*Article* **Research on the Optimization Method of Safety Input Structure in Coal Mine Enterprise**

**Xiu-Zhi Shi, Jin-Yun Zhu and Shu Zhang \***

School of Resources and Safety Engineering, Central South University, Changsha 410083, China **\*** Correspondence: shuzhang303@gmail.com

**Abstract:** In order to study the application of the Cobb-Douglas production function on the optimization of safety inputs and further reduce accident losses, two safety input structures of a coal mine enterprise were constructed using literature, and the weight order of each safety input indicator was determined by the entropy weight method (EWM) and the analytical hierarchy process (AHP). The Cobb-Douglas production function was used to calculate the accident loss function of the safety input structure, and the accident loss function was obtained by multiple regression analysis. The optimal configuration of safety inputs was obtained by fitting the accident loss function. Finally, the optimal loss and mean squared error (MSE) of the corresponding functions of the two safety input structures were compared. The results show that the optimal configuration of Safety Input Structure 2 is better than that of Safety Input Structure 1, and the MSE of Safety Input Structure 2 is less than that of Safety Input Structure 1. The research results demonstrate that coal enterprises can find more significant indicators by refining the safety input structure and increasing monetary resources for more crucial indicators of safety input to effectively minimize accident loss and boost economic benefits, and to test the quality of safety input structures' regression function using MSE.

**Keywords:** safety input; Cobb-Douglas production function; structure optimization; safety input structure; comprehensive empowerment

#### **1. Introduction**

Due to the continued expansion of China's economy and society, personal safety, particularly the life and health of enterprise workers, has received increasing attention in recent years. Coal, which accounts for more than half of China's total energy supply, will continue to be the primary energy source for a long time. Various safety accidents occur occasionally in large-scale coal mines. According to statistics, the total number of coal mine accident fatalities in China is 1–2 times higher than that of other coal-producing countries. The coal mine fatality rate per million tons in 2017 was 0.06, which is twice that of the United States and 1.5 times that of Germany [1,2]. One of the most essential approaches to managing the frequency of accidents successfully is to ensure sufficient safety input. Because the development of businesses is always focused on lowering costs and increasing benefits, the money businesses invest in safety is limited. Safety input is characterized by latency and invisibility [3]. As a result of limited safety funds, how to enhance the utilization rate of safety funds, maximize the protection of employees' lives and maintain normal corporate production is a subject to be explored.

Numerous researchers have conducted extensive studies of the utilization of safety input funds. Some researchers established a safety input model and used data analysis of the safety input–output function or the safety input–output relative outcomes of each scheme to determine the ideal safety solution. For example, Jiang et al. [2] used a type of entropy-close to ideal solution (TOPSIS) to establish a safe input scheme evaluation model and calculate the safety input's best configuration. Song et al. [4] used the order relation method and expert scoring method to determine the weight, and the

**Citation:** Shi, X.-Z.; Zhu, J.-Y.; Zhang, S. Research on the Optimization Method of Safety Input Structure in Coal Mine Enterprise. *Processes* **2022**, *10*, 2497. https://doi.org/10.3390/ pr10122497

Academic Editors: Francisco Ronay López-Estrada, Guillermo Valencia-Palomo and Anthony Rossiter

Received: 14 October 2022 Accepted: 15 November 2022 Published: 24 November 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Cobb-Douglas (C-D) production function to construct the safety input–output function relationship. Guo et al. [5] comprehensively worked out the optimal scheme of safety input allocation by using grey correlation degree analysis, partial correlation analysis and the C-D production function model. Based on the literature, Ye et al. [6] identified widely used safety input indicators and used data envelopment analysis to build a safety output efficiency assessment model and evaluate the efficiency methods of various safety inputs and outputs. Various academics have also proposed some strategies for identifying safety input indicators: Wang et al. [7] used cluster analysis to pick the primary indications of safety input. By studying the rationale and accuracy of the assessment indicators of safety input, Li et al. [8] created an evaluation indicator system comprising three Level Two and 16 Level Three indicators. From the standpoint of a safety system, Zhang et al. [9] built a safety input assessment indicator system. Xin [10] converted the safety decision in engineering to a mathematical problem, establishing three robust optimization models using robust optimization technology, thereby providing safety input decision-making models for different enterprise risk scenarios. Chao [11] presented the economic optimization model and relative optimization algorithm based on CBA and CEA for the process industry. Roy and Gupta [12] created a nonlinear optimization model for safety input. Son et al. [13] established a safety input optimization model that aims to minimize the overall safety input expenditure of coal mining enterprises. Compared with foreign countries' research on industry enterprises, many studies of coal mine enterprises' safety input have been conducted in China. Although research on safety input has reached a relatively mature stage in China [14], China lags far behind the United States in terms of coal mine safety [15]; there is currently less relevant study of the influence of various safety input models on safety output.

To sum up, the current studies on the optimization of safety input structure have all established only one indicator structure [2,4–6,10–13] for optimization. In order to better increase the efficiency of safety inputs, this research uses two safety input structures for comparison for the first time. The two safety input structures are obtained by combining references to previous security input construction methods and models [2,16–18]. The research objectives are twofold: in contrast to the current situation, where only one indicator structure has been used, two safety input structures were used to verify that different indicator structures make different optimization assignments for safety input. While the C-D production function was used to construct a function model, the Mean Squared Error (MSE) of the two obtained regression functions were compared for the first time to illustrate that when selecting safety input indicators for modeling, the regression function that minimizes the MSE should be selected. The regression function with the lower MSE is not only closer to reality, but also obtained a better safety input assignment. The research discusses that when the C-D production function is used to model and the accident loss function equation is obtained by multiple linear regression, if the safety input indicator structure can be refined, the optimal solution of the accident loss function is lower and the coal mine enterprise's safety input efficiency also can be raised to a higher extent.

Thus, the analytic hierarchy process and the entropy weight method (AHP-EWM) is used to assign weight comprehensively to each indicator, and the C-D production function is applied to construct the accident loss function model. Finally, the optimal accident losses of functions are compared and the fitness quality of accident loss functions is examined using MSE. While constructing accident loss functions, the preventive safety input is assumed to be constant, minimizing the accident losses of the enterprise by changing the assignment of each safety input in the safety input structure. Simultaneously, two safety input structures of the same coal mine enterprise are constructed according to the various hierarch and the number of indicators of the preventive safety input, and the optimal accident loss and MSE of the two safety input structures are compared under the same preventive safety input expenditure.

#### **2. Construction of Safety Input Structure of Coal Mine**

#### *2.1. Establishment of a Hierarchical Structure of Safety Input in Coal Mine Enterprises*

At the moment, there is no consistent definition of safety input, and there are variances in the separation of various elements of safety input. Safety input [8,16] refers to the total amount of money spent by a company to ensure the safety and health of its personnel during the manufacturing process. Safety input [8,19] refers to the total of a country's or enterprise's safety-related costs, such as expenditures on safety measures, personal protective equipment and occupational illness prevention, among other things. The human, material and financial resources invested to manage hazard sources in the manufacturing process, prevent possible accidents and provide safe production conditions are referred to as safety input [8,20]. The term "safety input" [2] refers to the amount of human, material and financial resources expended during the production and operation processes to assure the safe production of enterprises, eliminate possible mishaps and lower the fatality rate. The State Administration of Work Safety has split the safety expenditures of coal enterprises into 10 categories in the Measures for the Administration of Extraction and Use of Enterprise Safety Production Expenses. According to Duan et al. [17], the safety expenditures of coal firms are divided into two categories: safety input and accident loss. In terms of enterprise safety input, coal mining businesses may be classified into a variety of safety input variables based on the current condition. Professor Mei [16] classified safety input into five categories: safety technology, industrial cleanliness, safety education, personal protective equipment, daily management and labor expenses. When researching the logic of safety input structure, Zhao et al. [18] separated safety input into first-level indicators and second-level indicators, with the first-level indicators including personnel, safety technology and safety management input.

