according to Eq. (5), then the normalized S, O and D weights are obtained.

3

4

Table 4.

Fuzzy complementary matrix parameter scale.

Scale	Instructions
0.5	Both elements are equally important
0.6	One element is slightly more important than the other
0.7	One element is significantly more important than the other
0.8	One element is much more important than the other
0.9	One element is more important than the other
0.1, 0.2, 0.3, 0.4	If element is compared with element to get the judgment , then element is compared with element to get the judgment

where, is the maximum eigenvalue of the fuzzy judgment matrix; is the number of matrix evaluation indicators.

5

where, is the consistency ratio; is a random consistency index, obtained according to the matrix order table, 0,0,0.52, 0.89, 1.12, …, 1.59.

Entropy weight method objective weight method

First of all, the original data matrix is constructed according to the expert scoring and evaluation results, the index standardization Eq. (7) is used to standardize the data of each element, and the standardized matrix is constructed. After standardization, the information entropy of each index is calculated through (8), the larger the entropy value, the smaller the impact on the result. Finally, the entropy of each index , the weight of each indicator is calculated by (9), the greater the weight, the more significant the impact of the evaluation.

6

where, is the evaluation value of the i-th evaluation object under the j-th evaluation index; m indicates the number of evaluation objects.

7

where, is the standardized value in the range of [0,1]. is the minimum value of the j-th evaluation index. is the maximum value of the j-th evaluation index.

8

where, is the information entropy value of the j-th evaluation index.

9

where, is the weight of the j-th evaluation index.

The membership function is defined based on the fuzzy number, which depends on the differential characteristics of each index. The membership function represents a fuzzy set of in the field U. Every element x in U has a membership degree corresponding to it, where , are called membership function, such as Eq. (10), (11).

10

where, is the lowest possible value (lower limit), is the highest possible value, and is the ultimate possible value (upper limit).

11

where, is the lowest possible value (lower limit), is the left platform limit value (middle value), is the right platform limit value (middle value), and is the limit possible value (upper limit).

Finally, The expert weights are combined with the fuzzy evaluation results of fault modes. The weighted average of each index S, O and D is calculated using Eq. (12). Then, the gravity center method is applied to defuzzify the comprehensive fuzzy number and obtain the final result. The FRPN formula is applied to calculate the comprehensive weights of S, O and D, the FRPN of each fault mode is calculated by (13), and ranked to identify high-risk fault mode.

12

where, is S, O and D correspond to the comprehensive fuzzy weighted average; is the expert weight; is the corresponding weights for S, O and D; is the S, O and D correspond to the fuzzy weighted average.

13

FTA-BN model construction method

FTA model construction and analysis

To improve the systematic and dynamic analysis capability of reliability, the FTA¹⁷ model is constructed. The high-risk failure mode determined by the FRPN results in "Fuzzy FMECA method" section, servers as input for intermediate events. The second-level events are identified from three aspects: software, hardware and operation. The causes of failure mode are then decomposed step by step to the smallest basic events combined with FFMECA analysis. The bottom event is linked to the fault cause in FMECA. The valid information in reliability analysis is reused to enhance the completeness of the analysis results and the accuracy of decision support. Finally, events at each level are connected through logical relations with gates (AND), OR gates (OR). Then, a complete FTA model is established based on the elevator fault data, and the prior probability of the bottom event in the FTA model is assigned. The structure function is constructed and simplified based on the logic gate formula, and the first-order and second-order minimum cut set are determined using Eq. (14). Quantitative analysis is then used to determine the structural importance of the bottom event and the probability importance of the basic event.

14

where, is the structural importance of the basic event ; is the minimum number of cut sets contained in the fault tree. is the minimum number of cut sets containing the -th basic event; is the number of fundamental events of the C smallest cut set of the -th fundamental event.

