A Novel Pythagorean Fuzzy Set–Based Risk-Ranking Method for Handling Human Cognitive Information in Risk-Assessment Problems

Abstract

With the rapid evolution of the information age and the development of artificial intelligence, processing human cognitive information has become increasingly important. The risk-priority-number (RPN) approach is a natural language-processing method and is the most widely used risk-evaluation tool. However, the typical RPN approach cannot effectively process the various forms of human cognitive information or hesitant information provided by experts in risk assessments. In addition, it cannot process the relative-weight consideration of risk-assessment factors. In order to fully grasp the various forms of human cognitive information provided by experts during risk assessment, this paper proposes a novel Pythagorean fuzzy set–based (PFS) risk-ranking method. This method integrates the PFS and the combined compromise-solution (CoCoSo) method to handle human cognitive information in risk-assessment problems. In the numerical case study, this paper used a healthcare waste-hazards risk-assessment case to verify the validity and rationality of the proposed method for handling risk-assessment issues. The calculation results of the healthcare waste-hazards risk-assessment case are compared with the typical RPN approach, intuitionistic fuzzy set (IFS) method, PFS method, and the CoCoSo method. The numerical simulation verification results prove that the proposed method can comprehensively grasp various forms of cognitive information from experts and consider the relative weight of risk-assessment factors, providing more accurate and reasonable risk-assessment results.

1. Introduction

With the rapid development of artificial intelligence, information processing has become more complex and diversified in meeting the actual needs of decision-making evaluations. Natural language processing is an important part of artificial intelligence. How human cognition information is processed will affect the evaluation results of multi-criteria decision-analysis (MCDA) problems. Risk analysis is the first step in identifying a potential accident scenario [1]. The failure modes and effects analysis (FMEA) method, which originated in the US military in the 1960s, is a preventive risk-assessment tool widely used for qualitative assessment in MCDA. The risk-priority number (RPN) approach in FMEA is mainly used for ranking possible risk-failure items. The RPN approach is a natural language-processing method which uses the product of the three risk-assessment factors—severity (S), occurrence (O), and detection (D)—to calculate the RPN value. The risk factors S, O, and D use a semantic term set of 1 to 10 to represent the level of risk. Many scholars have combined FMEA and different algorithms to deal with issues related to risk assessment in different industries. For example, Chang et al. [2] combined the hesitant fuzzy linguistic-term sets and the ordered weighted geometric operator to process the risk-assessment problems of extreme low-k dielectric integration. Aydin et al. [3] combined the best-worst method and the picture fuzzy set–based approach to solving the risk-assessment problems of the oil and gas industry. Huang et al. [4] combined interactive multi-criteria decision-making and the probabilistic linguistic term-sets method to process the risk-assessment problems of enterprise architecture and information systems. To this day, many scholars have extended the application of the FMEA method to solve decision-related issues in many different industries, such as aircraft maintenance and repair [5], zero-carbon measures [6], the self-service electric car [7], high temperature superconducting devices [8], clean energy development [9], the lead-acid battery [10], and risk analysis of submarine pipelines [11]. Although the typical RPN approach is simple to calculate and widely used in different fields, it cannot effectively deal with fuzzy information (FI), intuitionistic FI, and Pythagorean FI, which are types of human cognitive information.

To grasp uncertain information in daily life, Atanassov [12] first introduced the concepts of intuitionistic fuzzy sets (IFSs). The characterized IFSs use membership grade (MG) and non-membership grade (NMG) to describe fuzzy phenomena in daily life, limiting the sum of MG and NMG to less than or equal to 1. As an extension of the original fuzzy set and IFS, the Pythagorean fuzzy set (PFS) can provide a larger solution space and better deal with uncertain information and FI when solving practical MCDA problems. Meanwhile, the characterized PFSs use the MG and the NMG to describe fuzzy phenomena in daily life, limiting the square sum of MG and NMG to less than or equal to 1. To this day, PFSs have been widely used in many different industries and fields, such as cotton fabric selection [13], security threats of computers [14], childhood cancer risk assessment [15], lean manufacturing [16], internet finance service [17], and sustainable supplier evaluation problems [18].

The key point of MCDA problems lies in the way multiple criteria are evaluated. Many scholars use different research methods to solve complex MCDA problems, such as the combined compromise solution (CoCoSo) method [19], simple weighted sum product (simple WISP) method [20], soft analytic network process method [21], neutrosophic set method [22], analytical hierarchy-process (AHP) method [23], the Serbian term ‘VlseKriterijumska Optimizacija I Kompromisno Resenje’ (VIKOR) method [24], multiobjective optimization by ratio analysis plus the full multiplicative form (MULTIMOORA) method [25], and so on. The relative weight consideration of risk-assessment factors is the main key to influencing the risk-ranking results of the basic elements of risk failure. However, the typical RPN method ignores the weight considerations between different risk-assessment factors, resulting in incorrect evaluation results. In order to effectively handle the weight considerations between different risk-assessment factors of MCDA problems, Yazdani et al. [19] first proposed the concept of the CoCoSo method, which uses the arithmetic weighted sum, the geometric weighted sum, and three different appraisal score strategies for ranking possible alternatives. Bouraima et al. [26] combined the interval rough number, stepwise weight assessment ratio analysis, and the CoCoSo method to perform the sustainable transportation assessment of railway systems in West Africa. Extending the concept of the CoCoSo method, Zafaranlouei et al. [27] combined the base-criterion method and the fuzzy Z-numbers method to solve the problem of sustainable waste management. Chang [28] combined the CoCoSo method, subjective-objective weights consideration, and the 2-tuple linguistic representation method to handle supplier-selection issues. To this day, the CoCoSo method has been extended to handle MCDA problems in different domains, such as railway transportation systems [26], doctor selection [29], risk assessment of gas pipeline construction [30], third-party reverse logistics [31], and the application of unmanned aerial vehicles in traffic management [32].