In conjunction with existing safety input structure and references, the research constructed the evaluation indicator system of coal mine safety inputs using Jiang et al. [2] and the TOPSIS method based on the entropy weight of coal mine safety input decision analysis. The safety input of coal mining enterprises is divided into five major categories—safety science and technology, safety engineering, safety equipment, safety management and safety education and training—and each category can be further subdivided into one or more safety indicators. Enterprise safety input has an effect on an enterprise's safety output to a certain extent, which can include impairment and value-added output. The former relates to accident economic damage, while the latter refers to coal yield. To clearly demonstrate the link to the resulting safety input structure according to the nature of various kinds of safety inputs, the inputs of safety management and safety education and training were integrated into the input in people, and the input of safety technology, safety engineering and safety equipment were integrated into the input in objects.

#### *2.2. Construction of Safety Input Structure*

Based on the definition of coal mine safety input and safety cost in Duan et al. [17], given the definition of safety cost and the safety standards of the benchmark, the safety related costs incurred include two parts: preventive safety input and accident loss. The cost of safety input (which is also called total safety input) can be divided into preventive safety input and loss of safety input.

Preventive safety input, also known as safety input, refers to the safety input before the occurrence of an accident and the loss of safety input refers to the total loss caused by an enterprise accident, also known as accident loss. Total safety input includes both loss of safety input and preventive safety input. The following two safety input structures were constructed using the above hierarchical structure of safety input. Safety Input Structure 1 is more detailed than Safety Input Structure 2 and has three more safety input indicators, as shown in Figures 1 and 2.

**Figure 1.** Safety Input Structure 1 of coal mine enterprise.

**Figure 2.** Safety Input Structure 2 of coal mine enterprise.

#### **3. Method of Weight Determination of Safety Input Index**

*3.1. Weight Determination by the Analytic Hierarchy Process*

(1) Construction of a safety input structure comparison matrix

The safety input fund allocation generates countless solutions, and each indicator of safety input is greater than 0. The evaluation indicators of safety input were determined by safety input hierarchy. For each evaluation indicator, quantitative comparison to the expert scoring method was used to establish comparison matrixes, using mutual comparisons of two indicators to determine the relative degree of importance of each evaluation indicator with a structure comparison matrix of n × n. The relative importance value is shown in Table 1.

$$\mathbf{a}\_{\overline{\mathbf{i}}} = \mathbf{i}'\mathbf{s} \text{ relative importance}/\mathbf{j}'\mathbf{s} \text{ relative importance},\tag{1}$$

**Table 1.** Relative importance table.


(2) Indicator weight and consistency test

The normalization of the feature vector of the comparison matrix's maximum characteristic root max is designated as W, and the element W is the ordering weight of the elements at the same level to the relative importance of a component at the next level. This is known as hierarchical single ordering. The consistency indicator is calculated using the following formula:

$$\text{CI} = (\lambda\_{\text{max}} - \text{n}) / (\text{n} - 1), \tag{2}$$

where n denotes the number of matrix evaluation indicators. Based on the number of indicators in the judgment matrix, look for the average random consistency RI in Table 2.


The consistency ratio CR is calculated by the following formula:

$$\text{CR} = \text{CI}/\text{RI},\tag{3}$$

It is generally believed that when the consistency ratio is:

$$\mathbb{CR} < 0.1,\tag{4}$$

it indicates standard consistency, and its normalized feature vector can be used as the weight vector through the consistency test. Otherwise, the comparison matrix should be reconstructed and the value of aij adjusted.

#### *3.2. Weight Determination by the Entropy Weight Method*

The entropy weight method (EWM) is a thorough, objective approach to weight assignment that determines the indicator weight based on current data while minimizing the variation caused by subjective assignment. Entropy is a measure of uncertainty in information. The lower the entropy number, the more information and weight there is. The specific steps of this model are as follows:

(1) Data standardization. Standardized processing of data into dimensionless data. Positive indicators:

$$\text{xri} = \frac{\text{xri} - \min\{\text{x1} \cdot \dots \cdot \text{xm}\}}{\max\{\text{x1} \cdot \dots \cdot \text{xm}\} - \min\{\text{x1} \cdot \dots \cdot \text{xm}\}} \tag{5}$$

Matrix:

$$
\begin{pmatrix}
\mathcal{X}\_{11} & \cdots & \mathcal{X}\_{1n} \\
\vdots & \ddots & \vdots \\
\mathcal{X}\_{\overline{m}1} & \cdots & \mathcal{X}\_{\overline{m}\overline{n}}
\end{pmatrix}
$$

where, xij is the value of the j indicator of the i sample, x ij is the value of the j indicator of the standardized i sample, n is the number of indicators and m is the number of samples.

(2) Calculate the entropy value of the j indicator.

$$\text{Pi}j = \frac{\mathbf{x}'\_{ij}}{\sum\_{i=1}^{m} \mathbf{x}'\_{ij}} \tag{6}$$

$$\mathbf{e}\_{\dot{j}} = -\mathbf{k} \sum\_{i=1}^{m} P\_{\dot{i}\dot{j}} \ln \left( P\_{\dot{i}\dot{j}} \right) \tag{7}$$

j = 1, . . . , n

In the formula, Pij is the proportion of x'ij in i sample and

$$\mathbf{K} = 1/\ln(\mathbf{n}).\tag{8}$$

$$\text{(3)}\quad \text{Calculate the information entropy of j indicator.}$$

$$d\dot{\jmath} = 1 - e\dot{\jmath} \tag{9}$$

j = 1, . . . , n

In the formula, dj is the information entropy of j indicator.

(4) Determine the weight of each indicator βj.

$$\beta\_j = \frac{d\_j}{\sum\_{j=1}^n d\_j} \tag{10}$$

j = 1, . . . , n

In the formula, β<sup>j</sup> is the weight of j indicator.

#### *3.3. AHP-EWM for Comprehensive Weight Assignment*

The analytic hierarchy process (AHP) is highly subjective and based on specialists' practical experience. In contrast, EWM is based only on objective data, which has objective advantages but cannot be applied to the actual situation. To compensate for the lack of single weighting, the AHP-EWM are coupled to assure the dependability of evaluation outcomes. The comprehensive weight of AHP-EWM is calculated as:

$$\mathcal{W}\_{\dot{\jmath}} = \frac{\alpha\_{\dot{\jmath}} \beta\_{\dot{\jmath}}}{\sum\_{j=1}^{n} \alpha\_{\dot{\jmath}} \beta\_{\dot{\jmath}}} \tag{11}$$

In the formula, α<sup>j</sup> is the weight of the j indicator that was calculated by the analytic hierarchy process; β<sup>j</sup> is the weight of the j indicator that was calculated by the entropy weight method.

#### **4. Establishment of the Accident Loss Model Based on the C-D Production Function** *4.1. Cobb-Douglas Production Function*

The C-D production function, proposed by American mathematician C.W. Cobb and American economist Paul H. Douglas in the 1930s, is widely used in economic quantitative analysis and can be used to analyze the relationship between input and output under certain conditions of time.