FTA-BN network mapping and analysis

BN¹⁸ is a graphical model that uses Bayesian theorem of conditional probability (16) to represent causality. BN model can be visually represented as a directed acyclic graph (DAG). First, based on FTA model structure learning, the top event, middle event and bottom event in the fault tree are hardwired to the BN node. Logic gates (" and gates ", “or gates”) map to inter-node causality, making the dependencies between BN model nodes consistent with the FTA model. Secondly, based on the freight elevator fault data and expert experience, a prior probability P (Xi) is assigned to the root node for parameter learning. The connection strength between causal relationships is represented based on the input data and the conditional probability function (CPF) (15), as shown in Fig. 3.

15

Fig. 3.

Bayesian network structure schematic diagram.

where, is the topped event (leaf node); is the intermediate node and the root node of the Bayesian networks; is the probability of event occurrence; is the number of nodes in the BN.

Finally, based on the quantitative analysis results of FTA and BN, the main influencing factors of each failure mode are determined, and targeted improvement measures are proposed to reduce the impact of failure, to improve the reliability and the average time of normal operation.

Case study

In order to verify the effectiveness of the above proposed method based on FFMECA- FTA-BN in improving the reliability of the freight elevator system, a case of large-tonnage freight elevator systems in the textile dyeing industry of 5 to 10 T is studied.

Results of FFMECA and FRPN analysis

First, the LTEF fault data for 180 days of production and use were collected, which data is from the enterprise over a period of no more than ten years. In this study, four elevator experts from different directions were invited, and they conducted FMECA analysis to determine the failure mode, causes and effects, which identify 21 critical fault modes. Parameters S, O and D were scored and evaluated, and this was done by combining the fault data and expert experience. The scoring criteria are shown in Table 5. At the same time, in view of the differences in expert ability, the expert ability was evaluated according to the evaluation indicators of experts in Table 3. The normalized expert weight of each expert was calculated through (1), which was 0.214, 0.262, 0.310 and 0.214 respectively, as Table 6 shows.

Table 5.

Results of expert evaluation for 21 key failure modes.

NO	Expert 1	Expert 2	Expert 3	Expert 4	NO	Expert 1	Expert 2	Expert 3	Expert 4
FM 1	C, VL, L	H, VL, M	VS, VL, H	C, N, AC	FM 12	S, N, L	VS, VL, L	VS, VL, M	VS, VL, H
FM 2	SI, VF, AC	M, F, M	M, F, L	M, F, H	FM 13	S, N, M	VS, N, L	S, N, L	S, N, H
FM 3	M, F, H	SI, F, M	M, F, H	M, O, H	FM 14	S, I, L	SI, I, M	S, VL, H	S, L, AI
FM 4	S, O, AC	VS, L, M	H, I, H	S, M, H	FM 15	M, L, M	M, I, H	M, L, M	M, VL, L
FM 5	M, M, L	M, M, M	L, M, H	M, L, M	FM 16	L, VF, M	M, VF, AC	SI, VF, AC	M, F, H
FM 6	VS, VL, H	VS, L, M	VS, N, M	H, L, L	FM 17	C, N, AC	C, N, AC	VS, N, AC	H, VL, AC
FM 7	S, L, L	SI, VL, M	H, M, AI	VS, I, L	FM 18	C, L, H	S, O, H	SI, M, AC	VS, VL, H
FM 8	M, O, H	M, M, M	M, I, L	SI, O, M	FM 19	C, VL, L	VS, N, M	H, N, AC	VS, AI, H
FM 9	L, L, AC	NE, L, AC	NE, VL, AC	L, VL, AC	FM 20	S, I, L	S, N, AC	SI, N, H	VS, N, M
FM 10	M, N, H	SI, N, M	S, VL, H	S, N, L	FM 21	S, I, M	L, M, L	SI, M, H	L, L, AC
FM 11	SI, L, L	S, L, M	VS, VL, H	SI, I, L

Table 6.

Expert competency assessment results.