When risk-assessment operations are performed, cognitive information provided by experts is sometimes hesitant. However, the conventional risk-evaluation approach cannot handle hesitant cognitive information. To effectively overcome the limitation of the conventional risk-evaluation approach—such as the inability to handle FI, intuitionistic FI, and Pythagorean FI of human cognitive information, ignoring the relative-weight considerations between evaluation criteria, and the inability to process hesitant information from experts—this paper introduced a novel PFS-based risk-ranking method to handle human cognitive information in MCDA problems. The proposed PFS-based risk-ranking method uses the PFS method to simultaneously process complete information, FI, intuitionistic FI, and Pythagorean FI. For the risk-assessment factor consideration, the proposed PFS-based risk-ranking method uses the CoCoSo approach to consider the relative weight between risk-assessment factors. In processing hesitant information, the maximum operator, average operator, and minimum operator of the hesitation-interval value are used to calculate the risk-failure item ranking of different operators.

The remainder of this paper is structured as follows. In Section 2, we briefly review and introduce the basic concepts and operation rules related to IFSs, PFSs, and the CoCoSo method. Section 3 discusses the PFS-based risk-ranking approach for handling human cognitive information in risk-assessment problems. In Section 4, a practical healthcare waste-hazards risk-ranking example is used to illustrate the calculation process of the proposed risk-ranking method based on PFS and verify the effectiveness and correctness of the proposed method. Section 5 summarizes the conclusions and provides suggestions for future research.

2. Preliminary

This section briefly reviews the related definition, basic concepts, and algorithms of IFSs, PFSs, and the CoCoSo method.

2.1. Intuitionistic Fuzzy Sets and Pythagorean Fuzzy Sets

Atanassov [12] first introduced IFSs, which use the MG and the NMG to describe fuzzy phenomena in the real world. In IFSs, the numerical sum of the restricted MG and the NMG must be less than or equal to 1. To overcome the limitation of IFSs, Yager [33] proposed PFSs to deal with human cognitive information more realistically and flexibly. In the PFSs, the sum of the squares of MG and NMG must be less than or equal to 1. The related definitions and operation of IFSs and the PFSs are briefly introduced as follows:

Definition 1

[34]. If the U is the domain of discourse, IFS I in U is represented by the following equation:

$$I=\left\{<u,{\mu }_I\left(u\right),{\upsilon }_I\left(u\right)>|u\in U\right\}$$

(1)

where ${\mu }_I\left(u\right):U\to \left[0,1\right]$, ${\upsilon }_I\left(u\right):U\to \left[0,1\right]$, and $0\le {\mu }_I\left(u\right)+{\upsilon }_I\left(u\right)\le 1$ must be held. ${\mu }_I\left(u\right)$ and ${\upsilon }_I\left(u\right)$ indicate the MG and the NMG of the element $u\in U$, respectively. The degree of hesitation ${\pi }_I\left(u\right)$ is defined as ${\pi }_I\left(u\right)=1-{\mu }_I\left(u\right)-{\upsilon }_I\left(u\right)$, ${\pi }_I\left(u\right)\in \left[0,1\right]$. When ${\pi }_I\left(u\right)=0$, IFS I degenerates into the original fuzzy set.

Definition 2

[35]. Let $A=\left({\mu }_1,{\upsilon }_1\right)$ and $B=\left({\mu }_2,{\upsilon }_2\right)$ be two intuitionistic fuzzy numbers (IFNs); then, the operations laws of IFNs are defined as follows:

$$A\oplus B=〈{\mu }_1+{\mu }_2-{\mu }_1{\mu }_2,{\upsilon }_1{\upsilon }_2〉$$

(2)

$$A\otimes B=\left({\mu }_1{\mu }_2,{\upsilon }_1{+\upsilon }_2{-\upsilon }_1{\upsilon }_2\right)$$

(3)

Definition 3

[36]. Let $A_i=\left({\mu }_{Ai},{\upsilon }_{Ai}\right)$ be a collection of the IFN ($i=1,2,\dots ,n$), and $w={\left(w_1,w_2,\dots ,w_n\right)}^T$ the weights vector of $A_i$, $w_i\in \left[0,1\right]$, and ${\sum }_{i=1}^n{w}_i=1$. Then, the intuitionistic fuzzy weighted arithmetic averaging (IFWAA) operator is expressed as the formula below:

$$IFWAA\left(A_i\right)=\left(1-{\prod }_{i=1}^n{\left(1-{\mu }_i\right)}^{w_i},{\prod }_{i=1}^n{{\nu }_i}^{w_i}\right)$$

(4)

The score function $S\left(A_i\right)$ of the IFN $A_i=\left({\mu }_{Ai},{\upsilon }_{Ai}\right)$ is expressed as the formula below:

$$S\left(A_i\right)={\mu }_{Ai}-{\upsilon }_{Ai}$$

(5)

Definition 4

[37,38]. If U is the domain of discourse, PFS P in U is represented by the following equation:

$$P=\left\{<u,{\mu }_P\left(u\right),{\upsilon }_P\left(u\right)>|u\in U\right\}$$

(6)

where ${\mu }_P\left(u\right):U\to \left[0,1\right]$, ${\upsilon }_P\left(u\right):U\to \left[0,1\right]$, and $0\le {\left({\mu }_P\left(x\right)\right)}^2+{\left({\upsilon }_P\left(x\right)\right)}^2\le 1$ must be held. ${\mu }_P\left(u\right)$ and ${\upsilon }_P\left(u\right)$ indicate the MG and the NMG of the element $u\in U$, respectively. The degree of hesitation ${\pi }_P\left(u\right)$ is defined as ${\pi }_P\left(u\right)=\sqrt{{1-\left({\mu }_P\left(u\right)\right)}^2-{\left({\upsilon }_P\left(u\right)\right)}^2}$, ${\pi }_P\left(u\right)\in \left[0,1\right]$.

In order to ensure a simple algorithm, the real number pair $P=\left({\mu }_P,{\upsilon }_P\right)$ in the PFS is used, called the Pythagorean fuzzy number (PFN). PFN $P_i=\left({\mu }_{Pi},{\upsilon }_{Pi}\right)$ satisfies ${0\le \mu }_{Pi}\le 1$, ${0\le \upsilon }_{Pi}\le 1$, and $0\le {\left({\mu }_{Pi}\right)}^2+{\left({\upsilon }_{Pi}\right)}^2\le 1$. It must be noted that the original fuzzy sets and IFSs are only a special case of PFS.