The original form of the C-D production function is as follows:

$$\Upsilon = \mathbf{A} \mathcal{K}^{\mathfrak{a}} L^{\mathfrak{b}} \tag{12}$$

In the formula, Y is the total output, K is the total capital input, L is the total labor input, A is the constant determined by the technical conditions of production in a certain era, α is the capital elasticity coefficient and β is the labor elasticity coefficient.

#### *4.2. Establishment of Accident Loss Models*

When preventive safety input remains constant in a safety input model, total safety input depends on loss of safety input, and there is a certain functional relation between loss of safety input and each preventive safety input's safety input indicator. The word "model" means a set of relationships between two or more variables. These relationships can be expressed in terms of mathematical equations (34). The C-D production function [4,21,22] is used to simulate the functional relationship between safety input indicators and loss of safety input.

(1) According to the Safety Input Structure 1 of coal mine enterprises in Figure 1 and C-D production function, the accident loss multiple regression model S1 was constructed [4,21]:

$$B\_1 = \mathcal{F}(\mathbf{p1}, \mathbf{p2}) = \mathcal{A}\_1 P\_1^{a1} P\_2^{a2} \tag{13}$$

In the above formula, p1 was the input in people and p2 was the input in objects; α<sup>1</sup> and α<sup>2</sup> were the capital elasticity coefficient corresponding to p1 and p2, respectively; and B1 was the loss of safety input (accident loss) of Structure 1. At this point, Safety Input Structure 1's total safety input wasB+B1.

(2) According to the Safety Input Structure 2 of coal mine enterprises in Figure 2 and C-D production function, the accident loss multiple regression model S2 was constructed [4,21]:

$$B\_2 = \mathbf{F}(\mathbf{x1}, \mathbf{x2}, \mathbf{x3}, \mathbf{x4}, \mathbf{x5}) = \mathbf{A}\_2 \mathbf{x}\_1^{\pounds 1} \mathbf{x2}\_2^{\pounds 2} \mathbf{x}\_3^{\pounds 3} \mathbf{x4}\_4^{\pounds 4} \mathbf{x5}^{\pounds 5} \tag{14}$$

In the formula above, x1 represented the input of safety technology, x2 represented the input of safety engineering, x3 represented the input of safety equipment, x4 represented the input of safety management and x5 represented the input of safety education. β1, β2, β3, β<sup>4</sup> and β<sup>5</sup> corresponded to the capital elasticity coefficient of safety science and technology input, safety engineering input, safety equipment input, safety management input and safety education input, respectively. B2 was the loss of safety input (accident loss) of Structure 2. At this time, the total safety input of Structure 2 wasB+B2.

#### *4.3. Optimization of Safety Input Fund Assignmentf*

First, the data of the enterprise safety input and loss of safety input were processed into logarithmic form using Excel, and then multiple linear regression was performed using Matlab to determine the accident loss models S1 and S2 corresponding to Safety Input Structure 1 and 2. The functional relationship between each safety input indicator and the accident loss was obtained by multiple linear regression analysis [23]. The accident loss models S1 and S2 were obtained using the same statistics and modelling method. Between the two models, model S2 had more specific indicators than model S1. We then took the mean value of the preventive safety input of the coal mine enterprise and inserted it in the obtained function to calculate the minimum accident loss under constraints, to determine the optimal assignment of safety input and to compare the minimum values of accident loss.

#### *4.4. Comparison of Accident Loss Regression Functions by MSE*

Few studies have been conducted to validate the obtained functions for modeling using the C-D production function. MSE is a common metric for testing regression functions, and its value provides a visual indication of how well the function fits the actual data.

The mean squared error is the expected value of the squared difference between the parameter estimate and the true value of the parameter, denoted as MSE.

The MSE is calculated by the following formula:

$$MSE = \frac{1}{N} \sum\_{i=1}^{n} \left( y\_i - \mathcal{y}\_i \right)^2 \tag{15}$$

#### **5. Application Analysis of Method**

#### *5.1. Safety Input Data of a Coal Mine Mnterprise*

Simulated and analyzed using the safety input details of a large state-owned coal mine enterprise, Table 3 [24] shows the detailed statistical data of different safety element inputs and the accident economic loss of the state-owned coal mine enterprise from 2001 to 2010.

**Table 3.** Safety input and safety output statistics of a coal mine enterprise from 2001 to 2010.


*5.2. Weight Determination of the Safety Input Indicators*

(1) According to the Safety Input Structure 1 and Safety Input Structure 2 of coal enterprises, the expert scoring method was adopted to establish the comparison matrix B1-Pj and B2-Cj. The obtained results are shown in Tables 4 and 5, and the consistency test was conducted. According to the expert scoring results, the weight was calculated by the geometric evaluation method.

**Table 4.** Comparison matrix of B1-Pj.


**Table 5.** Comparison matrix of B2-Cj.


The consistency test revealed that:

$$\text{CR}\_1 = 0, \text{CR}\_2 = 0.008. \tag{16}$$

The consistency ratio CR1 and CR2 were less than 0.1, hence the comparison matrixes were consistent and the feature vectors equaled the weight vectors.

(2) The weight was determined by the entropy weight method. The results of the entropy weight method assignment obtained by Python programming calculations are shown in Tables 6 and 7:

**Table 6.** Model 1 Entropy weight method assignment results.


**Table 7.** Model 2 Entropy weight method assignment results.


(3) AHP-EWM comprehensive empowerment: Python programming was used to calculate the final weights of each safety input indicator. The results are shown in Tables 8 and 9:

**Table 8.** Comprehensive empowerment results of Model 1.


**Table 9.** Comprehensive empowerment results of Model 2.


In Safety Input Structure 1, the ranking of the relative importance of the comprehensive weight assignment was p1 > p2.

In Safety Input Structure 2, the ranking of the relative importance of the comprehensive weight assignment was safety management C4 > safety equipment C3 > safety engineering C2 > safety education C5 > safety science and technology C1.

#### *5.3. Analysis of the Extreme Value of the Accident Loss Function by Matlab Software*

(1) Before multivariate regression of the accident loss function, logarithms of Formulas (13) and (14) corresponding to Safety Input Structure 1 and Safety Input Structure 2, respectively, were taken and the results were as follows:

Model S1:

$$
\ln \mathcal{B}\_1 = \ln A\_1 + \beta\_1 \ln P\_1 + \beta\_2 \ln P\_2 \tag{17}
$$

Model S2:

$$\begin{aligned} \ln \mathcal{B}\_2 &= \ln A\_2 + \beta\_1 \ln X\_1 + \beta\_2 \ln X\_2 + \\ \beta\_3 \ln X\_3 &+ \beta\_4 \ln X\_4 + \beta\_5 \ln X\_5 \end{aligned} \tag{18}$$

(2) The datasets from 2001 to 2010 were sorted into pairs by Excel, and multiple linear regression was performed on Equations (17) and (18) by Matlab programming [25]. After obtaining the parameters, the accident loss functions were obtained as follows:

Model S1:

$$B\_1 = 14338.64 P\_1^{-0.2496} P\_2^{-0.107968} \tag{19}$$

Model S2:

$$B\_2 = 24247.7x\_1^{-0.081823}x\_2^{-0.1618}x\_3^{-0.2544}x\_4^{-0.3363}x\_5^{-0.1191} \tag{20}$$

(3) Taking accident loss minimization as the goal, the average safety input of 4159.157 million RMB over the 10-year period 2001–2010 and the minimum economic loss of

950.463 million RMB as the condition, the Matlab software was used to calculate the extreme values of Equations (19) and (20). The minimum values and corresponding safety input indicators values of the accident loss functions were determined. The results of the two models were calculated as follows:

Minimum value of B1 was 910.894 million RMB, P1 was 1251.200 million RMB and P2 was 2907.900 million RMB.

