Using a Flexible Risk Priority Number Method to Reinforce Abilities of Imprecise Data Assessments of Risk Assessment Problems

Abstract

Risk priority number (RPN) is the most commonly used failure risk ranking method among all risk assessment methods. However, the traditional RPN method not only cannot handle incomplete information and hesitation information (such as hesitation information between the $s_5=\mathrm{L}\mathrm{o}\mathrm{w}$ and $s_6=\mathrm{M}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}$) provided by experts, but it also does not consider the objective weights of risk assessment factors. These reasons will lead to biased conclusions, causing decision makers to make wrong judgments. To address the limitations of the traditional RPN technique, the aim of this paper is to propose the flexible RPN assessment method under an uncertain environment. The flexible RPN assessment method is an extension of the traditional RPN technique. The flexible RPN method integrates the traditional RPN method and interval-valued 2-tuple weighted average method, and, simultaneously, considers subjective weights and objective weights. In the numerical verification, this study has adopted the example of stages of treatment planning for proton beam radiation therapy to verify the correctness and validity of the proposed flexible RPN technique. The numerical results show that the proposed flexible RPN technique not only handles the hesitation and incomplete information provided by experts but also considers the subjective and objective weights of risk assessment factors, providing more reasonable ranking results in the risk analysis.

1. Introduction

With the development of information technology and the arrival of the information age, the pursuit of high-quality systems or products has become the trend. Failure mode and effects analysis (FMEA) is one of the most commonly and widely used risk assessment tools. The purposes of FMEA are to provide early prevention of failure occurrences, reduce the impact of system failures on products, and improve product quality and market competitiveness. Therefore, many studies [1,2,3,4,5,6,7,8,9,10,11] have used the FMEA method to address practical issues related to risk assessment. Although the FMEA method is widely used for risk assessment, it still has several limitations, such as lacking consideration for the objective weight of risk factors and being unable to handle incomplete or uncertain information during information processing.

The traditional FMEA approach utilized risk priority number (RPN) to rank failure risk rating. The RPN score had the following risk assessment factors: severity (S), occurrence (O), and detection (D). The three risk assessment factors S, O, and D used an ordinal score of 1–10 to assess all of the potential failure modes. A higher RPN score means that the higher the risk level, the more the need for priority improvement measures. Because of the typical RPN method, simple calculations are widely used in various fields. For example, the ranking method based on ELECTRE was introduced by Liu, You, Chen, and Chen [12] to solve the issue of healthcare medical risk assessment in uncertain environments. Expanding the concept of data envelopment analysis (DEA), Chin et al. [13] developed interval DEA models to rank the failure modes risks. Chin et al. [13] used the geometric mean of the minimum and maximum risk of the failure mode as the overall risk of potential failure mode. Moreover, Yousefi et al. [14] introduced a robust DEA FMEA technique for the assessment of safety and environment risk prioritization in the automotive parts industry. They used S, O, and D indicators as input parameters, and duration of treatment and cost as output parameters to rank prioritization of safety and environmental risk. To increase the consideration of fuzzy information, Hosseinpour, Amirkhan, Rezaeian, and Doostideilami [15] combined the fuzzy decision-making trial and evaluation laboratory method with the fuzzy best–worst method to process risk assessment problems related to the food industry. Chang, Chen, and Liao [16] combined the 2-tuple fuzzy linguistic representation model with data envelopment analysis to process risk assessment problems of crawler cranes.

The typical RPN approach assumes that the three risk assessment factors, S, O, and D, are equally important [17,18,19]. However, the relative importance of these three risk assessment factors is not considered, which may lead to biased assessment results. Although the typical RPN approach has received great attention and has been used in the academic, industrial, and military fields, the typical RPN approaches have no consideration for the objective weight of risk assessment factors S, O, and D. Liu, Chan, and Ran [20] pointed out that to obtain more accurate and logical decision outcomes, subjective weights and objective weights must be simultaneously considered. Yu et al. [21] combined subjective and objective weight considerations to calculate risk factor weights, then integrated Fermatean fuzzy sets and the combined compromise solution method to address the failure risk assessment problems of liquefied natural gas storage tank leakage. Nowadays, many scholars combine the consideration of subjective and objective weights to solve decision-making problems in different fields [22,23,24,25,26,27,28,29,30,31,32,33].

Another limitation of the typical RPN approach is that it cannot handle incomplete or hesitant information during information processing. To process incomplete or hesitant information, the typical RPN approach will directly delete these messages and only process complete information. This technique will cause insufficient consideration of all available information, resulting in biased decision results. The interval value 2-tuple weighted average (IV2WA) method can solve the related problems of incomplete or hesitant information during the implementation of risk assessment. Akyuz and Celik [34], combining FMEA and interval type 2 fuzzy sets, proposed a new quantitative risk analysis method to conduct risk assessment in maritime transportation. Akyuz and Celik’s [34] research findings show that the incomplete information provided by local maritime authorities and poor organized coastline cleanup teams are the main causes of oil spills.

Recently, Zhang [35] introduced the IV2WA method as a useful tool to handle the multi-attribute group decision-making issues. Moreover, Zhang [35] introduced the accuracy function and the score function to compare the difference with the two interval-valued 2-tuples linguistic (IV2L) method. For the processing of incomplete information, the IV2WA method does not reference other experts to provide this failure mode’s useful information. This is due to the fact that the typical RPN approach cannot handle incomplete or hesitant information during information processing. In order to effectively address the limitations of the typical RPN approach, this paper proposes a flexible RPN method to enhance the ability to assess imprecise data in failure mode and effect analysis. For verification, the paper adopts the risk assessment of the stages of treatment planning for proton beam radiation therapy as a numerical example to validate the proposed flexible RPN method. The main advantages of the proposed flexible RPN method are as follows: for incomplete information processing, this study adopted other experts who provided useful and complete information to perform information filling of incomplete data. For aggregated information processing, the proposed flexible RPN method uses interval-valued 2-tuple information to handle complete, incomplete, and hesitation information during the information aggregation process. Moreover, the proposed method fully considers both subjective and objective weights of the risk assessment factors.

The remainder of this paper is organized as follows: Section 2 presents a brief introduction of FMEA, the interval 2-tuple linguistic model, the analytic hierarchy process (AHP) method, and the stepwise weight assessment ratio analysis (SWARA) method; Section 3 explains the theoretical background and calculation execution procedure of the proposed flexible RPN assessment method; Section 4 presents an illustrative example of stages of treatment planning for proton beam radiation therapy, which has been analyzed and compared with some common methods; and Section 5 offers a brief summary of this study and future possible research directions.

2. Preliminaries

This section features briefly reviewed definitions and operations of FMEA and the interval 2-tuple linguistic representation model (LRM), the AHP method, and the SWARA method, which are relevant to this study.