Definition 5

[35]. Let $C=\left({\mu }_1,{\upsilon }_1\right)$ and $D=\left({\mu }_2,{\upsilon }_2\right)$ be two PFNs; then, the operation laws of PFNs are defined as follows:

$$C\oplus D=\left(\sqrt{{\mu }_1^2+{\mu }_2^2-{\mu }_1^2{\mu }_2^2},{\upsilon }_1{\upsilon }_2\right)$$

(7)

$$C\otimes D=\left({\mu }_1{\mu }_2,\sqrt{{\upsilon }_1^2+{\upsilon }_2^2-{\upsilon }_1^2{\upsilon }_2^2}\right)$$

(8)

$$\lambda C=\left(\sqrt{1-{\left(1-{\mu }_1^2\right)}^{\lambda }},{\nu }_1^{\lambda }\right),\lambda >0$$

(9)

$$C^{\lambda }=\left({\mu }_1^{\lambda },\sqrt{1-{\left(1-{\nu }_1^2\right)}^{\lambda }}\right),\lambda >0$$

(10)

Definition 6

[39]. Let $C_i=\left({\mu }_{Ci},{\upsilon }_{Ci}\right)$ be a collection of the PFN ($i=1,2,\dots ,n$), and $w={\left(w_1,w_2,\dots ,w_n\right)}^T$ the weights vector of $C_i$, $w_i\in \left[0,1\right]$, and ${\sum }_{i=1}^n{w}_i=1$. Then, the Pythagorean fuzzy weighted arithmetic averaging (PFWAA) operator is expressed as the formula below:

$$PFWAA\left(C_i\right)=\left(\sqrt{{\sum }_{i=1}^n{w}_i{\mu }_{Ci}^2},\sqrt{1-{\sum }_{i=1}^n{w}_i\left(1-{\upsilon }_{Ci}^2\right)}\right)$$

(11)

Definition 7

[38,39]. The score function $S\left(C_i\right)$ and the accuracy function $G\left(C_i\right)$ of PFN $C_i=\left({\mu }_{Ci},{\upsilon }_{Ci}\right)$ is expressed as the formula below:

$$S\left(C_i\right)={\mu }_{C_i}^2-{\nu }_{C_i}^2,S\left(C_i\right)\in \left[-1,1\right]$$

(12)

$$G\left(C_i\right)={\mu }_{C_i}^2+{\nu }_{C_i}^2,G\left(C_i\right)\in \left[0,1\right]$$

(13)

If $C_1$ and $C_2$ are any of the two PFNs:

1.

If $S\left(C_1\right)>S\left(C_2\right)$, then $C_1>C_2$.

2.

If $S\left(C_1\right)<S\left(C_2\right)$, then $C_1<C_2$.

3.

If $S\left(C_1\right)=S\left(C_2\right)$, then

(a)

If $G\left(C_1\right)>G\left(C_2\right)$, then $C_1>C_2$.

(b)

If $G\left(C_1\right)<G\left(C_2\right)$, then $C_1<C_2$.

(c)

If $G\left(C_1\right)=G\left(C_2\right)$, then $C_1=C_2$.

2.2. CoCoSo Method

Yazdani et al. [19] first proposed the concept of the CoCoSo method to process MCDA problems. The basic principle of the CoCoSo method involves ranking potentially multiple alternatives using a combination of arithmetic weighting, geometric weighting, and three different aggregation strategies.

The detailed calculation steps and algorithm of the CoCoSo approach are expressed as follows [28,40]:

• Step 1. Construct an initial evaluation matrix ($x_{ij}$).

$x_{ij}$ is the evaluation information of alternative i and criterion j.

$$x_{ij}=\left[\begin{array}{ccc}{x}_{11}& x_{12}& \begin{array}{cc}\cdots & x_{1n}\end{array}\\ x_{21}& x_{22}& \begin{array}{cc}\cdots & x_{2n}\end{array}\\ \begin{array}{c}⋮\\ x_{m1}\end{array}& \begin{array}{c}⋮\\ x_{m2}\end{array}& \begin{array}{cc}\begin{array}{c}\ddots \\ \cdots \end{array}& \begin{array}{c}⋮\\ x_{mn}\end{array}\end{array}\end{array}\right],i=1,2,\dots ,m,j=1,2,\dots ,n$$

(14)

• Step 2. Calculate the normalized evaluation matrix ($y_{ij}$).

$$y_{ij}=\frac{x_{ij}-{min}_i{x}_{ij}}{{max}_i{x}_{ij}-{min}_i{x}_{ij}};\mathrm{for}\mathrm{the}\mathrm{performance}\mathrm{criterion}$$

(15)

$$y_{ij}=\frac{{max}_i{x}_{ij}-x_{ij}}{{max}_i{x}_{ij}-{min}_i{x}_{ij}};\mathrm{for}\mathrm{the}\mathrm{cost}\mathrm{criterion}$$

(16)

• Step 3. Calculate the arithmetic weighted sum of the comparability sequence ($S_i$) and the geometric weighted sum of the comparability sequence ($P_i$).

$$S_i={\sum }_{j=1}^n{w}_j{y}_{ij};i=1,2,\dots ,m;$$

(17)

$$P_i={\sum }_{j=1}^n{{(y}_{ij})}^{w_j};i=1,2,\dots ,m.$$

(18)

• Step 4. Use the three appraisal score strategies to calculate the relative weight of alternatives.

$$R_{ia}=\frac{P_i+S_i}{\sum _{i=1}^m\left(P_i+S_i\right)}$$

(19)

$$R_{ib}=\frac{S_i}{{min}_i{S}_i}+\frac{P_i}{{min}_i{P}_i}$$

(20)

$$R_{ic}=\frac{\xi S_i+{\left(1-\xi \right)P}_i}{{\xi ·max}_i{S}_i+{(1-\xi )·max}_i{P}_i},0\le \xi \le 1.$$

(21)

The $\xi$ value is decided by the decision-maker, usually set to $\xi =0.5$.

• Step 5. Determine the final ranking of possible alternatives.