Minimum value of B2 was 679.770 million RMB, x1 was 356.00 million RMB, x2 was 729.400 million RMB, x3 was 111.140 million RMB, x4 was 144.557 million RMB and x5 was 516.600 million RMB.

The results indicated that B1's minimum value > B2's minimum value.

The optimal results of comparing the two types of safety input structure indicates that the optimal accident loss of Safety Input Structure 2 was significantly less than the optimal accident loss of Safety Input Structure 1. Although these two optimization results were the optimal solution of two safety input structures that were modeled by the C-D production function, they were not necessarily the optimal solution of safety input in coal enterprises.

#### *5.4. MSE of Two Accident Loss Functions*

The MSE of the two accident loss functions was calculated separately by the MSE formula and the results were as follows:

Accident Loss Function 1's MSE of Model S1*:* MSE1 = 1763.681538

Accident Loss Function 2's MSE of Model S2: MSE2 = 392.7999477

The MSE1 of function 1 was found to be significantly larger than that of function 2. If the MSE is less, the more the function reflects the input–output situation of production; that is, the production function should minimize the MSE. Therefore, the accident loss function 2 should be chosen, and the corresponding safety input indicator structure 2.

#### *5.5. Discussion*

In the article, two safety input structures were constructed for the same coal mine, modeled using the C-D production function, and the analytic formula of the accident loss function was obtained by multiple regression analysis. The optimal outputs (least loss) and MSEs of the two accident loss functions were obtained and compared, respectively. The comparison reveals that the MSE of the loss function of the higher output (less loss) Safety Input Structure 2 is also smaller.

Generally, when constructing an accident loss function, we expect the extreme value of this function to be the least possible, which means that the company can save money on accident losses. The optimal accident loss for Safety Input Structure 2 was 679.770 million RMB and the optimal accident loss for Safety Input Structure 1 was 910.894 million RMB. When optimizing the safety input allocation of the company, we can give preference to Safety Input Structure 2. The MSE2 of Accident Loss Function 2 was 392.7999477 and the MSE1 of Accident Loss Function 1 was 1763.681538, yielding MSE2 < MSE1. The fitness of Accident Loss Function 2 was better than that of Accident Loss Function 1, and this result was consistent with the comparison of optimal accident losses.

To summarize: (1) When modeling with the C-D production function, the calculated optimal configuration of safety inputs will be different because of the different safety input structures constructed; (2) The quality of the fitness of the function can be checked by calculating the MSE of the function, and the function with a lower MSE value has a better fit and a better calculated optimal solution; and (3) At present, when using the C-D production function for the optimal solution, the safety input structure construction is considered less, and in the future, when using the C-D production function for modeling, it can be combined with structure optimization.

#### **6. Conclusions**

In this paper, the models S1 and S2 are compared. The two models use data from a coal mine enterprise to simulate the enterprise's accident loss function. In addition, AHP-EWM is employed to assign the weight of each safety input indicator of safety input structure, the C-D production function is utilized to simulate the coal enterprise accident loss function model and Matlab software is used for multiple regression analysis and solution. Due to the varied degrees of refinement of preventive safety input, the optimum solution of the accident loss model of safety input structure presents different results, hence the main conclusions can be summarized as follows.

Comparing safety input structure and two functions of the structure illustrates the more specific indicators, and more refined structures are better when modeling using the Cobb-Douglas model with limited data.

By comparing the MSE of two safety input structures and the optimal configuration of the corresponding accident loss function, we selected a better structure, which is safety input indicator structure 2. This indicates that Safety Input Structure 2 is an effective structure, so we can apply this method to identify the quality of the constructed structure.

Compared to modeling using only the C-D production function, the MSE can be used as a supplement to determine the degree of consistency of the regression function with regard to the actual situation, which provides a reference for the use of the C-D production function.

In the digital era in the future, enterprises can obtain specific internal safety inputrelated data and determine the best method of allocating safety funds through programming and modeling. Currently, technical conditions cannot be upgraded to improve the utilization of safety input funds effectively in order to help reduce accident losses.

The optimization of the coal firms' safety input structure still has to be strengthened. An in-depth study of the safety input structure of coal enterprises requires the use of more detailed safety input-related data. The statistics from a longer period of years can simulate more accurate accident loss function. In the future, coal enterprises can apply the refined safety input structure and the more accurate function to provide a scientific basis for the allocation and decision-making of safety input funds.

**Author Contributions:** Conceptualization, J.-Y.Z. and X.-Z.S.; methodology, J.-Y.Z.; validation, S.Z.; formal analysis, J.-Y.Z.; data curation, J.-Y.Z.; writing—original draft preparation, J.-Y.Z.; writing review and editing, J.-Y.Z. and X.-Z.S.; visualization, J.-Y.Z.; supervision, S.Z. and X.-Z.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Xiaofang Luo 1, Haolang He 2, Xu Zhang 2,3,\*, Yong Ma 2,3,\* and Xu Bai <sup>4</sup>**


**Abstract:** Currently, intelligent ships are still in the early stages of development in terms of autonomous navigation and autonomous berthing, so almost no source of fault data can be obtained. Conducting an in-depth analysis of the failure modes of intelligent ships is critical to optimizing the design of smart ships and ensuring their normal and safe navigation. In this paper, the fixed-weight Failure Mode Effects and Criticality Analysis (FMECA) is combined with the decision-making trial and evaluation laboratory (DEMATEL) method to analyze the failure modes and effects of intelligent ship positioning systems. This combined method not only overcomes the failure of traditional FMECA methods to differentiate between severity, incidence, and detection rates but also allows the correlation of failure causes to be analyzed, bringing the results of the analysis closer to reality. Through the expert scoring of failure modes, the failure modes of this system are risk-ranked, and the key failure causes of this system are identified. Correlations between the critical failure causes are then considered. According to the analysis results, the high-accuracy attitude sensor was identified as the subsystem with the highest level of risk. Unavoidable, unknown failures and environmental factors were found to be key factors in causing positioning system failures. The conclusions can provide a reference for the design of equipment safety for intelligent ship positioning systems.

**Keywords:** fixed-weight; FMECA; intelligent ship; positioning system; risk identification

#### **1. Introduction**

Due to the large potential for saving energy consumption on ships and reducing ship crew, the intelligent ship has become the focus of the industry and the trend in its development. According to the China Classification Society (CCS) [1], an intelligent ship requires the use of sensors, communications, the Internet of Things, and other technical means to automatically sense and obtain information and data about the ship itself, the marine environment, logistics, and ports. Intelligent operation is applied to ship navigation, management, maintenance, and cargo transportation. This makes intelligent ships safer and more environmentally friendly than traditional ships.