Project	Expert1	Expert2	Expert3	Expert4
al(1)	3	2	4	4
al(2)	4	3	3	1
al(3)	1	2	2	3
al(4)	3	2	4	1
Score	11	9	13	9
Weight	0.262	0.214	0.310	0.214

Next, based on the expert scoring results for S, O, and D and the expert capability weights , the weights for the S, O, and D parameters are calculated using both FAHP and the Entropy method. In the FAHP process, four experts performed pairwise comparisons of the S, O, and D parameters according to Fuzzy complementary matrix parameter scale. They constructing fuzzy reciprocal judgment matrices. These matrices were combined with the expert weights to form a comprehensive fuzzy matrix . The normalized weights for S, O, and D were calculated as 0.327, 0.408, and 0.264, respectively. For the Entropy method, the scoring results from Table 1 were converted into fuzzy numbers according to the definitions in Table 2. A fuzzy matrix was constructed and normalized. The entropy values and weights for each failure mode parameter were computed using Eqs. (8) and (9). The resulting normalized weights for S, O, and D were 0.288, 0.277, and 0.435, respectively. Finally, the comprehensive normalized weights for S, O, and D were determined using Eq. (2), yielding values of 0.292, 0.351, and 0.357, respectively.

16

Finally, based on the fuzzy number mapping results from the expert scoring, the comprehensive fuzzy numbers for the S, O, and D parameters of each failure mode are calculated using Eq. (12). The centroid method is then applied to defuzzify the comprehensive fuzzy numbers, yielding values , , and for S, O, and D, respectively. Using Eq. (13), the FRPN for each failure mode is calculated, and the FRPN values are used to rank the severity of the failure modes, as Table 7 shows. FRPNS with high importance of major fault modes can be extracted from the table and sorted in descending order: FM16 > FM18 > FM4 > FM3 > FM2 > FM17. A higher FRPN value indicates a greater risk posed by the failure mode to the system, which allows for the identification of six high-risk failure modes: gate structure failure, tractor failure, electromagnetic brake failure, guide mechanism failure, tractor rope failure and electrical control failure, and provides the basis for the subsequent root cause and transmission path analysis.

Table 7.

Fault mode gravity center method to solve fuzzy and FRPN results.

NO							FRPN	Ranking
FM 1	(8.118,9.118,9.69)	(1.786,2.786,3.786)	(4.572,5.572,6.834,7.62)	9.01	2.21	9.25	5.20	9
FM 2	(4.214,5.214,6.214)	(7.214,8.214,9.214)	(4.092,5.092,6.354,7.14)	5.21	8.21	5.67	6.30	5
FM 3	(4.262,5.262,6.262)	(6.786,7.786,8.786)	(5.214,6.214,7.476,8.476)	5.26	7.79	6.85	6.63	4
FM 4	(6.882,7.882,8.882)	(4.380,5.380,6.380)	(5.642,6.642,7.904,8.690)	7.88	5.38	7.22	6.68	3
FM 5	(3.690,4.690,5.690)	(4.572,5.572,6.572)	(3.502,4.502,5.978,6.978)	4.69	5.57	5.24	5.18	10
FM 6	(7.214,8.214,9.214)	(2.166,3.166,4.166)	(3.214,4.214,5.786,6.786)	8.21	3.17	5.00	4.92	12
FM 7	(6.572,7.572,8.572)	(3.572,4.572,5.572)	(1.524,2.214,3.476,4.476)	7.57	4.57	2.92	4.52	16
FM 8	(4.214,5.214,6.214)	(5.118,6.118,7.118)	(3.022,4.022,5.498,6.498)	5.21	6.12	4.76	5.34	7
FM 9	(1.428,2.428,3.428)	(2.476,3.476,4.476)	(8.000,9.000,10.000,10.000)	2.43	3.48	9.25	4.44	18
FM 10	(5.310,6.310,7.310)	(1.310,2.310,3.310)	(4.144,5.144,6.406,7.406)	6.31	2.31	5.78	4.30	20
FM 11	(5.882,6.882,7.882)	(2.904,3.904,4.904)	(3.074,4.074,5.336,6.336)	6.88	3.90	4.71	4.92	13
FM 12	(6.786,7.786,8.786)	(1.786,2.786,3.786)	(2.690,3.690,5.000,6.000)	7.79	2.79	4.35	4.41	19
FM 13	(6.262,7.262,8.262)	(1.000,2.000,3.000)	(2.498,3.498,4.712,5.712)	7.26	2.00	4.11	3.77	21
FM 14	(5.738,6.738,7.738)	(3.166,4.166,5.166)	(3.074,3.86,5.122,6.122)	6.74	4.17	4.54	4.95	11
FM 15	(4.000,5.000,6.000)	(3.048,4.048,5.048)	(3.358,4.358,5.882,6.882)	5.00	4.05	5.12	4.68	15
FM 16	(4.096,5.096,6.096)	(7.786,8.786,9.786)	(6.502,7.502,8.716,9.144)	5.10	8.79	7.97	7.24	1
FM 17	(8.166,9.166,9.69)	(1.214,2.214,3.214)	(8.000,9.000,10.000,10.000)	8.98	2.79	6.15	5.56	6
FM 18	(6.546,7.546,8.332)	(4.192,5.192,6.192)	(6.620,7.620,8.620,9.310)	7.48	5.19	8.04	6.75	2
FM 19	(7.738,8.738,9.524)	(1.214,2.000,3.000)	(4.764,5.764,7.026,7.716)	8.67	2.16	6.32	4.75	14
FM 20	(5.904,6.904,7.904)	(1.642,2.642,3.642)	(4.812,5.812,7.026,7.764)	6.90	2.00	6.89	4.47	17
FM 21	(3.834,4.834,5.834)	(4.358,5.358,6.358)	(5.118,6.118,7.118,7.904)	4.83	5.36	5.66	5.30	8