2.1. FMEA

The FMEA approach originated in the 1960s to solve issues related to safety and reliability requirements in the aerospace industry [36]. For its applicability to different purposes, different types of FMEA exist such as the process FMEA, design FMEA, machinery FMEA, service FMEA, and the system FMEA. Because the FMEA approach is easy for calculation and important for risk assessment, it is used by many internationally related standards such as ISO/TS 16949, IEC 60812, QS9000, BS 5760, MIL-STD-1629A, and SAE J1739 [3]. Today, a large amount of research combines FMEA with different calculation methods to address risk assessment issues in various fields. For example, Kan, Wei, Zhao, and Cao [37] combined the gray relational projection method and the VIKOR compromise ranking method (Serbian term: VlseKriterijumska Optimizacija I Kompromisno Resenje) to evaluate submarine pipeline risk. Liao, Hu, Zhang, Tang, and Banaitis [38] combined the best–worst method, the PROMETHEE-II method, and the hesitant fuzzy linguistic term set to assess the risk of the food cold chain.

The traditional FMEA method used severity (S), occurrence (O), and detection (D) of risk assessment factors to calculate RPN values.

Table 1. Semantic terms for the ratings for typical risk factors.

Levels	S	O	D
s1	ANI (Almost no impact)	ANH (Almost never happens)	EH (Extremely high)
s2	VSI (Very slight impact)	VVL (Very very low)	VVH (Very very high)
s3	VMI (Very minor impact)	VL (Very low)	VH (Very high)
s4	LI (Low impact)	RL (Relatively low)	MH (Medium High)
s5	RLI (Relatively low impact)	L (Low)	M (Medium)
s6	MI (Moderate impact)	M (Medium)	L (Low)
s7	MHI (Medium to high impact)	MH (Medium High)	RL (Relatively low)
s8	HI (High impact)	VH (Very high)	VL (Very low)
s9	VHI (Very high impact)	VVH (Very very high)	VVL (Very very low)
s10	EI (Extreme impact)	EH (Extremely high)	AWF (Almost will find)

shows the linguistic terms for the ratings of these three risk factors [12]. The RPN score is the product of the following three risk factors S, O, and D as indicated below:

$$\mathrm{R}\mathrm{P}\mathrm{N}=S\times O\times D$$

(1)

The failure risk has a higher RPN score, which is more serious and critical and should be given higher risk priorities. In fact, the FMEA method is mainly used to provide early preventive actions before risk occurs.

2.2. Interval 2-Tuple LRM

Extending the definition of 2-tuple fuzzy LRM [39], Zhang [40] introduced the interval 2-tuple LRM as a generalization of 2-tuple fuzzy LRM. Some basic definitions about 2-tuple fuzzy LRM and interval 2-tuple LRM are stated subsequently:

Definition 1 

([27,39]). Let $S=\left\{s_0,s_1,\dots ,s_g\right\}$ be an ordered set of linguistic terms, and $\beta \in [0,g]$ be the values that support the symbolic aggregation operations result; the translation function $∆$ is utilized to obtain the equivalent information of β.

$$\Delta :[0,g]\to S\times [-0.5,0.5)$$

(2)

$$\Delta (\beta )=(s_i,\alpha ),\mathrm{with}\left\{\begin{array}{cc}{s}_i,& i=\mathrm{round}(\beta )\\ \alpha =\beta -i,& \alpha \in [-0.5,0.5)\end{array}\right.$$

(3)

where s_i has the label index closest to the index β, round (·) is the operation of the usual round, and α is the value of symbolic translation.

Definition 2 

([27,40]). Let $S=\left\{s_0,s_1,\dots ,s_g\right\}$ be an ordered set of linguistic terms. The interval 2-tuple linguistic terms consist of two tuples, expressed by $\left[\left(s_i,{\alpha }_1\right),\left(s_j,{\alpha }_2\right)\right]$, where $i\le j$. $s_i$ and $s_j$ represent the linguistic term, and ${\alpha }_1$ and ${\alpha }_2$ are two crisp numbers representing the symbolic translation. It expresses the equivalent information as the interval 2-tuple and interval value $\left[{\beta }_1,{\beta }_2\right]$ (${\beta }_1\le {\beta }_2{and\beta }_1,{\beta }_2\in [0,1]$) as the following function:

$$∆\left[{\beta }_1,{\beta }_2\right]=\left[\left(s_i,{\alpha }_1\right),\left(s_j,{\alpha }_2\right)\right]with\left\{\begin{array}{cc}\begin{array}{c}{s}_i,\\ s_j,\\ \begin{array}{c}{\alpha }_1={\beta }_1-i/g\\ {\alpha }_2={\beta }_2-i/g\end{array}\end{array}& \begin{array}{c}i=round({\beta }_1·g)\\ j=round({\beta }_2·g)\\ \begin{array}{c}{\alpha }_1\in [-0.5/g,0.5/g)\\ {\alpha }_2\in [-0.5/g,0.5/g)\end{array}\end{array}\end{array}\right.$$

(4)

Definition 3 

([27,40]). Let $S=\left\{s_0,s_1,\dots ,s_g\right\}$ be an ordered set of linguistic terms and  $\left[\left(s_i,{\alpha }_1\right),\left(s_j,{\alpha }_2\right)\right]$  be the IV2L. There is always a function  ${∆}^{-1}$, such that  $\left[\left(s_i,{\alpha }_1\right),\left(s_j,{\alpha }_2\right)\right]$can be converted to interval value  $[{\beta }_1,{\beta }_2]$  (${\beta }_1\le {\beta }_2{and\beta }_1,{\beta }_2\in [0,1]$) as the following function:

$${∆}^{-1}\left[\left(s_i,{\alpha }_1\right),\left(s_j,{\alpha }_2\right)\right]=\left[{\displaystyle \frac{i}{g}}+{\alpha }_1,{\displaystyle \frac{j}{g}}+{\alpha }_2\right]$$

(5)

When   $s_i=s_j$  and  ${\alpha }_1={\alpha }_2$, then the IV2L returns to a traditional 2-tuple linguistic.

Definition 4 

([35,40]). Let $S=\left\{s_0,s_1,\dots ,s_g\right\}$ be an ordered set of linguistic terms. For an IV2L  $A=\left[\left(s_i,{\alpha }_1\right),\left(s_j,{\alpha }_2\right)\right]$, then, the accuracy function  $H\left(A\right)$  and score function  $S\left(A\right)$  are explained by the following functions, respectively:

$$H\left(A\right)=\left(j-i\right)/g+{\alpha }_2-{\alpha }_1$$

$$S\left(A\right)=\left(i+j\right)/\left(2g\right)+({\alpha }_1+{\alpha }_2)/2$$

where $0\le H\left(A\right)\le 1$ and $0\le S\left(A\right)\le 1$.

Definition 5 

([27,40]). Let $S=\left\{s_0,s_1,\dots ,s_g\right\}$ be an ordered set of linguistic terms.  $A=\left[\left(s_i,{\alpha }_1\right),\left(s_j,{\alpha }_2\right)\right]$  and  $B=\left[\left(s_k,{\alpha }_3\right),\left(s_l,{\alpha }_4\right)\right]$  are two IV2L models:

• If  $S\left(A\right)=S\left(B\right)$, then:

  • • (1) 

      If  $H\left(A\right)<H\left(B\right)$, then  $A>B$;

      (2) 

      If  $H\left(A\right)>H\left(B\right)$, then  $A<B$;

      (3) 

      If $H\left(A\right)=H\left(B\right)$, then $A=B$.