The technology of intelligent ships includes intelligent navigation, autonomous berthing, status detection, and fault diagnosis. Currently, research work in intelligent navigation is focused on collision avoidance and algorithmic optimization of path planning. Agnieszka [2] proposes a new deterministic approach using the concept of a trajectory database to calculate the safe, optimal path of a ship, considering its dynamic properties and static and dynamic obstacles. He et al. [3] proposed an open-water intelligent navigation decision method capable of dynamically adapting to system residual errors and random maneuvers of the target vessel. By utilizing the velocity obstacle (VO) theory and dynamic collision avoidance mechanism, the vessel is able to navigate autonomously in an open water environment with multiple static and dynamic objects. Since autonomous berthing requires

**Citation:** Luo, X.; He, H.; Zhang, X.; Ma, Y.; Bai, X. Failure Mode Analysis of Intelligent Ship Positioning System Considering Correlations Based on Fixed-Weight FMECA. *Processes* **2022**, *10*, 2677. https://doi.org/10.3390/ pr10122677

Academic Editors: Francisco Ronay López-Estrada and Guillermo Valencia-Palomo

Received: 31 October 2022 Accepted: 9 December 2022 Published: 12 December 2022

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high ship maneuverability, the existing research work is mainly focused on berthing control tasks and controller design. Lee et al. [4,5] used fuzzy control and LOS algorithms to experiment with a 4 m-long boat to solve the problem of small boat position and navigation accuracy to achieve side booster-assisted berthing. Mizuno et al. [6–8] solved the problem of automatic berthing under uncertainty disturbances using an artificial neural network approach.

Although the technology has received a lot of attention, it still requires development. There are still many unresolved issues, and the weaknesses of intelligent ships cannot be clarified. Intelligent ships, compared with traditional ships, involve more fields and disciplines, and their components are more numerous and complex. The harsh conditions at sea lead to extremely violent ship movements. The equipment, components, signal transmission, and structures on ships are, therefore, also extremely vulnerable to failure [9]. According to the data published by Allianz Global Corporate & Specialty (AGCS) [10], more than 1000 ships of more than 100 gross tons have been lost in the last decade. More than one-third of shipping accidents are caused by mechanical damage or failure, out of more than 20,000 reported in the last 10 years. It is conceivable that the failure of intelligent ship equipment, which relies on equipment and information transfer, can also suffer from similar serious consequences. Currently, there is little data available on the failure of intelligent ship equipment. Therefore, research on risk identification related to equipment must be conducted in order to prevent or mitigate potential hazards. Through Failure Mode Effects and Criticality Analysis (FMECA), finding the weak points of equipment and proposing targeted repair and maintenance strategies can significantly improve the reliability of intelligent ships.

An intelligent ship mainly includes a hull structure, power propulsion system, positioning and navigation system, control system, communication system, and interaction system. Among them, the positioning system plays an extremely significant role in normal operation. Its functions involve ship positioning, external environment sensing and platform state sensing, providing the necessary data sources for motion decision and control as well as its own state monitoring. The causes of the failure of certain subsystems and components in positioning systems have been investigated. Alan et al. [11] studied the effects of data reliability and human error on the Automatic Identification System (AIS) in the positioning system and showed that many input errors in the navigation state are due to personnel memory errors or negligence in performing the required operations. Pallotta [12] and Tsou [13] et al. studied the impact of data redundancy on AIS. Philipp et al. [14] investigated the effects of antenna height and environmental changes on information transmission. If the system fails, serious consequences will occur, such as intelligent navigation deviation, autonomous berthing and unberthing failure, and even collision between the ship and obstacles. Therefore, it is necessary to analyze its failure mode to ensure its safety. However, relevant risk identification research, especially for positioning system equipment, has not been reported. This paper intends to focus on equipment failure mode and its impact on the positioning system.

One of the main methods for failure mode analysis is the FMECA method. The key to failure analysis by the FMECA method is usually based on three risk parameters: severity (*S*), occurrence (*O*), and detection (*D*). The magnitude of the risk priority number (RPN), the product of the three, measures the severity of potential system problems. By prioritizing the high-risk failure modes and guiding maintenance management strategies. Although the traditional FMECA method is widely used, it is still criticized for its many drawbacks. Different values of *S*, *O*, and *D* can get the same RPN value, which is theoretically the same priority as the two, but in practice, the priority of the two risk ranks is different.

Therefore, many enhanced versions have been proposed in the literature. Some scholars have proposed measuring risk in more dimensions. For example, Carmignani [15] suggested the use of a fourth parameter, profitability, in the RPN calculation. Bevilacqua et al. [16] proposed that the RPN can consist of a weighted sum of six parameters (safety, importance of the machine to the process, maintenance cost, failure frequency, downtime, and operating conditions). Other studies combine FMECA methods with other methods. Zammori et al. [17] combined FMEA with analytical network process (ANP) techniques [18] to consider the possible interactions between the main causes of failure. Silvia et al. [19] proposed a method combining reliability analysis and a multi-criteria decision-making approach to improve the maintenance activities of complex systems. Some scholars combine FMECA with Analytic Hierarchy Process (AHP) to solve the problem that the traditional FMECA method cannot distinguish the different weights of risk factors by giving different weights to the evaluation parameters so that the risk ranking of failure modes is closer to the actual results. Braglia [20] proposed the analysis hierarchical method (AHP) [21] to compare pairs of potential failure causes by assuming the classical risk factors *S*, *O*, and *D* and the expected cost caused by the failure as criteria. Xiao et al. [22] proposed a weighted RPN evaluation method, which multiplied the RPN value with the weight parameter representing the importance of the fault causes in the system and then ranked them. Zhang et al. [23] proposed a new method for FMECA failure mode ranking based on incentive variable weight AHP. Li et al. [24] proposed a fixed-weight FMECA method. The method considers that the scales of *S*, *O*, and *D* and their weights are different and designs a normalization method to convert *S*, *O*, and *D* to the same scales as their weights and then generates the RPN of the cause of the failure as well as the failure mode. This method not only improves the problem of different weights of *S*, *O*, and *D* but also solves the sorting problems with the same RPN values. Due to the insufficient ability of traditional AHP to deal with fuzziness, many scholars use fuzzy theory to solve this problem. Many researchers have proven the effectiveness and superiority of fuzzy theory in dealing with fuzzy information. Luqman et al. [25] proposed an FMEA risk assessment technology based on TPFNs and DGMA. Akram et al. [26] proposed a mixed solution of TOPSIS and ELECTRE I with Pythagorean fuzzy information, using the Pythagorean fuzzy weighted average operator to aggregate their independent evaluations into group evaluations. Some studies also combine fuzzy theory and traditional AHP methods to manage the lack of information acquisition on complex problems, such as Liu et al. [27].

However, these FMECA methods do not consider the correlation between the failure factors, making the results obtained from the analysis somewhat one-sided. Many existing studies have proposed solutions to the problem of correlation of structural reliability, mainly involving integral methods [28,29] and numerical simulation (Monte Carlo method) [30]. The integral method could solve the multidimensional integration problem, but the procedure is complicated and not practical when the system composition is large. The Monte Carlo method uses a huge number of samples to simulate the variables obeying the desired distribution, and the more simulations there are, the higher the accuracy. However, it requires a lot of time and computing capacity and is less efficient [31]. Unlike structural faults, equipment faults do not allow for the construction of limit state equations. It is, therefore, difficult to apply reliable indicator vector methods for accurate correlation assessment. In this paper, the decision-making trial and evaluation laboratory (DEMATEL) method is used for the computational analysis of equipment failure correlations. The DEMATEL method was first proposed by Gabus and Fontela [32] of the Battelle Memorial Association in Geneva and aimed to analyze the causal relationships between the elements of a complex system and the degree of mutual influence [33].