The comprehensive evaluation importance index ($R_i$) values are considered to determine the final ranking of possible alternatives. Higher $R_i$ values represent the higher performance of the possible alternative.

$$R_i=\sqrt[3]{R_{ia}·R_{ib}·R_{ib}}+\frac{1}{3}\left(R_{ia}+R_{ib}+R_{ib}\right)$$

(22)

3. Proposed Novel Pythagorean Fuzzy Set–Based Risk-Ranking Method

Accurate risk assessment results use limited resources to prevent the occurrence of possible malfunctioning projects, ensuring that the product or system achieves the expected function. Risk-assessment problems are considered multi-expert and MCDA problems. The processing method of human cognitive information is a key problem in risk assessment, influencing the risk-ranking results. Due to its simple calculation, the RPN approach is the most widely used risk-assessment method. However, it can only handle complete information from experts and cannot handle FI, intuitionistic FI, and Pythagorean FI provided by experts. In addition, the RPN approach cannot handle hesitant information from experts and process the relative weight between possible evaluation criteria. These gaps can lead to incorrect risk-ranking results. To ensure correct and effective processing of risk ranking problems, this paper integrates PFSs and the CoCoSo method in handling human cognitive information. PFSs can process human cognitive information from a wide range of MCDA problems, unlike IFSs and typical fuzzy sets. The proposed novel PFS-based risk-ranking method uses the CoCoSo method to consider the relative weight of possible evaluation criteria. In dealing with hesitant information provided by experts, the proposed method uses the maximum operator, average operator, and minimum operator of the hesitation-interval value to perform calculations of different operators.

The proposed novel PFS-based risk-ranking method calculation has 7 steps; the flowchart is presented in Figure 1.

• Step 1. Determine the composition of the risk-evaluation team and confirm all possible failure items.

According to the expertise and experience of the experts, form a cross-disciplinary risk-evaluation team in which team members can discuss all possible failure items together.

• Step 2. Determine the risk levels of the three risk factors (S, O, and D) for each failed item.

Each expert determines the risk levels of the three risk factors (S, O, and D) for each failed item based on their past experience. The conversion between the risk levels, linguistic terms, and the PFN is shown in

Table 1. Linguistic terms of the PFN.

Levels	Linguistic Terms	PFN
s1	Extremely very low (EVL)	(0.05, 1.00)
s2	Extremely low (EL)	(0.15, 0.95)
s3	Very low (VL)	(0.25, 0.85)
s4	Low (L)	(0.35, 0.75)
s5	Medium (M)	(0.45, 0.65)
s6	Medium high (MH)	(0.55, 0.55)
s7	High (H)	(0.65, 0.45)
s8	Very high (VH)	(0.75, 0.35)
s9	Extremely high (EH)	(0.85, 0.25)
s10	Extremely very high (EVH)	(0.95, 0.15)

.

For example, in Table 1, an expert determines the risk level of the risk factors (S, O, and D) as s1, for which the corresponding linguistic term is extremely very low (EVL). Then the corresponding PFN is (0.05, 1.00).

• Step 3. Convert hesitant information provided by the experts.

To deal with the hesitation information that experts provide, use the maximum operator, average operator, and minimum operator of the hesitation-interval value to replace the original hesitant information.

• Step 4. Aggregate all information from different experts.

Use Equation (11) to aggregate all information from different experts.

• Step 5. Defuzzify and normalize the Pythagorean FI.

Use Equation (12) to defuzzify the initial PFS decision matrix. Then, use Equations (15) and (16) to normalize the PFS decision matrix.

• Step 6. Calculate the $R_{ia}$, $R_{ib}$, $R_{ic}$, and the $R_i$ values of different operators.

Based on the maximum operator, average operator, and minimum operator, use Equations (19)–(22) to calculate the $R_{ia}$, $R_{ib}$, $R_{ic}$, and the $R_i$ values for the different operators, respectively.

• Step 7. Determine the risk-ranking results of all possible failure items and provide a decision-making reference.

The $R_i$ values from large to small indicate the results of the risk ranking of all possible failure projects. This risk-ranking result provides a reference for decision-making.

Figure 1. The flowchart of the proposed novel PFS-based risk-ranking method.

Figure 1. The flowchart of the proposed novel PFS-based risk-ranking method.

4. Numerical Case Study

4.1. Case Description

This study uses the healthcare waste-hazards risk-assessment case (adapted from [41]) to illustrate the processing procedure of the proposed novel PFS-based risk-ranking approach and to verify the validity and rationality of the proposed approach for handling the risk-assessment issues. The healthcare waste-hazards risk-assessment case includes 15 types of medical waste from Sultan Qaboos University Hospital in Oman. The healthcare waste-hazards risk-assessment team comprises three experts (P1, P2, and P3). Each expert, based on Table 1 and their experience, determines the risk levels (s1, s2 to s10) of the three risk factors (S, O, and D) for each failed item (see

Table 2. The risk level of each failure item risk factor S, O, and D.

Item	Major Risks and Hazards Items in Medical Waste Management	Experts	S	O	D
1	Wrong garbage classification	P1	s4	s9	s3
		P2	s5	s10	s2
		P3	s5	s9	s3
2	Does not contain polluting waste	P1	s6	s5	s5
		P2	s6	s4	s5
		P3	s6	s4	s5
3	Failure to use personal protective equipment when handling infectious waste	P1	s6	s7	s8
		P2	s7	s7	s9
		P3	s7	s8	s8
4	Clutter and lack of safety cabinets	P1	s5	s4	s6
		P2	s5	s5	s4
		P3	s6	s5	s5
5	Incorrect disposal of chemical waste	P1	s7	s4	s5
		P2	s8	s3–s4	s5–s6
		P3	s7	s5	s4
6	Incorrect storage method of chemical raw materials	P1	s5	s6	s6
		P2	s4–s5	s6–s7	s5–s6
		P3	s4	s6	s5
7	No timely management mechanism for chemical spills	P1	s6	s4	s7
		P2	s6–s7	s3	s7–s8
		P3	s7	s3	s7
8	Reloading of syringes	P1	s8	s3	s7
		P2	s9	s4	s7
		P3	s9	s3	s6
9	Accidents cause bags to overflow and leak	P1	s6	s5	s4
		P2	s5	s5	s3
		P3	s5	s4	s3
10	Water pipe is rusted	P1	s4	s4	s2
		P2	s4	s3	s1
		P3	s4	s4	s1
11	Contaminated water from immunocompromised patients	P1	s9	s4	s2
		P2	s9	s3	s4
		P3	s10	s3	s3
12	Exposure to patients’ blood and body fluids causes an occupational hazard for healthcare worker	P1	s9	s7	s7
		P2	s9	s7	s6
		P3	s9	s6	s5
13	Autoclave leaks	P1	s8	s5	s7
		P2	s8	s6	s6
		P3	s8	s7	s6
14	The temperature of the incinerator does not meet the requirements	P1	s5	s5	s9
		P2	s5	s5-s6	s10
		P3	s6	s4	s9
15	Sharps box overfill practice	P1	s5	s4	s6
		P2	s5	s4	s7
		P3	s5	s3	s6

).