      • If  $S\left(A\right)>S\left(B\right)$, then$A>B$;
      • If $S\left(A\right)<S\left(B\right)$, then $A<B$.

Definition 6 

([27,40]). Let $X=\{\left[\left(s_1,{\alpha }_1\right),\left(t_1,{\epsilon }_1\right)\right],\left[\left(s_2,{\alpha }_2\right),\left(t_2,{\epsilon }_2\right)\right],$  $\dots ,\left[\left(s_n,{\alpha }_n\right),\left(t_n,{\epsilon }_n\right)\right]\}$  be a IV2L set, and  $w={(w_1,w_2,\dots ,w_n)}^T$  be a corresponding related weights vector, with  ${\sum }_{i=1}^n{w}_i=1$  and  $w_i\in [0,1]$  ($i=1,2,\dots ,n$). The interval value 2-tuple weighted average (IV2WA) operator is expressed as:

$$IV2WA\left(X\right)=∆\left[{\sum }_{i=1}^n{w}_i{∆}^{-1}\left(s_i,{\alpha }_i\right),{\sum }_{i=1}^n{w}_i{∆}^{-1}\left(t_i,{\epsilon }_i\right)\right]$$

(6)

Definition 7 

([27]). Let $\tilde{a}=\left[\left(s_1,{\alpha }_1\right),\left(t_1,{\epsilon }_1\right)\right]$ and  $\tilde{b}=\left[\left(s_2,{\alpha }_2\right),\left(t_2,{\epsilon }_2\right)\right]$  be two IV2L models; the  $\tilde{a}$  distance from  $\tilde{b}$  is defined as:

$$d\left(\tilde{a},\tilde{b}\right)=∆\left[{\displaystyle \frac{1}{2}}\left(\left|{∆}^{-1}\left(s_1,{\alpha }_1\right)-{∆}^{-1}\left(s_2,{\alpha }_2\right)\right|+\left|{∆}^{-1}\left(t_1,{\epsilon }_1\right)-{∆}^{-1}\left(t_2,{\epsilon }_2\right)\right|\right)\right]$$

(7)

2.3. AHP Method

The analytic hierarchy process (AHP) method is a subjective weight calculation method. It uses an initial pairwise comparison matrix (X) to calculate the subjective weights of evaluation criteria [41].

$$X=\left[\begin{array}{cccc}1& x_{12}& \cdots & x_{1n}\\ {\displaystyle \frac{1}{x_{12}}}& 1& \cdots & x_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\displaystyle \frac{1}{x_{1n}}}& {\displaystyle \frac{1}{x_{2n}}}& \cdots & 1\end{array}\right]$$

(8)

According to the initial pairwise comparisons matrix, the maximum eigenvalue (${\lambda }_{max}$) is calculated, and the consistency index (CI) and consistency ratio (CR) are used to perform a consistency test [42,43]. If CR ≤ 0.1, it indicates that the initial pairwise comparisons matrix is reasonable.

$$Xw={\lambda }_{max}w,{\sum }_{j=1}^n{w}_j=1$$

(9)

$$CI={\displaystyle \frac{{\lambda }_{max}-n}{n-1}}$$

(10)

$$CR={\displaystyle \frac{CI}{RI}}$$

(11)

2.4. SWARA Method

The stepwise weight assessment ratio analysis (SWARA) method is an objective weight calculation method. It was first introduced by Keršuliene, Zavadskas, and Turskis [44], and uses stepwise and ratio values of different evaluation criteria to calculate the objective weights of evaluation criteria [45].

The calculation steps of the SWARA method are as follows [46,47].

(1)

The aggregated evaluation criterion values, $t_j$, are sorted in descending order.

(2)

Calculate the comparative coefficient, $k_j$.

$$k_j=\left\{\begin{array}{c}\begin{array}{cc}1& j=1\end{array}\\ \begin{array}{cc}{t}_j+1& j>1\end{array}\end{array}\right.$$

(12)

(3)

Calculate the initial weights of the evaluation criteria, $l_j$.

$$l_j=\left\{\begin{array}{c}\begin{array}{cc}1& j=1\end{array}\\ \begin{array}{cc}{\displaystyle \frac{l_j-1}{k_j}}& j>1\end{array}\end{array}\right.$$

(13)

(4)

Calculate the objective weights of the evaluation criteria, $w_j$.

$$w_j^{OW}={\displaystyle \frac{l_j}{{\sum }_{j=1}^n{l}_j}},j=1,2,3,\dots ,n$$

(14)

3. Proposed Novel Flexible Risk Assessment Method

Today’s enterprise environment, with highly intensive competition, makes risk assessment a key issue for the sustainable development of an enterprise. It will affect how much capital of the company will be invested to prevent the occurrence of risks. In addition, the correctness of the risk assessment results will affect the quality of the products and market competitiveness. The RPN of FMEA method is one of the most commonly used practical approaches among all risk assessment methods. However, the traditional RPN method cannot handle incomplete information or hesitation information provided by experts. Moreover, the traditional RPN approach does not consider the subjective and objective weights of risk assessment factors S, O, and D. These restrictions of the traditional RPN technique will affect the risk assessment outcome, causing misjudgment by decision makers. To overcome these restrictions, this paper proposed a flexible RPN method, which integrates the IV2WA method and considers the integrated weights of risk assessment factors to handle issues related to risk assessment. A flow diagram of the proposed flexible RPN method is depicted in Figure 1.

The procedure of the proposed flexible RPN approach is outlined as follows:

• Step 1: Determine the team members for risk assessment.

This will be based on the different backgrounds and experience to form a cross-disciplinary risk assessment team.

• Step 2: Identify the potential failure cause, failure mode, and failure effect.

Cross-disciplinary risk assessment team members discuss possible failure items and identify the potential failure cause, failure mode, and failure effect of these failure items.

• Step 3: Use the linguistic information to identify the possible level of failure modes.

Every risk assessment team member uses the linguistic information to identify the level of failure modes and the risk assessment factors. The linguistic terms’ rating of three risk assessment factors is from s₁ to s₁₀, as shown in Table 1.

• Step 4: Convert information into interval-valued 2-tuple information.

The interdisciplinary experts use 10 levels of linguistic terms to provide their linguistic assessments as follows:

$$\begin{gathered} S=\{s_1=\mathrm{A}\mathrm{l}\mathrm{m}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e},s_2=\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t},s_3=\mathrm{S}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t},s_4=\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{y}\mathrm{l}\mathrm{o}\mathrm{w},s_5=\mathrm{L}\mathrm{o}\mathrm{w}, \\ {s}_6=\mathrm{M}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e},s_7=\mathrm{M}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h},s_8=\mathrm{H}\mathrm{i}\mathrm{g}\mathrm{h},s_9=\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{y}\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h},s_{10}=\mathrm{H}\mathrm{a}\mathrm{z}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{o}\mathrm{u}\mathrm{s}\} \end{gathered}$$

(1)

If the information is a certain linguistic term such as “$\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{y}\mathrm{l}\mathrm{o}\mathrm{w}$”, this can be expressed as $\left[{(s}_4,0\right),(s_4,0)]$.