Combining the above issues, considering the complex structural composition and many failure modes of the positioning system of intelligent ships, this paper uses a combination of fixed-weight FMECA and DEMATEL to study the system. On the one hand, it improves the problem of unreasonable distribution of *S*, *O*, and *D* weights in the traditional FMECA method, and on the other hand, it also considers the correlation between failure modes and improves the problem of mutual independence among failure causes. The results are closer to reality and increase the credibility of the failure mode analysis.

The remainder of the paper is organized as follows: Section 2 introduces the fixedweight FMECA method. Section 3 performs FMECA analysis on the positioning system

to obtain critical failure modes. Section 4 analyzes the correlation between the key failure modes, and Section 5 gives the conclusions.

#### **2. Method**

#### *2.1. Fixed-Weight FMECA Approach to Modeling*

FMECA includes Failure Mode and Effects Analysis (FMEA) and Criticality Analysis (CA), which is an analysis technique based on failure modes and targeting the effects or consequences of failures. FMECA is performed by finding all possible failures of the product, analyzing them according to the failure mode, determining the impact of each failure on the operation of the product, and identifying the hazards of the failure mode in the order of the RPN. RPN is the risk prioritization number, whose value is equal to the product of the values of severity (*S*), occurrence (*O*) and detection (*D*), calculated by the following formula:

$$RPN = S \times O \times D \tag{1}$$

Although the traditional FMECA method is widely used in production practice, it still has many drawbacks. For example, the same weights are assigned to severity, occurrence, and detection. However, in practice, the weights of the three in the system are not exactly equal. For some irreparable systems, the weights of factors *S* and *O* should be higher than the weights of *D*. In this paper, the fixed-weight FMECA method [24] is used to eliminate the above effects. This FMECA method generates the RPN of component failure caused by using the severity, incidence, and detection rates of each item and their relative weights. Considering that the scales of each factor (severity, incidence, and detection rate) and their weights are [1, 10] and [0, 1], *S*, *O*, and *D* are converted to the same scales as their weights before calculating the RPN.

Denote *βSi*, *βOi* and *βDi* as the average value of the severity, occurrence and detection of the fault cause, *i*, given by the expert. The weights of the factors: *ψ* = *KS KO KD* ! . *KS*, *KO*, and *KD* are the weights for severity, occurrence and detection, respectively. Thus, the raw values of severity, occurrence, and detection given by the experts can be expressed as follows:

$$
\begin{bmatrix}
\beta\_{S1} & \beta\_{S2} & \cdots & \beta\_{Si} & \cdots & \beta\_{Sn} \\
\beta\_{O1} & \beta\_{O2} & \cdots & \beta\_{Ci} & \cdots & \beta\_{On} \\
\beta\_{D1} & \beta\_{D2} & \cdots & \beta\_{Di} & \cdots & \beta\_{Dn}
\end{bmatrix}
\tag{2}
$$

Denote

$$
\xi\_{Kij} = \frac{\beta\_{K\dot{i}}}{\beta\_{K\dot{j}}} \tag{3}
$$

where, *K* represents severity, occurrence, and detection.

Therefore, the comparison matrix is attained as:

⎡ ⎢ ⎢ ⎢ ⎣ *ξK*<sup>11</sup> *ξK*<sup>12</sup> ··· *ξK*1*<sup>n</sup> ξK*<sup>21</sup> *ξK*<sup>22</sup> ··· *ξK*2*<sup>n</sup>* . . . . . . ... . . . *ξKn*<sup>1</sup> *ξKn*<sup>2</sup> ··· *ξKnn* ⎤ ⎥ ⎥ ⎥ ⎦ (4)

The normalized matrix is defined as:

$$\begin{bmatrix} \Phi\_{K11} & \Phi\_{K12} & \cdots & \Phi\_{K1n} \\ \Phi\_{K21} & \Phi\_{K22} & \cdots & \Phi\_{K2n} \\ \vdots & \vdots & \ddots & \vdots \\ \Phi\_{Kn1} & \Phi\_{Kn2} & \cdots & \Phi\_{Knn} \end{bmatrix} \tag{5}$$

*φKn*<sup>1</sup> *φKn*<sup>2</sup> ··· *φKnn*

$$\phi\_{Kij} = \frac{\mathfrak{F}\_{Kij}}{\sum\_{i=1}^{n} \mathfrak{F}\_{Kij}} \tag{6}$$

where

$$^{182}$$

The adjusted value of index *K* of failure cause *i* is defined as:

$$\gamma\_{Ki} = \frac{\sum\_{j=1}^{n} \Phi\_{Kij}}{n} \tag{7}$$

According to Equations (6) and (7),*γKi* ∈ [0, 1], which is the same as the scale of the weight vector of indices *ψ* = *KS KO KD* ! .

Hence, the weighted RPN of failure cause *i* is defined as:

$$RPN\_i = \boldsymbol{\psi} \times \boldsymbol{\Gamma}\_i = \begin{bmatrix} \mathbf{K}\_S & \mathbf{K}\_O & \mathbf{K}\_D \end{bmatrix} \begin{bmatrix} \gamma\_{Si} \\ \gamma\_{Oi} \\ \gamma\_{Di} \end{bmatrix} \tag{8}$$

The method first transforms the absolute values of *S*, *O*, and *D* into the same values as their weight scales, and the *S*, *O*, and *D* transformed values take values in the range [0, 1]. The larger the original *S*, *O*, and *D* values will still be larger after transformation and will not affect the order of RPN. The importance of the cause of failure does not change. When using the fixed-weight FMECA method, the calculation of RPN involves the *S*, *O*, and *D* values as well as the weights. However, the range of the two values is different, [1, 10] and [0, 1], respectively. This could bias the final calculated RPN. The consistent scale conversion of both before calculating the RPN prevents each type of parameter from affecting the results more than the other.

#### *2.2. Selection of Weights*

For intelligent ships, we should pay more attention to the consequences of their failure occurrences and the value of severity. This is because failures with low incidence can still occur, and detectable failures can still occur. The occurrence of faults can affect certain functions of intelligent ships and even lead to major accidents if certain critical units are affected. Therefore, the weight value of importance should be the largest among the three. At the level of occurrence and detection, the likelihood of failure occurrence is significantly more important than detection. Whether a fault occurs or not is the combined result of the physical properties of the system and the internal and external effects, and it does not change with whether the fault is detectable or not, so the occurrence degree should be given more weight than the detection degree. The selection of risk evaluation index weights should follow the principle that the severity degree is greater than the occurrence is greater than the detection.

According to the basic guideline that the selection of risk evaluation index weights should follow that the severity degree is greater than the occurrence degree is greater than the detection degree, and according to reference [23], combined with the risk characteristics of the intelligent ship positioning system, the failure mode analysis conducted in this paper selects the following weight vector:

$$
\psi = \begin{bmatrix} 0.40 & 0.35 & 0.25 \end{bmatrix} \tag{9}
$$

#### *2.3. DEMATEL Method*

The reason for the correlation assessment of hazardous units is that failure modes that are closely linked to other failures may lead to more severe consequences than relatively independent failure modes. Therefore, identifying correlations between failure modes can lead to more reliable results. The main steps of the DEMATEL method are as follows:

(1) Establishing assessment criteria. The degree of correlation between the assessment elements is quantified by means of expert scoring. The assessment scale ranges from 0 to 4, in the order of no impact, very low impact, low impact, high impact, and very high impact. The scoring values are entered into a direct impact matrix (10).

$$A = \begin{bmatrix} 1 & a\_{12} & a\_{13} & \cdots & a\_{1(n-1)} & a\_{1n} \\ a\_{21} & 1 & a\_{23} & \cdots & a\_{2(n-1)} & a\_{2n} \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ a\_{(n-1)1} & a\_{(n-1)2} & a\_{(n-1)3} & \cdots & 1 & a\_{(n-1)n} \\ a\_{n1} & a\_{n2} & a\_{n3} & \cdots & a\_{n(n-1)} & 1 \end{bmatrix} \tag{10}$$

where *aij* indicates the degree of influence of factor *i* on factor *j*.