4.2. Solution of Typical RPN Method

The key advantage of the RPN method lies in its simple computation. Hence, it is currently the most widely applied risk-evaluation tool in different fields. The typical RPN approach takes the product of three possible risk factors—S, O, and D—as the basis for risk ranking all possible failure items.

Because the typical RPN method [42] cannot handle the hesitant information provided by Expert 2, this study used only the average risk level provided by Expert 1 and Expert 3 to calculate the aggregated S, O, and D values. Based on Table 2, the RPN value was calculated using the product of the three possible risk factors and then sorting all potential failure items, as shown in

Table 3. The RPN value of all potential failure items.

ID	S	O	D	RPN	Ranking
1	4.5	9	3	121.500	11
2	6	4.5	5	135.000	10
3	6.5	7.5	8	390.000	1
4	5.5	4.5	5.5	136.125	9
5	7	4.5	4.5	141.750	8
6	4.5	6	5.5	148.500	7
7	6.5	3.5	7	159.250	6
8	8.5	3	6.5	165.750	5
9	5.5	4.5	3.5	86.625	13
10	4	4	1.5	24.000	15
11	9.5	3.5	2.5	83.125	14
12	9	6.5	6	351.000	2
13	8	6	6.5	312.000	3
14	5.5	4.5	9	222.750	4
15	5	3.5	6	105.000	12

.

4.3. Solution of Typical IFS Method

Atanassov [12] introduced the IFS method, which uses MG and NMG to handle information problems of individuals expressing cognitive information. However, in this study, the typical IFS method could not handle the hesitant information provided by Expert 2. Based on Table 2, Equation (4) was used to aggregate information on the S, O, and D values provided by Expert 1 and Expert 3. Then, Equation (5) was used to calculate the $S\left(A_i\right)$ value and the ranking results for each failure item. The results are shown in

Table 4. The $S\left(A_i\right)$ value for each failure item.

ID	S	O	D	IFWAA Value	$$\mathit{S}\left({\mathit{A}}_{\mathit{i}}\right)$$ Value	Ranking
1	(0.402, 0.598)	(0.850, 0.150)	(0.250, 0.750)	(0.593, 0.407)	0.187	7
2	(0.550, 0.450)	(0.402, 0.598)	(0.450, 0.550)	(0.471, 0.529)	−0.058	11
3	(0.603, 0.397)	(0.704, 0.296)	(0.750, 0.250)	(0.692, 0.308)	0.383	2
4	(0.503, 0.497)	(0.402, 0.598)	(0.503, 0.497)	(0.471, 0.529)	−0.058	11
5	(0.650, 0.350)	(0.402, 0.598)	(0.402, 0.598)	(0.500, 0.500)	0.000	9
6	(0.402, 0.598)	(0.550, 0.450)	(0.503, 0.497)	(0.488, 0.512)	−0.023	10
7	(0.603, 0.397)	(0.302, 0.698)	(0.650, 0.350)	(0.541, 0.459)	0.081	8
8	(0.806, 0.194)	(0.250, 0.750)	(0.603, 0.397)	(0.614, 0.386)	0.227	6
9	(0.503, 0.497)	(0.402, 0.598)	(0.302, 0.698)	(0.408, 0.592)	−0.184	14
10	(0.350, 0.650)	(0.350, 0.650)	(0.101, 0.899)	(0.276, 0.724)	−0.448	15
11	(0.913, 0.087)	(0.302, 0.698)	(0.202, 0.798)	(0.636, 0.364)	0.272	5
12	(0.850, 0.150)	(0.603, 0.397)	(0.561, 0.439)	(0.703, 0.297)	0.407	1
13	(0.750, 0.250)	(0.561, 0.439)	(0.603, 0.397)	(0.648, 0.352)	0.296	3
14	(0.503, 0.497)	(0.402, 0.598)	(0.850, 0.150)	(0.645, 0.355)	0.291	4
15	(0.450, 0.550)	(0.302, 0.698)	(0.550, 0.450)	(0.443, 0.557)	−0.114	13

.

4.4. Solution of Typical PFS Method

The typical PFS method [33] uses MG, NMG, and hesitation degree to handle information problems of individuals expressing cognition information. The typical PFS method can handle comprehensive information considerations more than the fuzzy set and IFS methods. This method is closer to the human cognitive information expression mode. However, in this study, the PFS method could not handle the hesitant information provided by Expert 2. Based on Table 2, Equation (11) was used to aggregate information in the S, O, and D values provided by Expert 1 and Expert 3. Then, Equations (12) and (13) were employed to calculate the $S\left(C_i\right)$ value, $G\left(C_i\right)$ value, and the ranking results for each failure item. The results are shown in

Table 5. The $S\left(C_i\right)$ value, $G\left(C_i\right)$ value, and the ranking results for each failure item.