Results of FTA-BN analysis of high- hazard failure mode

The main threats to LTFE come from high-frequency, high-hazard failures. These failures are primarily concentrated on the failure modes identified in the FFMECA analysis. By conducting root cause analysis on these failure modes, system reliability can be effectively improved. The FTA modeling process is as follows: The six high-risk failure modes identified in "Results of FFMECA and FRPN analysis" section are treated as intermediate events(T_i), with LTFE failure being the top event(T). Secondary events A_i and B_i are constructed from the hardware, software, and functional aspects. Some of the bottom events X_i are derived from the failure causes identified in the failure mode analysis. The others are supplemented using literature, expert experience, and failure data. This approach links and complements the FMECA and FTA analysis processes to enhance the systematic nature of the analysis. The final FTA model is shown in Fig. 4. A total of 497 LTFE fault data from ten years were collected for the bottom events of the fault tree. The occurrence frequency and failure rate were determined based on the bottom events at each level of the fault tree, and the data for subsequent quantitative fault tree and BN analysis is shown in Tables 8 and 9.

Fig. 4.

FTA based on FFMECA high-hazard faults.

Table 8.

Upper-level risk events in the LTFE FTA model.

Code	Event	Code	Event	Code	Event
T	Freight elevator system failure	A 4	Traction rope wear	B 2	Rail bracket failure
T 1	Tractor failure	A 5	Guide shoe failure	B 3	Sensor failure
T 2	Traction rope failure	A 6	Rail and bracket failure	B 4	Actuator hardware failure
T 3	Guiding mechanism failure	A 7	Brake structure failure	B 5	Control program failure
T 4	Electromagnetic brake failure	A 8	Brake control failure	B 6	Self-locking failure
T 5	Electrical control failure	A 9	Control hardware failure	B 7	Interlock mechanism failure
T 6	Gate structure fault	A 10	Control software failure	B 8	Door panel failure
A 1	Gearbox failure	A 11	Synchronous drive fault	B 9	Threshold failure
A 2	Traction structure failure	A 12	Actuator failure
A 3	Traction rope failed	B 1	Rail failure

Table 9.

LTFE FTA model bottom events and probability statistics.