(2) The direct relationship matrix is normalized by Equations (11) and (12) such that each value of the matrix lies between [0, 1].

$$S = \max\_{1 \le i \le n} \sum\_{j=1}^{N} a\_{ij} \tag{11}$$

$$K = \frac{A}{S} \tag{12}$$

(3) The total impact matrix *M* is obtained by Equation (13), where *I* denotes the unit array.

$$M = K(I - K)^{-1} \tag{13}$$

(4) The sum of each column and each row of the total impact matrix is calculated by Equations (14) and (15), denoted as D and R, respectively.

$$D\_i = \left[\sum\_{i=1}^n m\_{ij}\right]\_{1\times n} \tag{14}$$

$$\mathcal{R}\_{\bar{i}} = \left[ \sum\_{j=1}^{n} m\_{i\bar{j}} \right]\_{n \times 1} \tag{15}$$

where, *M* = *mij*, *i*, *j*=1, 2, . . . , *n*.

(5) Perform the calculation of *Ri* + *Dj*, *Ri* − *Dj*. *Ri + Dj* indicates the extent to which factor *i* plays a role in the problem, with a positive *Ri* − *Dj* indicating that factor *i* assigns influence to other problems and a negative *Ri* − *Dj* indicating that factor *i* receives influence from other factors.

#### **3. Fixed-Weight FMECA of Positioning Systems**

#### *3.1. Positioning System Introduction*

The intelligent ship positioning system is built for the needs of digitalization, networking, visualization, and intelligence in ship positioning. It can realize the rapid circulation of ship information and effective management of ships by using the BeiDou positioning system, automatic control, and other technologies, combined with the data update of dynamic changes in ocean climate, to conduct emergency command of ocean ships.

The positioning system includes five subsystems: a high-precision attitude sensor, the BeiDou positioning system, the electronic chart display information system (ECDIS), the automatic identification system (AIS), and a mobile communication receiver. The structure diagram of the positioning system is shown in Figure 1. The high-precision attitude sensor is used to capture dynamic reference signals. The BeiDou positioning system is used to receive satellite positioning signals and, in conjunction with ECDIS, to precisely locate the ship's position at sea. The AIS and mobile communication receiver are responsible for sending and receiving to the ships and shore stations in the nearby waters so that the neighboring ships and shore stations can grasp the dynamic and static information of all the ships in the nearby sea and can immediately talk to each other for coordination. They

can also calculate the voyage heading and take necessary avoidance actions to effectively ensure the safety of ship navigation.

**Figure 1.** Schematic diagram of positioning system structure.

#### *3.2. FMECA of Positioning Systems*

Based on the composition and working principle of the positioning system and the human factors and environmental influences identified in scholarly research, this section will analyze the reliability of the intelligent ship positioning system based on the fixedweight FMECA method [24]. The scoring of each failure mode and failure cause of the intelligent ship positioning system is based on the evaluation table, and the results are shown in Table 1. Detailed failure causes, transformed values for severity, occurrence and detection, and risk number ranking for each failure mode are also given in Table 2.

**Table 1.** Evaluation table of failure model risk evaluation indicators.


**Table 2.** FMECA of the positioning system of intelligent ships.


**Table 2.** *Cont.*









In this section, five subsystems of the intelligent ship with a total of 111 fault causes are analyzed, as shown in Figure 2. The RPN values of each fault cause in the positioning system and its RPN share in the respective unit are given in Figure 3.

**Figure 2.** RPNs' share of each subsystem of the positioning system.

The FMECA table shows that, overall, the ECDIS is the most important system for positioning systems (RPN of 0.3122), followed by high-precision attitude sensor (0.2359), BeiDou positioning system (0.1768), AIS (0.1766), and mobile communication receivers (0.0985). On the one hand, the RPN values of the fixed-weight FMECA method are additive, and systems with more failure modes will have larger RPN values. In terms of RPN values of subsystems, ECDISs, and high-precision attitude sensors have numerous failure modes, both of which occupy more than half of the RPN and are important subsystems that cause the failure of positioning systems. On the other hand, to determine the critical failure mode for locating the system, it is important to consider not only the magnitude of the RPN value but also the average value of the RPN. Analyzing the importance of the failure from an average perspective can better evaluate the risk ranking of the system failure causes. The mean values of RPN for the causes of failure for these five systems were 0.0092, 0.0107, 0.0093, 0.0098, and 0.0055, respectively.

The total RPN value of the ECDIS is higher than that of the high-precision attitude sensor, while the average value of RPN is lower than that of the high-precision attitude sensor, indicating that although the failure modes of the ECDIS are many, the severity of their failure consequences is smaller compared to that of the high-precision attitude sensor. The average RPN of mobile communication receivers is much lower than other subsystems, and the risk level is low. Combining the results of the RPN values as well as the average RPN values, the high-precision attitude sensor was identified as the riskiest subsystem of the positioning system, followed by the ECDIS, AIS, and BeiDou, and finally, the mobile communication receiver.

#### *3.3. Critical Failure Cause Analysis*

The identification of critical failure causes is related to the development and planning of restorative and preventive measures. And the identification of critical failure units mainly lies in the selection of risk thresholds. Lorenzo et al. [34] proposed a new RPN threshold estimation method for FMECA. The method requires the following:


**Figure 3.** RPNs value and percentage of each cause of failure in the positioning system.

In this section, the critical cause identification schematic of the intelligent ship positioning system equipment is drawn according to the RPN threshold estimation method proposed by Lorenzo et al., as shown in Figure 4. The 28 failure causes in the top 25% of the RPN value ranking of the intelligent ship positioning system are identified, plus two of the pending failure causes are identified, and a total of 30 failure causes are identified as critical failure causes, as shown in Table 3.

**Figure 4.** Box diagram of the positioning system RPN.



The high-precision attitude sensor relies on multiple precision sensing units and contains 12 critical failure causes. Not only does this subsystem have many failure modes, but the consequences of failure are severe, and the risk level of the equipment is extremely high. The main causes of high precision attitude sensor failure are design-related factors (interference torque due to friction, resonance), environmental factors (magnetic field interference, temperature and humidity effects), and unavoidable unknown failures (hardware failure, device wear and tear). Serious causes of failure are interference caused by environmental factors such as F1 (electromagnetic interference, 1.34% of RPN), G1 (humidity factor, 1.33%), and I1 (magnetic field interference, 1.31%). The three-axis gyroscope, which in turn concentrates most of the key failure causes (A1–G1), is the key unit of the high-precision attitude sensor. This subsystem is related to whether the intelligent ship can accurately

identify the surrounding environment. This affects the capability of the intelligent ship to complete berthing and unberthing and may even lead to the ship colliding with the shore wall. Close attention should be paid to this subsystem.