ID	S	O	D	PFWAA Value	$\mathit{S}\left({\mathit{C}}_{\mathit{i}}\right)$ Value	$\mathit{G}\left({\mathit{C}}_{\mathit{i}}\right)$ Value	Ranking
1	(0.403, 0.702)	(0.850, 0.250)	(0.250, 0.850)	(0.562, 0.653)	−0.110	0.742	7
2	(0.550, 0.550)	(0.403, 0.702)	(0.450, 0.650)	(0.472, 0.637)	−0.183	0.628	11
3	(0.602, 0.502)	(0.702, 0.403)	(0.750, 0.350)	(0.687, 0.423)	0.293	0.652	1
4	(0.502, 0.602)	(0.403, 0.702)	(0.502, 0.602)	(0.472, 0.637)	−0.183	0.628	11
5	(0.650, 0.450)	(0.403, 0.702)	(0.403, 0.702)	(0.499, 0.629)	−0.147	0.645	8
6	(0.403, 0.702)	(0.550, 0.550)	(0.502, 0.602)	(0.489, 0.621)	−0.147	0.625	9
7	(0.602, 0.502)	(0.304, 0.802)	(0.650, 0.450)	(0.541, 0.605)	−0.073	0.658	6
8	(0.802, 0.304)	(0.250, 0.850)	(0.602, 0.502)	(0.597, 0.597)	0.000	0.712	5
9	(0.502, 0.602)	(0.403, 0.702)	(0.304, 0.802)	(0.411, 0.707)	−0.330	0.668	14
10	(0.350, 0.750)	(0.350, 0.750)	(0.112, 0.975)	(0.293, 0.832)	−0.606	0.778	15
11	(0.901, 0.206)	(0.304, 0.802)	(0.206, 0.901)	(0.562, 0.707)	−0.183	0.815	10
12	(0.850, 0.250)	(0.602, 0.502)	(0.559, 0.559)	(0.683, 0.457)	0.257	0.675	2
13	(0.750, 0.350)	(0.559, 0.559)	(0.602, 0.502)	(0.642, 0.479)	0.183	0.642	3
14	(0.502, 0.602)	(0.403, 0.702)	(0.850, 0.250)	(0.616, 0.553)	0.073	0.685	4
15	(0.450, 0.650)	(0.304, 0.802)	(0.550, 0.550)	(0.446, 0.675)	−0.257	0.655	13

.

4.5. Solution of the Typical CoCoSo Method

Yazdani et al. [19] first proposed the CoCoSo method, which combines the different weight-calculation methods to solve the MCDA problem in different domains. However, in this study, the CoCoSo approach could not handle the hesitant information provided by Expert 2. Based on Table 2, Equation (11) was used to aggregate the MG information on the S, O, and D values provided by Expert 1 and Expert 3. The results are presented in Table 5.

Based on Table 5, the original S, O, and D values were normalized using Equations (15) and (16). Then, Equations (17) and (18) were used to compute the $S_i$ and the $P_i$ values based on the normalized S, O, and D values. The results are presented in

Table 6. The $S_i$ and the $P_i$ values of each failure item.

ID	S	O	D	${\mathit{S}}_{\mathit{i}}$ Value	${\mathit{P}}_{\mathit{i}}$ Value
1	0.096	1.000	0.187	0.428	1.408
2	0.363	0.255	0.458	0.359	1.469
3	0.457	0.753	0.865	0.692	1.825
4	0.277	0.255	0.529	0.354	1.452
5	0.544	0.255	0.395	0.398	1.514
6	0.096	0.500	0.529	0.375	1.429
7	0.457	0.090	0.729	0.425	1.469
8	0.819	0.000	0.664	0.494	1.254
9	0.277	0.255	0.261	0.264	1.334
10	0.000	0.167	0.000	0.056	0.382
11	1.000	0.090	0.128	0.406	1.354
12	0.907	0.587	0.606	0.700	1.838
13	0.725	0.515	0.664	0.635	1.784
14	0.277	0.255	1.000	0.511	1.585
15	0.181	0.090	0.594	0.288	1.286

.

Based on Table 6, Equations (19)–(22) were used to calculate the $R_{ia}$, $R_{ib}$, $R_{ic}$, and the $R_i$ values and the ranking results for each possible failure item, as expressed in

Table 7. The $R_i$ value and the ranking results for each failure item.

ID	${\mathit{R}}_{\mathit{i}\mathit{a}}$	${\mathit{R}}_{\mathit{i}\mathit{b}}$	${\mathit{R}}_{\mathit{i}\mathit{c}}$	${\mathit{R}}_{\mathit{i}}$	Ranking
1	0.066	11.391	0.723	4.877	7
2	0.066	10.306	0.720	4.485	11
3	0.091	17.232	0.992	7.262	2
4	0.065	10.172	0.712	4.428	12
5	0.069	11.132	0.753	4.818	8
6	0.065	10.499	0.711	4.544	10
7	0.068	11.509	0.746	4.945	6
8	0.063	12.184	0.689	5.120	5
9	0.058	8.251	0.630	3.648	14
10	0.016	2.000	0.172	0.905	15
11	0.063	10.856	0.693	4.652	9
12	0.091	17.414	1.000	7.336	1
13	0.087	16.102	0.953	6.816	3
14	0.075	13.344	0.826	5.689	4
15	0.057	8.562	0.620	3.750	13

.

4.6. Solution of the Proposed Novel Pythagorean Fuzzy Set–Based Risk-Ranking Method

In risk assessment problems, how human cognitive information is processed will directly affect the possible risk-ranking results of risk-failure projects. Therefore, this study integrated the PFSs and the CoCoSo method to handle human cognitive information in MCDA problems. In dealing with the hesitant cognitive information proposed by experts, this method uses the maximum operator, average operator, and minimum operator of the hesitant cognitive information to calculate the corresponding PFS information. Based on Table 2, Equation (11) was used to aggregate all expert-provided (Expert 1, Expert 2, and Expert 3) information in the S, O, and D values. Then, Equation (12) was used to defuzzify the Pythagorean FI.

Based on the defuzzified Pythagorean FI provided by all experts, Equations (15) and (16) were used to normalize the original S, O, and D values. The results are presented in

Table 8. Normalizing the defuzzified Pythagorean FI.