(2)

If the information is a certain linguistic term interval such as “$\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{y}\mathrm{l}\mathrm{o}\mathrm{w}-\mathrm{L}\mathrm{o}\mathrm{w}$”, this can be expressed as $\left[{(s}_4,0\right),(s_5,0)]$.

(3)

If the information is incomplete information, use the information of the maximum operator, minimum operator, or average operator to replace it.

• Step 5: Calculate the subjective weight of risk assessment factors.

The subjective weights ($w_j^{SW}$) of risk assessment factors j are calculated using the AHP method.

• Step 6: Calculate the objective weight of risk assessment factors.

The objective weights ($w_j^{OW}$) of risk assessment factors j are calculated using the SWARA method.

• Step 7: Calculate the integrated weights for the three risk assessment factors.

The integrated weights of the jth criteria $\left(w_j^I\right)$ can be calculated by the following equation:

$$w_j^I=\lambda ·w_j^{SW}+(1-\lambda )·w_j^{OW}$$

(15)

where $w_j^{SW}$ and $w_j^{OW}$ represent the subjective weight and objective weight, respectively. The value of the preference parameter $\left(\lambda \right)$ represents the subjective preferences of experts. If experts cannot determine the value of the preference parameter $\left(\lambda \right)$, 0.5 is typically used as the baseline.

• Step 8: Calculate the total performance values of different operators.

Use the maximum operator, minimum operator, and average operator to calculate the total performance values.

• Step 9: Rank the total performance values of different operators.

Rank the total performance values of the failure modes from highest to lowest for the different operators.

• Step 10: Identify the risk level ranking for the failure modes.

According to the calculation result of Step 9, use the fuzzy majority rule to identify the ranking of risk levels in ascending order.

4. Empirical Analysis

In this section, this paper uses an example of stages of treatment planning for proton beam radiation therapy (adapted from [12]) to explain the correctness and rationality of the proposed flexible RPN method.

4.1. Problem Description

Proton beam radiotherapy is a novel irradiation medical technique. Although proton beam radiotherapy can greatly improve the results after treatment, it also has some possible failure risks in the treatment planning stage. This paper probed the most critical failure modes of the stages of treatment planning for proton beam radiation therapy. There were nine failure modes as shown in

Table 2. Planning stage FMEA of proton beam radiotherapy.

Items	Potential Failure Mode	Potential Failure Cause	Potential Failure Effect
No. 1	Outdated manifestations of anatomy	Related to time delay for anatomical changes	Dose distribution mistake/dose delivery mistake
No. 2	Inaccurate delineation	Human mistake	Dose distribution mistake
No. 3	Incorrect Hounsfield unit numbers’ manual assignment	Human mistake or lack of documentation for referring clinicians	Dose distribution mistake
No. 4	The couch origin definition of coordinates is wrong (mall amount, in units of 2–3 mm)	Human mistake	Unexpected normal tissue irradiated and loss of clinical target volume
No. 5	Wrong gantry angle selection/bed frame rotation: the beam stops on a dangerous organ	Operators lack skills	Plan robustness low (uncertain range)
No. 6	Wrong gantry angle selection/bed frame rotation: the light beam passes through unstable tissue	Operators lack skills	Plan robustness low (uncertain range)
No. 7	Wrong selection of physical beam model and/or computing grid	Human mistake due to time pressure or lack of skills	Dose distribution mistake
No. 8	Field isocenter definition is wrong	Human mistake	Dose delivery mistake
No. 9	Approval of wrong plan	Human mistake, communication between operators fails	Dose delivery mistake

[12]. The FMEA assessment team consisted of five interdisciplinary experts (P1, P2…, P5) to evaluate the most critical failure modes for the proton beam radiation therapy. The backgrounds of the five experts include risk management, radiation oncology, medical physics, radiation dosimetry, and radiation protection. The five interdisciplinary experts’ assigned weights were equal. According to Table 1, five experts determined the pairwise comparisons matrix of evaluation criteria, as shown in

Table 3. The pairwise comparisons matrix of evaluation criteria.

		S	O	D
S	P1	1	2	2
	P2	1	3	3
	P3	1	2	2
	P4	1	2	2
	P5	1	3	3
O	P1	1/2	1	1
	P2	1/3	1	1
	P3	1/2	1	1
	P4	1/2	1	1
	P5	1/3	1	1
D	P1	1/2	1	1
	P2	1/3	1	1
	P3	1/2	1	1
	P4	1/2	1	1
	P5	1/3	1	1

, and evaluated the possible range of failure modes for three risk factors, as shown in

Table 4. Linguistic evaluation of the possible range of failure.

Items	S					O					D
	P1	P2	P3	P4	P5	P1	P2	P3	P4	P5	P1	P2	P3	P4	P5
No. 1	HI	HI	HI	HI	HI	VL	RL	VL-RL	VL	VVL	RL	VL	VL	VL	VVL
No. 2	MI	MI	MI	MHI	RLI	RL	VL	RL	L	RL	L-RL	L	M	L	L
No. 3	MHI	MI	MHI	MHI	*	RL	RL	RL	RL	*	RL	L	RL	VL	RL
No. 4	LI	RLI	RLI-MI	RLI	RLI	RL-L	RL	RL	RL	L	L	L	L	L	L
No. 5	HI	HI	HI	HI	HI	VL	RL	RL-L	RL	RL	VH	MH	M	VH	MH
No. 6	HI	VHI	HI	MHI	HI	VL	RL	VL	VL	VL	MH	M	M-L	M	M
No. 7	MHI	HI	MHI	MI	HI	RL	RL	RL	RL	RL	M	M	M	M	M
No. 8	RLI	RLI	RLI-MI	RLI	*	RL	VL	VL	VL	*	VL	VL	VL	VL	RL
No. 9	HI	VHI	HI	MHI	HI	VL-RL	VL	VL	VL	VL	L	M	M	M	MH

“*” represents missing or nonexistent data.

[12].

4.2. Solving the Case by the RPN Method

The traditional RPN method used the three risk assessment factors to indicate the failure risk scores. A higher RPN score means a higher risk level and the need to give resource priority to prevent the occurrence of malfunction. The traditional RPN technique can only handle complete information for the risk factors of failure modes. Therefore, it only considers complete information (P2 and P4 provided), and does not apply hesitation information (P1 and P3 provided) and incomplete information (P5 provided) in the risk assessment of proton beam radiotherapy. According to Table 1 and Table 3, use Equation (1) to calculate the RPN scores of every failure mode for the proton beam radiotherapy, as shown in

Table 5. Traditional RPN for the proton beam radiotherapy.

Failure Modes	S	O	D	RPN
No. 1	8	4	8	256
No. 2	7	4	6	168
No. 3	7	4	7	196
No. 4	5	4	6	120
No. 5	8	4	4	128
No. 6	8	4	5	160
No. 7	7	4	5	140
No. 8	5	3	8	120
No. 9	8	3	5	120

.

The risk assessment factors of S, O, and D belong to an ordered scale. However, the traditional RPN method uses the product of three risk assessment factors to calculate the RPN score. The multiplication calculation is not meaningful for ordinal scales.