Code	Event	F	P(Xi)	Code	Event	F	P(Xi)
X 1	Lubrication failure	27	5.43%	X 28	Environmental corrosion	44	8.85%
X 2	Gear wear	7	1.41%	X 29	Sensor failure	72	14.49%
X 3	Cooling failure at high temperatures	14	2.82%	X 30	Signal communication anomaly	15	3.02%
X 4	Motor winding failure	3	0.60%	X 31	Position sensor	33	6.64%
X 5	Power supply failure	19	3.82%	X 32	Speed sensor	19	3.82%
X 6	The coupling is damaged	20	4.02%	X 33	Load sensor	6	1.21%
X 7	Bearing failure	11	2.21%	X 34	Circuit aging	13	2.62%
X 8	Installation deviation	19	3.82%	X 35	The hardware environment corrodes and overheats	27	5.43%
X 9	The tow rope is broken	4	0.80%	X 36	CPU failure	8	1.61%
X 10	Uneven tension slip	12	2.41%	X 37	Memory failure	5	1.01%
X 11	Environmental corrosion	38	7.65%	X 38	I/O module fault	6	1.21%
X 12	Over or under lubricated	44	8.85%	X 39	Signal processing error	2	0.40%
X 13	Uneven tension	49	9.86%	X 40	Serial communication error	5	1.01%
X 14	Material defect	7	1.41%	X 41	The hardware environment	17	3.42%
X 15	Rope core wear	3	0.60%	X 42	Sensor failure	119	23.94%
X 16	Misalignment during installation	27	5.43%	X 43	Structural failure	23	4.63%
X 17	Lubrication chamber aging	19	3.82%	X 44	Self-closing device adjustment failure	39	7.85%
X 18	Loose fasteners due to vibration	79	15.90%	X 45	Electrical fault	47	9.46%
X 19	Environmental corrosion	53	10.66%	X 46	The interlock switch fails	41	8.25%
X 20	Damage from external impact	61	12.27%	X 47	Interlock mechanism blocking	26	5.23%
X 21	Installation deviation	2	0.40%	X 48	Impingement deformation	84	16.90%
X 22	Rail manufacturing defect	6	1.21%	X 49	Environmental corrosion, aging	37	7.44%
X 23	Under weld	7	1.41%	X 50	Knife and door lock fail	17	3.42%
X 24	Material defect	2	0.40%	X 51	Door guide failure	5	1.01%
X 25	External impact failure	5	1.01%	X 52	Pulley wear	29	5.84%
X 26	Blocked by foreign dust	8	1.61%	X 53	Level induction failure	16	3.22%
X 27	Electromagnetic coil insulation aging	16	3.22%	X 54	The threshold foreign body stuck	29	5.84%

The structure function is built from the bottom-up method based on the logical gate relationships in the FTA. The first-order and second-order minimal cut sets are determined through qualitative analysis. The structural importance of bottom events is analyzed using Eq. (16). Based on the FTA model framework, the structure of the BN is learned. The bottom events are classified into “failure” and “normal” states, corresponding to "0–1" variables. A BN model consistent with the FTA model is then constructed. The parameter learning for the BN is based on the prior probabilities of the root nodes P(X_i) as the input. Backward reasoning is performed on the events in the FTA model through the logical relationships between nodes, and the calculation of the posterior probabilities and importance of each network node is considered.

Finally, the FTA and BN inference results are combined for comparative analysis, as shown in Fig. 5. When FTA structure is of the same importance, the extreme value of BN posterior probability and sensitivity analysis results is from the same bottom event. The bottom event with high posterior probability of BN is selected as the key influencing factor, such as X₃, X₁₃. When the FTA structure importance is different and does not coincide with the results of BN posterior probability and sensitivity analysis, the key influencing factors are determined by comparing the ratio of FTA structure importance with BN posterior probability, such as X₁₈, X₄₈. When the same bottom event takes the highest value, the bottom event is selected as the key influencing factor, such as X₂₈ and X₃₅. According to this method, the key factors of high-risk failure modes are as follows: impingement deformation, cooling failure at high temperatures, blocked by foreign dust, Environmental corrosion, uneven tension and the hardware environment corrodes and overheats.

Fig. 5.

Bayesian network posterior probability and FTA structure importance comparison.