ID	Operator	S	O	D	${\mathit{S}}_{\mathit{i}}$ Value	${\mathit{P}}_{\mathit{i}}$ Value
1		0.125	1.000	0.133	0.419	1.394
2		0.375	0.167	0.436	0.326	1.408
3		0.500	0.667	0.870	0.679	1.818
4		0.250	0.222	0.436	0.303	1.383
5	Maximum operator	0.625	0.167	0.436	0.409	1.500
	Average operator	0.625	0.139	0.415	0.393	1.469
	Minimum operator	0.625	0.111	0.393	0.376	1.434
6	Maximum operator	0.125	0.500	0.523	0.383	1.456
	Average operator	0.094	0.472	0.501	0.356	1.406
	Minimum operator	0.062	0.444	0.480	0.329	1.347
7	Maximum operator	0.500	0.000	0.740	0.413	1.177
	Average operator	0.469	0.000	0.718	0.396	1.160
	Minimum operator	0.438	0.000	0.697	0.378	1.141
8		0.875	0.000	0.653	0.509	1.265
9		0.250	0.222	0.220	0.231	1.275
10		0.000	0.056	0.000	0.019	0.265
11		1.000	0.000	0.176	0.392	1.082
12		0.938	0.556	0.567	0.687	1.822
13		0.750	0.444	0.610	0.601	1.747
14	Maximum operator	0.250	0.278	1.000	0.509	1.583
	Averaging operator	0.250	0.250	1.000	0.500	1.567
	Minimum operator	0.250	0.222	1.000	0.491	1.550
15		0.188	0.056	0.610	0.284	1.249

. Then, Equations (17) and (18) were used to compute the $S_i$ and the $P_i$ values, as shown in Table 8.

Based on Table 8, Equations (19)–(22) were used to calculate the $R_{ia}$, $R_{ib}$, $R_{ic}$, and $R_i$ values of different operators. Then, the $R_i$ values from large to small were ranked, which are the ranking results for each failure item. The results are presented in

Table 9. The ranking results of different operators using the proposed method.

ID	Maximum Operator					Averaging Operator					Minimum Operator
	${\mathit{R}}_{\mathit{i}\mathit{a}}$	${\mathit{R}}_{\mathit{i}\mathit{b}}$	${\mathit{R}}_{\mathit{i}\mathit{c}}$	${\mathit{R}}_{\mathit{i}}$	Ranking	${\mathit{R}}_{\mathit{i}\mathit{a}}$	${\mathit{R}}_{\mathit{i}\mathit{b}}$	${\mathit{R}}_{\mathit{i}\mathit{c}}$	${\mathit{R}}_{\mathit{i}}$	Ranking	${\mathit{R}}_{\mathit{i}\mathit{a}}$	${\mathit{R}}_{\mathit{i}\mathit{b}}$	${\mathit{R}}_{\mathit{i}\mathit{c}}$	${\mathit{R}}_{\mathit{i}}$	Ranking
1	0.068	27.913	0.723	10.680	7	0.069	27.913	0.723	10.683	6	0.069	27.913	0.723	10.686	6
2	0.065	22.926	0.691	8.905	11	0.066	22.926	0.691	8.907	11	0.066	22.926	0.691	8.910	10
3	0.094	43.530	0.995	16.470	2	0.095	43.530	0.995	16.473	2	0.095	43.530	0.995	16.478	2
4	0.063	21.582	0.672	8.412	12	0.064	21.582	0.672	8.414	12	0.064	21.582	0.672	8.417	12
5	0.072	27.777	0.761	10.686	6	0.071	26.768	0.742	10.313	7	0.069	25.746	0.722	9.932	7
6	0.069	26.169	0.733	10.089	9	0.067	24.528	0.702	9.480	10	0.064	22.853	0.668	8.854	11
7	0.060	26.769	0.634	10.159	8	0.059	25.748	0.620	9.789	8	0.058	24.726	0.605	9.417	9
8	0.067	32.288	0.707	12.171	5	0.067	32.288	0.707	12.174	5	0.068	32.288	0.707	12.177	5
9	0.057	17.274	0.600	6.814	14	0.057	17.274	0.600	6.817	14	0.057	17.274	0.600	6.819	14
10	0.011	2.000	0.113	0.842	15	0.011	2.000	0.113	0.842	15	0.011	2.000	0.113	0.842	15
11	0.055	25.265	0.588	9.573	10	0.056	25.265	0.588	9.576	9	0.056	25.265	0.588	9.578	8
12	0.094	43.960	1.000	16.625	1	0.095	43.960	1.000	16.629	1	0.096	43.960	1.000	16.633	1
13	0.088	39.081	0.936	14.847	3	0.089	39.081	0.936	14.851	3	0.090	39.081	0.936	14.854	3
14	0.079	33.482	0.834	12.765	4	0.078	32.923	0.824	12.560	4	0.078	32.359	0.813	12.354	4
15	0.058	20.075	0.611	7.806	13	0.058	20.075	0.611	7.808	13	0.059	20.075	0.611	7.810	13

.

4.7. Analysis and Discussion of Calculation Results

To verify the validity and rationality of the proposed PFS-based risk-ranking approach, this paper used the healthcare waste-hazards risk-assessment case to test and verify the results of different algorithm approaches. The five different calculation methods used in this study included the typical RPN method [42], the typical IFS method [12], the typical PFS method [33], the CoCoSo method [19], and the proposed PFS-based risk-ranking method. Noteworthily, the same data (Table 2) were used among the different calculation methods. The results of risk-ranking after their respective calculations are shown in

Table 10. The possible failure items ranking results using different algorithm approaches.