4.3. Solving the Case by the 2-Tuple RPN Method

The 2-tuple RPN method [48] uses the 2-tuple fuzzy linguistic representation method to express semantic information in the process of aggregated semantic information. The 2-tuple RPN method, like the traditional RPN approach, cannot handle hesitation information and incomplete information. Therefore, for assessment of the failure risk for the proton beam radiotherapy, it only considered P2 and P4, which provided complete information.

The arithmetic mean of 2-tuple $\overline{x}$ is explained as follows [48]:

$$\overline{x}=∆\left({\sum }_{j=1}^n{\displaystyle \frac{1}{n}}{∆}^{-1}(r_j,{\alpha }_j)\right)=∆\left({\displaystyle \frac{1}{n}}{\sum }_{j=1}^n{\beta }_j\right)$$

(16)

where $x=\{\left(r_1,{\alpha }_1\right),\left(r_2,{\alpha }_2\right),\dots .,\left(r_n,{\alpha }_n\right)\}$ is a set of 2-tuples.

According to Table 1 and Table 4, use Equation (16) to calculate the 2-tuple RPN values of every failure mode for the proton beam radiotherapy, as shown in

Table 6. Two-tuple RPN values of the proton beam radiotherapy.

Failure Modes	S	O	D	2-Tuple RPN
No. 1	(s8, 0)	(s4, −0.5)	(s8, 0)	(s7, −0.500)
No. 2	(s7, −0.5)	(s4, 0)	(s6, 0)	(s6, −0.500)
No. 3	(s7, −0.5)	(s4, 0)	(s7, 0)	(s6, −0.167)
No. 4	(s5, 0)	(s4, −0.5)	(s6, 0)	(s5, −0.167)
No. 5	(s8, 0)	(s4, 0)	(s4, −0.5)	(s5, 0.167)
No. 6	(s8, 0)	(s4, −0.5)	(s5, 0)	(s6, −0.500)
No. 7	(s7, 0)	(s4, 0)	(s5, 0)	(s5, 0.333)
No. 8	(s5, 0)	(s3, 0)	(s8, 0)	(s5, 0.333)
No. 9	(s8, 0)	(s3, 0)	(s5, 0)	(s5, 0.333)

.

4.4. Solving the Case by the IV2WA Method

Because the traditional FMEA approach cannot handle incomplete or hesitant information during information processing, the IV2WA method [35] can overcome these limitations. This method is capable of handling complete, incomplete, and hesitant information simultaneously. For incomplete information processing, the IV2WA method uses the linguistic interval between the minimum and maximum risk linguistic rating terms to replace incomplete information. All information provided by experts is handled by the interval-valued information to assess the failure risk of the proton beam radiotherapy. According to Table 1 and Table 3, use Equation (6) to calculate the overall performance values of every failure mode for the proton beam radiotherapy, as shown in

Table 7. IV2WA values of the proton beam radiotherapy.

Failure Modes	P1	P2	P3	P4	P5	Overall Performance Values
No. 1	Δ([0.600, 0.600])	Δ([0.667, 0.667])	Δ([0.633, 0.667])	Δ([0.633, 0.633])	Δ([0.633, 0.633])	Δ([0.633, 0.642])
No. 2	Δ([0.533, 0.567])	Δ([0.500, 0.500])	Δ([0.500, 0.500])	Δ([0.600, 0.600])	Δ([0.500, 0.500])	Δ([0.533, 0.542])
No. 3	Δ([0.600, 0.600])	Δ([0.533, 0.533])	Δ([0.600, 0.600])	Δ([0.633, 0.633])	Δ([0.300, 0.900])	Δ([0.592, 0.592])
No. 4	Δ([0.467, 0.500])	Δ([0.467, 0.467])	Δ([0.500, 0.533])	Δ([0.500, 0.500])	Δ([0.533, 0.533])	Δ([0.483, 0.500])
No. 5	Δ([0.467, 0.467])	Δ([0.533, 0.533])	Δ([0.567, 0.600])	Δ([0.500, 0.500])	Δ([0.533, 0.533])	Δ([0.517, 0.525])
No. 6	Δ([0.500, 0.500])	Δ([0.600, 0.600])	Δ([0.533, 0.567])	Δ([0.500, 0.500])	Δ([0.533, 0.533])	Δ([0.533, 0.542])
No. 7	Δ([0.533, 0.533])	Δ([0.567, 0.567])	Δ([0.533, 0.533])	Δ([0.500, 0.500])	Δ([0.567, 0.567])	Δ([0.533, 0.533])
No. 8	Δ([0.567, 0.567])	Δ([0.533, 0.533])	Δ([0.533, 0.567])	Δ([0.533, 0.533])	Δ([0.300, 0.900])	Δ([0.542, 0.550])
No. 9	Δ([0.567, 0.600])	Δ([0.567, 0.567])	Δ([0.533, 0.533])	Δ([0.500, 0.500])	Δ([0.500, 0.500])	Δ([0.542, 0.550])

.

4.5. Solving the Case by the Proposed Flexible RPN Method

The first step in the risk assessment case for proton beam radiation therapy using the proposed flexible RPN method was to determine the team members for the assessment. Next, they identified the potential causes of failure, failure modes, and failure effects (Table 2). Each risk assessment team member determined the pairwise comparisons matrix for the evaluation criteria, as shown in Table 3, and each risk assessment team member then used linguistic information to identify the levels of failure modes and the risk assessment factors (Table 4).

Step 4: Convert information into interval-valued 2-tuple information.

According to Table 3, the possible range of failure information can be converted into interval-valued 2-tuple information. To handle incomplete information, the information of the maximum operator, minimum operator, and average operator can be used. The information from the maximum operator, minimum operator, and average operator reflects a decision maker’s current degree of optimism, pessimism, and neutrality.

Use the maximum operator, minimum operator, and average operator to calculate the overall performance values; that is, the aggregated S, O, and D values of different operators. The results are shown in

Table 8. The aggregated S, O, and D values of different operators.