Improvement measures and result analysis

Based on the above analysis, the high-risk failure modes of LTFE and their main influencing factors are identified, and corresponding improvement and optimization measures are also proposed. The key cause of the gate structure fault is the high-temperature environment. Currently, two measures are suggested to improve the gate structure fault: (1) Enhance the motor radiator performance, increase the dust protection net, and establish a cleaning cycle; (2) Improve the equipment room’s ventilation structure and connect it with the well air exchange system to reduce humidity and temperature. The remaining fault modes and their improvement measures are listed in Table 10, and the initial result improvements of above fault modes are given. These analysis method can also be applied to complex mechanical systems operating in special working conditions, such as the chemical industry and explosion-proof fields, providing a standardized framework for reliability design.

Table 10.

High hazard fault mode targeted improvement and optimization measures.

NO	Failure mode	Fault cause	Fault effect	Improvement measure
FM 18	Tractor failure	Cooling failure at high temperatures	Local: Traction machine overheating damage, abnormal operation or can not run Overall: Abnormal operation of the freight elevator system vibration, abnormal speed or unable to operate	FM18_1. Add dust isolation net, develop cleaning cycle specifications, reduce the impact of environmental foreign matter on heat dissipation; FM18_2. Improve the ventilation structure of the machine room, connect the well dehumidification ventilation system, and reduce the temperature and humidity of the machine room
FM 16	Gate structure fault	Impingement deformation	Local: The freight elevator door is not working properly Overall: The freight elevator system cannot complete the transportation function	FM16_1. Design a vehicle detection system based on machine vision and connect it with the vehicle control system to realize collision warning and emergency stopping FM16_2. A blocking device is set outside the floor door to prevent vehicle impact through machine vision detection and vehicle path prediction algorithm
FM 4	Electromagnetic braking fault	Blocked by foreign dust	Local: The braking ability of the brake is weakened or cannot be started or stopped Overall: The freight elevator system can not stop or locate the floor normally, and the basic function is lost	FM4_1. Design brake shield, add dust shield and cooling device; FM4_2. Formulate regular cleaning specifications to remove the oil, dust and foreign matter generated by the printing and dyeing industry such as cotton, linen and silk on the attached surface
FM 3	Guide mechanism failure	Environmental corrosion	Local area: The guiding effect is weakened, and the wear of synergistic parts is increased Overall: the freight elevator operation resistance increases, abnormal sound, shutdown and other faults	FM3_1. Optimize structural materials, welded joints and surface coatings; FM3_2. Design the dehumidification system, design the water level detection and discharge device in the pit, and design the integrated ventilation device between the well and the car room
FM 2	Traction rope wear	Uneven tension	Local: the running resistance of the traction rope increases, and the wear of the synergistic parts increases Overall: Abnormal vibration or stop operation of the freight elevator system	FM2_1. Design tension detection and self-adjusting device for fixed end of rope head; FM2_2. Design the car floor partition weight detection mechanism to assist in adjusting the placement of goods to reduce the impact of partial load
FM 21	Electrical control fault	The hardware environment corrodes and overheats	Local: The control cabinet stops data processing and signal transmission Overall: the freight elevator system is out of control and has a serious failure	FM21_1. Design the liquid cooling device of the electric control cabinet to ensure the thermal performance requirements of the core components; FM21_2. The packaging structure of the electric control cabinet is designed to prevent water vapor from entering the corrosive aging electronic components;

Based on the above analysis and improvement measures, a redundancy design for the system’s FTA model is carried out, and taking event T₁, T₂ and T₃ as an example, a redundant fault tree model is constructed, as shown in Fig. 6. Monte Carlo simulation is performed based on this model to compare the system reliability under no redundancy, K/N redundancy, and active redundancy. The analysis confirms that the redundancy design improves reliability compared to the no redundancy case, and the active redundancy performs the best, with the MTBF simulation result of 1360.93 h, as shown in Fig. 7. The reliability change of the redundancy design over time is shown in Fig. 8, the system reliability gradually decreases with time, which decline rate accelerates between 10 and 20 years. The increase in redundancy improves reliability, this effect is significant between 5 and 15 years. The system can maintain high reliability until the 15th year, meeting the reliability requirements.