ID	Typical RPN Method [42]		Typical IFS Method [12]		Typical PFS Method [33]			Typical CoCoSo Method [19]		Proposed Method
										Maximum Operator		Averaging Operator		Minimum Operator
	RPN Value	Ranking	$\mathit{S}\left({\mathit{A}}_{\mathit{i}}\right)$ Value	Ranking	$\mathit{S}\left({\mathit{C}}_{\mathit{i}}\right)$ Value	$\mathit{G}\left({\mathit{C}}_{\mathit{i}}\right)$ Value	Ranking	${\mathit{R}}_{\mathit{i}}$ Value	Ranking	${\mathit{R}}_{\mathit{i}}$ Value	Ranking	${\mathit{R}}_{\mathit{i}}$ Value	Ranking	${\mathit{R}}_{\mathit{i}}$ Value	Ranking
1	121.500	11	0.187	7	−0.110	0.742	7	4.877	7	10.680	7	10.683	6	10.686	6
2	135.000	10	−0.058	11	−0.183	0.628	11	4.485	11	8.905	11	8.907	11	8.910	10
3	390.000	1	0.383	2	0.293	0.652	1	7.262	2	16.470	2	16.473	2	16.478	2
4	136.125	9	−0.058	11	−0.183	0.628	11	4.428	12	8.412	12	8.414	12	8.417	12
5	141.750	8	0.000	9	−0.147	0.645	8	4.818	8	10.686	6	10.313	7	9.932	7
6	148.500	7	−0.023	10	−0.147	0.625	9	4.544	10	10.089	9	9.480	10	8.854	11
7	159.250	6	0.081	8	−0.073	0.658	6	4.945	6	10.159	8	9.789	8	9.417	9
8	165.750	5	0.227	6	0.000	0.712	5	5.120	5	12.171	5	12.174	5	12.177	5
9	86.625	13	−0.184	14	−0.330	0.668	14	3.648	14	6.814	14	6.817	14	6.819	14
10	24.000	15	−0.448	15	−0.606	0.778	15	0.905	15	0.842	15	0.842	15	0.842	15
11	83.125	14	0.272	5	−0.183	0.815	10	4.652	9	9.573	10	9.576	9	9.578	8
12	351.000	2	0.407	1	0.257	0.675	2	7.336	1	16.625	1	16.629	1	16.633	1
13	312.000	3	0.296	3	0.183	0.642	3	6.816	3	14.847	3	14.851	3	14.854	3
14	222.750	4	0.291	4	0.073	0.685	4	5.689	4	12.765	4	12.560	4	12.354	4
15	105.000	12	−0.114	13	−0.257	0.655	13	3.750	13	7.806	13	7.808	13	7.810	13

. The main differences in the different calculation methods considering information patterns and the relative weights of risk-assessment factors are shown in

Table 11. The main differences in different calculation methods considering information pattern.

Calculation Methods	The Information Pattern that Can Be Processed				Consideration of Hesitant Cognitive Information	Consideration of Relative Weights of Risk Assessment Factors
	Complete Information	FI	Intuitionistic FI	Pythagorean FI
Typical RPN method [42]	Yes	No	No	No	No	No
Typical IFS method [12]	Yes	Yes	Yes	No	No	No
Typical PFS method [33]	Yes	Yes	Yes	Yes	No	No
CoCoSo method [19]	Yes	No	No	No	No	Yes
Proposed method	Yes	Yes	Yes	Yes	Yes	Yes

.

According to the evaluation results in Table 10 and Table 11, the proposed novel PFS-based risk-ranking method has several advantages. First, the information-pattern processing: Both the typical RPN and the CoCoSo methods can handle only information patterns of complete information and cannot handle those of FI, intuitionistic FI, and the Pythagorean FI. On the other hand, the IFS method can process information patterns of complete information, FI, and intuitionistic FI. However, it cannot handle the information patterns of the Pythagorean FI. Both the typical PFS method and the proposed PFS-based risk-ranking approach can simultaneously handle information patterns of complete information, FI, intuitionistic FI, and Pythagorean FI. Therefore, the evaluation results indicate that the proposed PFS-based risk-ranking approach can more fully and accurately express human cognitive information in the real world.

The second advantage of the proposed novel PFS-based risk-ranking method is the hesitant cognitive-information processing. The typical RPN, typical IFS, typical PFS, and the CoCoSo calculation methods cannot process hesitant cognitive information provided by experts, which may result in biased evaluation results. On the other hand, the proposed novel Pythagorean fuzzy set–based risk-ranking method uses the maximum operator, average operator, and minimum operator to handle hesitant cognitive information provided by experts and can fully grasp all evaluation information.

The third advantage of the proposed novel PFS-based risk-ranking method lies in its capability of considering the relative weights of risk-assessment factors. The RPN, IFS, and PFS calculation methods do not consider the relative weights of risk-assessment factors, which may cause biased evaluation results. Both the CoCoSo approach and the proposed novel PFS-based risk-ranking method use the three different appraisal-score strategies to calculate the relative weight of risk-assessment factors, showing the difference in the importance of risk-assessment factors.

5. Conclusions

How human cognitive information is processed is a key issue in risk assessment. With the advent of artificial intelligence, processing human cognitive information has become more important because it will directly affect the results of decision-making evaluations. Because the typical RPN method is relatively simpler to calculate than other risk-assessment methods, it is currently the most widely used risk-assessment tool. However, the typical RPN method is also accompanied by many limitations, such as the inability to handle FI, intuitionistic FI, and Pythagorean FI of human cognition. Further, the RPN method cannot process the relative weights between evaluation criteria, which may lead to biased evaluation results.

Therefore, to strengthen the ability of risk assessments, this paper integrates the PFSs and the CoCoSo method to process the risk-ranking problems of potential failure items. PFSs provide a wider solution space than the original fuzzy set and IFSs and can comprehensively deal with all kinds of uncertain information in real life. The CoCoSo method can consider different weight-calculation approaches to handle the relative weights between different risk-assessment factors. In dealing with hesitation information provided by experts, the maximum operator, average operator, and minimum operator are applied to compute the risk-ranking results of all possible failure projects. This risk-ranking result provides a reference for decision-making.

The proposed novel Pythagorean fuzzy set–based risk-ranking method contains the following advantages:

(1)

The proposed method can simultaneously process complete information, FI, intuitionistic FI, and Pythagorean FI.

(2)

The proposed method can handle hesitant cognitive information.

(3)

The proposed method takes into account the relative weights of risk-assessment factors.

Although the proposed novel Pythagorean fuzzy set-based risk-ranking method can effectively handle human cognitive information in risk-assessment problems, however, the proposed method still has the limitation that it cannot handle the incomplete information provided by experts. Future research directions can expand upon the discussion of how to deal with the incomplete information provided by experts. Subsequent researchers can extend the proposed method to deal with decision-making issues in different fields, such as green energy planning, investment selection, performance evaluation, reliability evaluation, and supplier selection. They can also use different algorithms to calculate the subjective and objective weights between evaluation criteria and explore the influence of different combinations of subjective and objective weights on the evaluation results.