Failure Modes	Maximum Operator			Minimum Operator			Average Operator
	S	O	D	S	O	D	S	O	D
No. 1	Δ([0.800, 0.800])	Δ([0.300, 0.320])	Δ([0.800, 0.800])	Δ([0.800, 0.800])	Δ([0.300, 0.320])	Δ([0.800, 0.800])	Δ([0.800, 0.800])	Δ([0.300, 0.320])	Δ([0.800, 0.800])
No. 2	Δ([0.600, 0.600])	Δ([0.400, 0.400])	Δ([0.580, 0.600])	Δ([0.600, 0.600])	Δ([0.400, 0.400])	Δ([0.580, 0.600])	Δ([0.600, 0.600])	Δ([0.400, 0.400])	Δ([0.580, 0.600])
No. 3	Δ([0.680, 0.680])	Δ([0.400, 0.400])	Δ([0.700, 0.700])	Δ([0.660, 0.660])	Δ([0.400, 0.400])	Δ([0.700, 0.700])	Δ([0.675, 0.675])	Δ([0.400, 0.400])	Δ([0.700, 0.700])
No. 4	Δ([0.480, 0.500])	Δ([0.400, 0.420])	Δ([0.600, 0.600])	Δ([0.480, 0.500])	Δ([0.400, 0.420])	Δ([0.600, 0.600])	Δ([0.480, 0.500])	Δ([0.400, 0.420])	Δ([0.600, 0.600])
No. 5	Δ([0.800, 0.800])	Δ([0.380, 0.400])	Δ([0.380, 0.380])	Δ([0.800, 0.800])	Δ([0.380, 0.400])	Δ([0.380, 0.380])	Δ([0.800, 0.800])	Δ([0.380, 0.400])	Δ([0.380, 0.380])
No. 6	Δ([0.800, 0.800])	Δ([0.320, 0.320])	Δ([0.480, 0.500])	Δ([0.800, 0.800])	Δ([0.320, 0.320])	Δ([0.480, 0.500])	Δ([0.800, 0.800])	Δ([0.320, 0.320])	Δ([0.480, 0.500])
No. 7	Δ([0.720, 0.720])	Δ([0.400, 0.400])	Δ([0.500, 0.500])	Δ([0.720, 0.720])	Δ([0.400, 0.400])	Δ([0.500, 0.500])	Δ([0.720, 0.720])	Δ([0.400, 0.400])	Δ([0.500, 0.500])
No. 8	Δ([0.500, 0.540])	Δ([0.340, 0.340])	Δ([0.780, 0.780])	Δ([0.500, 0.520])	Δ([0.320, 0.320])	Δ([0.780, 0.780])	Δ([0.500, 0.525])	Δ([0.325, 0.325])	Δ([0.780, 0.780])
No. 9	Δ([0.800, 0.800])	Δ([0.300, 0.320])	Δ([0.500, 0.500])	Δ([0.800, 0.800])	Δ([0.300, 0.320])	Δ([0.500, 0.500])	Δ([0.800, 0.800])	Δ([0.300, 0.320])	Δ([0.500, 0.500])

.

• Step 5: Calculate the subjective weight of risk assessment factors.

According to Table 3, and using Equations (8) and (9) to calculate the subjective weight of risk assessment factors, the results are shown in

Table 9. Subjective, objective, and average weights of different operators.

	Maximum Operator			Minimum Operator			Average Operator
	S	O	D	S	O	D	S	O	D
Subjective weight	0.545	0.227	0.227	0.545	0.227	0.227	0.545	0.227	0.227
Objective weight	0.585	0.102	0.313	0.581	0.103	0.316	0.584	0.102	0.314
Average weight	0.565	0.165	0.270	0.563	0.165	0.272	0.565	0.165	0.271

.

• Step 6: Calculate the objective weight of risk assessment factors.

According to Table 4, and using Equations (12)–(14), calculate the objective weight of risk assessment factors; the results are shown in Table 9.

• Step 7: Calculate the integrated weights for the three risk assessment factors.

According to the results of Step 5 and Step 6, use Equation (15) to compute the integrated weights for the three risk assessment factors. The value of the preference parameter (λ) is typically set at the baseline of 0.5; calculation results are as shown in Table 9.

For example, when the subjective weight ($w_j^{SW}$) of risk assessment factor S is 0.545, the objective weight ($w_j^{OW}$) of risk assessment factor S is 0.585 in the maximum operator, and by Equation (10), it is found that

$$w_j^I=\lambda ·w_j^{SW}+\left(1-\lambda \right)·w_j^{OW}=0.5\times 0.545+(1-0.5)\times 0.585=0.565$$

• Step 8: Calculate the total performance values of different operators.

According to Table 8 and Table 9, use Equation (6) to calculate the total performance values of every failure mode for the proton beam radiotherapy, as shown in

Table 10. Fuzzy majority rule ranking of the proton beam radiotherapy.

Failure Modes	Maximum Operator		Minimum Operator		Average Operator		Fuzzy Majority Rule
	Overall Performance Values	Ranking	Overall Performance Values	Ranking	Overall Performance Values	Ranking
No. 1	Δ([0.749, 0.751])	1	Δ([0.749, 0.751])	1	Δ([0.749, 0.751])	1	1
No. 2	Δ([0.573, 0.580])	8	Δ([0.573, 0.580])	7	Δ([0.573, 0.580])	8	8
No. 3	Δ([0.658, 0.658])	2	Δ([0.646, 0.646])	4	Δ([0.655, 0.655])	3	3
No. 4	Δ([0.509, 0.523])	9	Δ([0.509, 0.523])	9	Δ([0.509, 0.523])	9	9
No. 5	Δ([0.626, 0.628])	5	Δ([0.626, 0.628])	5	Δ([0.626, 0.628])	5	5
No. 6	Δ([0.651, 0.657])	4	Δ([0.651, 0.657])	3	Δ([0.651, 0.657])	4	4
No. 7	Δ([0.619, 0.619])	6	Δ([0.619, 0.619])	6	Δ([0.619, 0.619])	6	6
No. 8	Δ([0.571, 0.595])	7	Δ([0.569, 0.581])	8	Δ([0.570, 0.584])	7	7
No. 9	Δ([0.655, 0.657])	3	Δ([0.655, 0.657])	2	Δ([0.655, 0.657])	2	2

.

• Step 9: Rank the total performance values of different operators.
• Step 10: Identify the risk level ranking for the failure modes.

For different operators, the overall performance value of the failure mode is ranked from highest to lowest; the results are shown in Table 10. Based on the ranking results of the maximum operator, minimum operator, and average operator, the fuzzy majority rule is used to determine the risk level ranking in ascending order; the results are shown in Table 10.

For example, when the S value is Δ([0.800, 0.800]), the O value is Δ([0.300, 0.320]), and the D value is Δ([0.800, 0.800]), in the maximum operator of failure modes No. 1 (Table 8), the corresponding weights of risk assessment factors S, O, and D are 0.585, 0.102, and 0.313, respectively. The overall performance values of the failure modes are No. 1 (Table 10) as Δ([0.749, 0.751]).

$$\mathrm{I}\mathrm{V}2\mathrm{W}\mathrm{A}\left(X\right)=\Delta \left(\left[0.585\times 0.800+0.102\times 0.300+0.313\times 0.800,0.585\times 0.800+0.102\times 0.300+0.313\times 0.800\right]\right)=\mathsf{\Delta }\left(\right[0.749,0.751\left]\right)$$

4.6. Comparison and Analysis

In order to prove the effectiveness and rationality of the proposed flexible RPN method, an illustration of the treatment planning stages for proton beam radiation therapy is performed in Section 4, using the same input data (Table 1, Table 2, Table 3 and Table 4) to compare the results of four different calculation methods (the traditional RPN method, 2-tuple RPN approach, IV2WA method, and flexible RPN method).

Table 11. Risk ranking results’ comparison of the different risk ranking methods.