The BeiDou positioning system is one of the cores of the positioning system, which consists of BeiDou satellites, ground base stations, receivers, terminal processors, displays and other units. The system includes 5 key causes of failure. The main causes of BeiDou positioning system failures are design-related factors (poorly designed software functions) and unavoidable unknown failures (circuit damage, hardware failure). The serious causes of failure are O1 (processor hardware damage, 1.15%), O2 (circuit failure, 1.16%), and T1 (display the IPC part of the board damage, 1.18%). In general, BeiDou positioning system units are not prone to problems [35] (the BeiDou navigation system is the responsibility of the state and has a low probability of failure).

ECDIS is mainly used to accurately display the position of ships at sea. It consists of image display, text display, processor, data storage, and various data interfaces, including 7 key fault causes. The causes of system failure include unavoidable unknown faults (failure of frequency synthesis module, circuit damage), environmental factors (magnetic field interference, poor sea conditions), and human factors (wrong setting of operating parameters). The serious causes of failure are AE2 (failure of frequency synthesis module, 1.16%) and AF1 (damage to AC/AD module, 1.16%). Among them, the radar interface concentrates more critical failure causes and is the key unit of the electronic chart system. The majority of the remaining failures are minor, easy to detect and repair, and do not significantly affect the overall positioning system.

AIS consists of a VHF receiver/transmitter and an AIS information processor, and each part of the interface contains six critical failure causes. The system relies on the reception and processing of GPS data, so in practice, the failure of the working unit itself and the loss of GPS data caused by external interference are the main factors causing the system to fail. The faults with high RPN are the lack of GPS position signal caused by AT1 (GPS data distributor of VHF/TDMA receiver unit has a fault or poor connection, 1.19%). Overall, the system has a high failure rate, but failures occur less frequently, are easier to detect, and are less likely to cause very serious effects.

A mobile communication receiver is used to receive and send communication information. It consists of an antenna, filter, mixer, demodulator, CPU, and peripheral circuits and none of the failure causes are identified as critical failure causes. The unit of this subsystem was low in precision, and the basic failure modes were divided into circuit damage and capacitor breakdown due to the power supply and the aging and substandard quality of the equipment in the unit itself. In practice, receiver failures are less frequent and can be easily inspected and repaired. And temporary damage to the mobile communication receiver does not seriously affect the work of the entire positioning system and is the least dangerous in failure mode analysis.

From the analysis results, it is clear that the main reasons for the failure of the positioning system of intelligent ships are unavoidable unknown failures, environmental factors, and design-related factors.

The unavoidable unknown failures are mainly circuit failures, hardware damage, and parts damage. In order to cope with such failures, the control of key and fragile parts of intelligent ships should be strengthened, and good-quality parts should be used. At the same time, fragile parts and units should be checked regularly. It will also be beneficial to strengthen the research on condition monitoring, fault warning, and diagnosis of intelligent ships and improve the inspection system of intelligent ships to ensure the normal use of intelligent ship units and parts.

Intelligent ship positioning systems are highly susceptible to the influence of the surrounding environment. Interference from magnetic fields and other factors in the environment can easily affect the use of the positioning system and lead to deviations in positioning, which can easily lead to dangerous situations. In order to reduce the influence of the environment on intelligent ship navigation, research on extreme environmental conditions should be strengthened, especially the influence of magnetic field disturbance, strong wind, and strong waves on intelligent ship equipment.

Failures caused by design-related factors include interference torque due to gyroscope component friction, vibration, and poorly designed positioning system software functions. To avoid such failures, the design process should be improved, and the development of software should be enhanced.

#### **4. Correlation Analysis of Critical Failure Cause**

The complexity of intelligent ship systems leads to numerous failure modes; therefore, calculating the correlation between each two failure modes would take a lot of effort, and, in addition, further analysis of low-risk failure modes would be of limited significance. Therefore, only the critical failure causes obtained in Section 3.3 are subject to correlation analysis. The units analyzed are shown in Table 4. After scoring by three experts based on the steps in Section 2.3, the total relationship matrix was calculated, as shown in Table 5. The influence degree of factors is shown in Table 6.

**Table 4.** Analyzed units.


**Table 5.** The matrix of total relation.


**Table 6.** The influence of the degree of factors.


In this study, we set the threshold (k) in the total relationship matrix to 0.2. 0.2 is the most appropriate value obtained from the attempt. The causality diagram of the total relationship obtained according to k > 0.2 is shown in Figure 5.

**Figure 5.** The causal diagram of total relation (k > 0.2).

The results of the analysis show that there is a high degree of correlation between the gyroscope, accelerometer, and electronic compass. That is, the failure of any one of the three components has the probability of leading to the failure of the other two units. All three belong to the same high-precision attitude sensor unit, which is a high-precision unit for monitoring the ship's attitude and is highly susceptible to interference from the external environment. The Arm processor, display, power interface, and text display are another group of units with a high degree of relevance. These four failure modes mainly concern the compass positioning system as well as the electronic charting system. The VFH transmitter and VHF receiver are more influenced by each other than by the other hazard units. The navigation system interface and the radar system interface weakly influenced other units and were barely influenced by other units, being two more independent units.

This section further determines that the high-precision attitude sensor is the most dangerous subsystem of the positioning system. It not only has many failure modes and serious consequences but also has a high degree of correlation between failure modes. It is easy to break down. Failure of this system will result in inadequate ship positioning accuracy, posing a serious safety hazard for intelligent navigation and autonomous berthing operations, and it should be given high priority. Regular servicing of similar high-precision components can effectively improve the reliability of smart ships.

Equipment failures on smart ships do not occur in isolation; each failure that occurs may lead to the occurrence of another. The correlation between failure modes shows that in reliability analysis, where failure modes have a cascade relationship with each other, we cannot simply consider the failure modes as independent of each other. To obtain reliable results, correlations between failure modes must be taken into account. The combined approach can be used not only for equipment failure analysis of smart ships but also for failure analysis of other marine engineering equipment.

#### **5. Conclusions**

This paper considers the shortcomings of the traditional FMECA method in that the weights of severity, occurrence, and detection are unreasonably assigned, and the correlation between failure modes is taken into account. The failure modes of the positioning system of an intelligent ship are analyzed using a combination of fixed-weight FMECA

and DEMATEL, and the failure causes of the failure modes are identified. The following conclusions were reached.


The relevant conclusions can provide a reference for the maintenance of intelligent ship positioning system equipment. The safety of intelligent ships in navigation can be ensured by reducing the possibility of malfunctioning or reducing the severity of damage caused by intelligent ship equipment.

However, this paper analyzes the intelligent ship positioning system by fixed-weight FMECA using only one weight assignment method. In practice, the intelligent ship positioning system is a complex system integrated with multiple components. The failure mechanism and failure characteristics of the system itself and its components vary greatly. In the future, it is expected to consider variable and floating risk evaluation index weights, combine real data information, and select specific weights for different systems and components for FMECA analysis.

**Author Contributions:** Conceptualization, X.L., X.Z. and X.B.; methodology, X.L.; software, X.L.; validation, X.L. and H.H.; formal analysis, X.L.; investigation, H.H. and X.Z.; resources, X.Z. and Y.M.; data curation, H.H.; writing—original draft preparation, H.H.; writing—review and editing, X.Z.; visualization, X.B.; supervision, X.Z. and Y.M.; project administration, Y.M.; funding acquisition, X.B. and X.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially supported by the Key R & D Projects in Guangdong Province (No. 2020B1111500001), the National Natural Science Foundation of China (42276225, 52001112), the Natural Science Foundation of Jiangsu Province (Grants 389 No. BK20211342), and the National Key Research and Development Program of China (No. 2022YFC2806300).

**Institutional Review Board Statement:** This study does not involve any institutional review issues.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