Fig. 6.

Fault tree model of redundant design.

Fig. 7.

MTBF simulation results of redundant design.

Fig. 8.

Reliability curve with time and redundancy.

In order to highlight the advantages of this method. As shown in Fig. 9, compared with the traditional FMECA method, the FMECA method with fuzzy theory can effectively reduce the bias from individual subjectivity and single-factor influences in the traditional FMECA. The six failure modes identified by FRPN do not fully match the results from the four traditional FMECA evaluations. For example, in the FM₁₈ Sensor failure, RPN-E4 did not identify this failure mode as high-risk. This was due to the lack of elevator operational information, leading to assessment bias. Introducing fuzzy theory can effectively eliminate the influence of the evaluator’s subjectivity and identify high-risk failure modes that align with actual operating conditions.

Fig. 9.

Comparison of LTFE fuzzy FMECA and FMECA results.

Finally, the prototype improved using this method is currently operating normally. To further validate the effectiveness of this method, reliability prediction is performed through Monte Carlo simulation and compared with traditional analysis methods such as FMECA-FTA and DFTA-BN. Over the first 10 years of LTFE operation, the average reliability is approximately 94%, which meets actual usage requirements and relatively outperforms similar analysis methods. In the 10–15 years period (the maximum service life of LTFE is 15 years, and Fig. 10 compares data only within the effective service life), due to data limitations and the complexity of real-world conditions, the reliability is slightly lower than that predicted by DFTA-BN, but still meets expectations and usage requirements, as shown in Fig. 10.

Fig. 10.

Comparison of the results of this study’s method with similar methods.

Conclusions

This paper proposes a systematic reliability analysis method based on FFMECA-FTA-BN. Fuzzy theory is introduced to improve the FMECA method, which helps identify potential failure modes and causes in large-tonnage freight elevators in the textile printing and dyeing industry. It also establishes S, O, D parameters, criteria for expert capability evaluation, and feature-fault mapping rules. Based on the system, subsystem, and component structure, 21 main failure modes are identified, and after expert evaluation and FRPN calculation, six fault modes with the greatest impact on the system are obtained. A fault tree model is constructed based on these six fault modes, and the minimum cut set is found and structural importance is calculated. The fault tree model is then mapped to a BN model, and bidirectional reasoning is applied to all events in the FTA model to assess their impact and sensitivity. Finally, key influencing factors for high-risk failure modes are identified through comparative analysis. The case study results show that the main fault modes are gate structure faults, tractor failures, electromagnetic braking faults, guide mechanism failures, traction rope wear, and electrical control faults, and they are the main weaknesses of the large-tonnage freight elevator system in the textile printing and dyeing industry. The key influencing factors are impingement deformation, cooling failure at high temperatures, blockage by foreign dust, environmental corrosion, uneven tension, and hardware environment corrosion and overheating, and specific measures is taken to optimize design or prevention. Finally, a comparison with traditional and existing enhanced methods was conducted, verifying that the proposed method provides improvements. System reliability was evaluated using Monte Carlo simulation and confidence interval estimation, revealing that reliability remains above 94% during the first ten years of operation. Additionally, in this case, the overall workload for reliability analysis was reduced by more than 53.74%, demonstrating that the method supports efficient reliability design for complex mechatronic systems to adapt to special operating conditions.

This research takes a large number of data processing and analysis tasks manually, and system simulation analysis is operated independently. In the future work, the intelligent technologies, such as MBSE, deep learning, etc., is perhaps introduced to reduce the analysis workload and improve the systematic analysis process.

Author contributions

J.Y. Conceptualization, Methodology, Software. J.Y. Data curation, Writing—Original draft preparation. J.Y. Data processing and analysis. J.Y. Writing—Reviewing and Editing. C.L. Improve language quality. J.Y. J.Z. C.L. Z.C. C.H. All authors reviewed the manuscript.

Funding

This work was supported in the Shaoxing Science and Technology. Plan Project (Grant no. 2022B41004).

Data availability

The datasets used and analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.