Failure Modes	Traditional RPN Method		2-Tuple RPN Method		IV2WA Method		Proposed Method
	RPN	Ranking	2-Tuple RPN	Ranking	Overall Performance Values	Ranking	Maximum Operator	Minimum Operator	Average Operator	Fuzzy Majority Rule
No. 1	256	1	(s7, −0.500)	1	Δ([0.633, 0.642])	1	Δ([0.749, 0.751])	Δ([0.749, 0.751])	Δ([0.749, 0.751])	1
No. 2	168	3	(s6, −0.500)	3	Δ([0.533, 0.542])	5	Δ([0.573, 0.580])	Δ([0.573, 0.580])	Δ([0.573, 0.580])	8
No. 3	196	2	(s6, −0.167)	2	Δ([0.592, 0.592])	2	Δ([0.658, 0.658])	Δ([0.646, 0.646])	Δ([0.655, 0.655])	3
No. 4	120	7	(s5, −0.167)	9	Δ([0.483, 0.500])	9	Δ([0.509, 0.523])	Δ([0.509, 0.523])	Δ([0.509, 0.523])	9
No. 5	128	6	(s5, 0.167)	8	Δ([0.517, 0.525])	8	Δ([0.626, 0.628])	Δ([0.626, 0.628])	Δ([0.626, 0.628])	5
No. 6	160	4	(s6, −0.500)	3	Δ([0.533, 0.542])	5	Δ([0.651, 0.657])	Δ([0.651, 0.657])	Δ([0.651, 0.657])	4
No. 7	140	5	(s5, 0.333)	5	Δ([0.533, 0.533])	7	Δ([0.619, 0.619])	Δ([0.619, 0.619])	Δ([0.619, 0.619])	6
No. 8	120	7	(s5, 0.333)	5	Δ([0.542, 0.550])	3	Δ([0.571, 0.595])	Δ([0.569, 0.581])	Δ([0.570, 0.584])	7
No. 9	120	7	(s5, 0.333)	5	Δ([0.542, 0.550])	3	Δ([0.655, 0.657])	Δ([0.655, 0.657])	Δ([0.655, 0.657])	2

shows the comparison of the risk ranking results of different risk ranking methods. The main differences in the factors considered between the traditional RPN method, the 2-tuple RPN approach, the IV2WA method, and the flexible RPN method are shown in

Table 12. The main differences lie in the factors considered by different risk ranking methods.

Method	Factors Consideration
	Process the Hesitation Information	Process the Incomplete Information	Objective Weight Consideration of Risk Assessment Factors
The RPN method	No	No	No
The 2-tuple RPN approach [36]	No	No	No
The IV2WA method [32]	Yes	Yes	No
Proposed method	Yes	Yes	Yes

.

The traditional RPN approach and 2-tuple RPN method can only handle complete information and cannot process incomplete and hesitation information. If the information provided by the expert contains some hesitation information or incomplete information, all of the information provided by this expert will not be considered. Clearly, these two methods do not fully consider the information provided by all experts; hence, they will generate biased assessment results. However, both the IV2WA method and the proposed flexible RPN method use the linguistic interval to handle complete, incomplete, and hesitation information. Specifically, for incomplete information processing, the IV2WA method uses the linguistic interval between minimum and maximum risk linguistic rating terms to replace incomplete information. Moreover, the IV2WA method considers more information than both the traditional RPN approach and the 2-tuple RPN method. However, for the processing of incomplete information, the IV2WA method does not refer to other experts on this failure mode for the provision of useful information. The proposed flexible RPN method uses the maximum operator, minimum operator, and average operator of other experts to accommodate this failure mode, thus deriving useful information to calculate the overall performance values. Therefore, all useful information in the risk assessment process can be considered by the proposed flexible RPN method.

Furthermore, the traditional RPN method, 2-tuple RPN method, and IV2WA approach do not consider subjective and objective weights; they only consider the subjective weight of risk assessment factors. These approaches will cause biased evaluation results and inform wrong decision making. However, the proposed flexible RPN method fully considers all subjective weights and objective weights between the risk assessment factors, S, O, and D. In its consideration of objective weights, the proposed flexible RPN method applies the statistical distance method to calculate the objective weights of risk assessment factors. The risk ranking comparison between the proposed flexible RPN method and other different risk ranking methods is as shown in Figure 2.

4.7. Sensitivity Analysis

This article uses different combinations of subjective and objective weights to conduct a sensitivity analysis of risk assessment in the treatment planning stage of proton beam radiotherapy. The sensitivity analysis uses six different $\lambda$ values ($\lambda =0.0$, $\lambda =0.2$, $\lambda =0.4$, $\lambda =0.6$, $\lambda =0.8$, $\lambda =1.0$) to compare the failure risk ranking results of different failure modes, as shown in

Table 13. Risk ranking results for different α values.

α Value	$\mathit{\alpha }=0.0$				$\mathit{\alpha }=0.2$
Failure Modes	Maximum Operator	Minimum Operator	Average Operator	Fuzzy Majority Rule	Maximum Operator	Minimum Operator	Average Operator	Fuzzy Majority Rule
No. 1	Δ([0.749, 0.751])	Δ([0.749, 0.751])	Δ([0.749, 0.751])	1	Δ([0.735, 0.737])	Δ([0.736, 0.739])	Δ([0.737, 0.739])	1
No. 2	Δ([0.573, 0.580])	Δ([0.573, 0.579])	Δ([0.573, 0.580])	8	Δ([0.567, 0.573])	Δ([0.568, 0.574])	Δ([0.569, 0.575])	7
No. 3	Δ([0.658, 0.658])	Δ([0.646, 0.646])	Δ([0.655, 0.655])	3	Δ([0.649, 0.649])	Δ([0.639, 0.639])	Δ([0.648, 0.648])	3
No. 4	Δ([0.510, 0.523])	Δ([0.510, 0.523])	Δ([0.510, 0.523])	9	Δ([0.504, 0.518])	Δ([0.506, 0.520])	Δ([0.505, 0.520])	9
No. 5	Δ([0.624, 0.628])	Δ([0.624, 0.626])	Δ([0.625, 0.627])	5	Δ([0.621, 0.623])	Δ([0.621, 0.624])	Δ([0.622, 0.624])	5
No. 6	Δ([0.649, 0.657])	Δ([0.649, 0.656])	Δ([0.651, 0.657])	4	Δ([0.643, 0.649])	Δ([0.643, 0.649])	Δ([0.644, 0.650])	4
No. 7	Δ([0.619, 0.619])	Δ([0.618, 0.618])	Δ([0.618, 0.618])	6	Δ([0.613, 0.613])	Δ([0.613, 0.613])	Δ([0.614, 0.614])	6
No. 8	Δ([0.572, 0.595])	Δ([0.570, 0.582])	Δ([0.570, 0.585])	7	Δ([0.562, 0.585])	Δ([0.560, 0.572])	Δ([0.561, 0.575])	8
No. 9	Δ([0.654, 0.657])	Δ([0.654, 0.656])	Δ([0.655, 0.657])	2	Δ([0.646, 0.649])	Δ([0.647, 0.649])	Δ([0.647, 0.650])	2
α value	$\alpha =0.4$				$\alpha =0.6$
Failure modes	Maximum operator	Minimum operator	Average operator	Fuzzy majority rule	Maximum operator	Minimum operator	Average operator	Fuzzy majority rule
No. 1	Δ([0.724, 0.727])	Δ([0.724, 0.727])	Δ([0.723, 0.726])	1	Δ([0.711, 0.714])	Δ([0.711, 0.714])	Δ([0.712, 0.715])	1
No. 2	Δ([0.564, 0.570])	Δ([0.564, 0.569])	Δ([0.563, 0.569])	7	Δ([0.559, 0.564])	Δ([0.559, 0.564])	Δ([0.559, 0.565])	7
No. 3	Δ([0.643, 0.643])	Δ([0.631, 0.631])	Δ([0.640, 0.640])	3	Δ([0.635, 0.635])	Δ([0.624, 0.624])	Δ([0.633, 0.633])	4
No. 4	Δ([0.501, 0.516])	Δ([0.501, 0.516])	Δ([0.501, 0.515])	9	Δ([0.497, 0.511])	Δ([0.497, 0.512)	Δ([0.497, 0.512)	9
No. 5	Δ([0.619, 0.622])	Δ([0.618, 0.621])	Δ([0.618, 0.621])	5	Δ([0.615, 0.619])	Δ([0.614, 0.618])	Δ([0.616, 0.619])	5
No. 6	Δ([0.638, 0.643])	Δ([0.637, 0.643])	Δ([0.637, 0.643])	4	Δ([0.631, 0.636])	Δ([0.630, 0.635])	Δ([0.631, 0.636])	3
No. 7	Δ([0.610, 0.610])	Δ([0.609, 0.609])	Δ([0.609, 0.609])	6	Δ([0.605, 0.605])	Δ([0.605, 0.605])	Δ([0.606, 0.606])	6
No. 8	Δ([0.554, 0.577])	Δ([0.551, 0.562])	Δ([0.551, 0.565])	8	Δ([0.544, 0.567])	Δ([0.541, 0.552])	Δ([0.542, 0.556])	8
No. 9	Δ([0.640, 0.643])	Δ([0.640, 0.643])	Δ([0.640, 0.643])	2	Δ([0.632, 0.636])	Δ([0.632, 0.635])	Δ([0.633, 0.636])	2
α value	$\alpha =0.8$				$\alpha =1.0$
Failure modes	Maximum operator	Minimum operator	Average operator	Fuzzy majority rule	Maximum operator	Minimum operator	Average operator	Fuzzy majority rule
No. 1	Δ([0.698, 0.702])	Δ([0.698, 0.702])	Δ([0.698, 0.702])	1	Δ([0.686, 0.690])	Δ([0.686, 0.690])	Δ([0.686, 0.690])	1
No. 2	Δ([0.554, 0.559])	Δ([0.554, 0.559])	Δ([0.554, 0.559])	7	Δ([0.549, 0.554])	Δ([0.549, 0.554])	Δ([0.549, 0.554])	7
No. 3	Δ([0.628, 0.628])	Δ([0.617, 0.617])	Δ([0.625, 0.625])	4	Δ([0.620, 0.620])	Δ([0.609, 0.609])	Δ([0.618, 0.618])	4
No. 4	Δ([0.493, 0.508])	Δ([0.493, 0.508])	Δ([0.493, 0.508])	9	Δ([0.489, 0.504])	Δ([0.489, 0.504])	Δ([0.489, 0.504])	9
No. 5	Δ([0.612, 0.616])	Δ([0.611, 0.616])	Δ([0.612, 0.616])	5	Δ([0.609, 0.613])	Δ([0.609, 0.613])	Δ([0.609, 0.613])	5
No. 6	Δ([0.624, 0.629])	Δ([0.624, 0.629])	Δ([0.624, 0.629])	3	Δ([0.618, 0.622])	Δ([0.618, 0.622])	Δ([0.618, 0.622])	2
No. 7	Δ([0.601, 0.601])	Δ([0.601, 0.601])	Δ([0.601, 0.601])	6	Δ([0.597, 0.597])	Δ([0.597, 0.597])	Δ([0.597, 0.597])	6
No. 8	Δ([0.536, 0.558])	Δ([0.532, 0.543])	Δ([0.532, 0.546])	8	Δ([0.527, 0.549])	Δ([0.522, 0.533])	Δ([0.523, 0.537])	8
No. 9	Δ([0.625, 0.629])	Δ([0.625, 0.629])	Δ([0.625, 0.629])	2	Δ([0.618, 0.622])	Δ([0.618, 0.622])	Δ([0.618, 0.622])	2

.

Based on the results in Table 13, we can see that the failure risk ranking results for the failure modes are the same when the λ values are 0.2 and 0.4, as well as when they are 0.6 and 0.8. When the $\lambda$ values are 0.0 and 0.2, the failure risk ranking results of the failure modes are only slightly different. Specifically, the ranking positions of failure modes No. 2 and No. 8 are inter-changed. When the $\lambda$ values are 0.8 and 1.0, the ranking results of the failure risks for failure modes No. 1, No. 2, No. 3, No. 4, No. 5, No. 7, No. 8, and No. 9 are consistent.

5. Conclusions and Future Work

Risk prevention and effective risk control are the most important issues for the sustainable operation of enterprises. Many international standards apply the RPN method to rank failure risk prevention under limited resources. However, the traditional RPN approach has some restrictions. These restrictions limit the typical RPN method from handling issues of incomplete information and hesitant information in information processing. Moreover, most risk assessment methods have not considered the objective weights of risk assessment factors (S, O, and D), which can lead to deviations in results. In order to overcome these restrictions of the typical RPN approach, this study proposed a novel flexible RPN approach to enhance risk assessment abilities.

This paper used the risk assessment of the stages of treatment planning for proton beam radiation therapy as a numerical example to verify the correctness of the proposed flexible RPN method. The risk ranking result of failure modes from the proposed flexible RPN method is No. 1 > No. 9 > No. 3 > No. 6 > No. 5 > No. 7 > No. 8 > No. 2 > No. 4. This differs from the RPN method, which is No. 1 > No. 3 > No. 2 > No. 6 > No. 7 > No. 5 > No. 4, No. 8, No. 9. It also differs from the 2-tuple RPN approach [36], which is No. 1 > No. 3 > No. 2, No. 6 > No. 7, No. 8, No. 9 > No. 5 > No. 4, and from the IV2WA method [32], which is No. 1 > No. 3 > No. 8, No. 9 > No. 2, No. 6 > No. 7 > No. 5 > No. 4.

The contributions of the proposed flexible RPN method as applied research are as follows:

(1)

The proposed flexible RPN approach is not limited to considering only the risk assessment factors, S, O, and D. It can consider extra risk factors depending on the product or system characteristics.

(2)

The proposed flexible RPN method can handle incomplete information and hesitant information during the aggregated process of information processing.

(3)

The proposed flexible RPN method considers the subjective and objective weights among risk assessment factors, thereby avoiding deviation in results.

Although the proposed flexible RPN method can handle incomplete and hesitant information in risk assessment problems, it does not process different types of fuzzy cognitive information, such as intuitionistic fuzzy set information, Pythagorean fuzzy information, Fermatean fuzzy information, picture fuzzy set information, and spherical fuzzy information. Future researchers may explore the processing of these different types of fuzzy cognitive information for risk assessment problems. Future research directions can use the proposed flexible RPN method to solve ranking issues in different fields such as talent selection, resource allocation, supplier selection, and material selection, to mention but a few. On the other hand, subsequent researchers may also employ gray relational analysis to address the imprecise data in risk assessment problems.