A novel risk assessment model based on failure mode and effect analysis and probabilistic linguistic ELECTRE II method

Abstract

Failure mode and effect analysis is a powerful risk analysis tool in engineering management. When properly conducted, FMEA can make a huge contribution to reduce costs. The traditional FMEA ranks the failure modes on the basis of Risk Priority Number, which is defined as the multiplication of three risk factors. However, this method has always been criticized for it can’t handle the situation where the information given is uncertain or ambiguous. In order to extend the application of FMEA under the fuzzy environment, in this paper, we proposed a novel risk assessment model known as probabilistic linguistic ELECTRE II method to rank failure modes based on FMEA. To realize this goal, probabilistic linguistic term sets (PLTSs) that consider both the hesitant information and probabilistic information are introduced to depict decision maker’s cognitive information. To better use the PLTSs in the decision-making process, some important information measures are defined, and a method to obtain the combined weight based on entropy weight of PLTSs is also proposed. Subsequently, we establish a score-deviation based PLTS-ELECTRE II model to study FMEA as a multi-criteria group decision-making problem. Finally, we successfully apply this model in a nuclear reheat valve system and the effectiveness of the proposed method is verified by sensitivity analysis and comparative analysis.

Keywords

Risk management outranking probabilistic linguistic term set combined weight hybrid distance measure

1 Introduction

Failure Mode and Effect Analysis (FMEA) is an engineering technique used to define, identify, and eliminate known and/or potential failures, problems, errors, and so on [1, 2]. It’s a safe and specialized tool that has played an important role in numerous professional fields, such as reliable management [3], quality management [4], risk management [5] and so on. When properly performed, FMEA can make a huge contribution to reduce costs. To conduct an FMEA, a cross-functional team of decision-makers (DMs) must be established initially. As brainstorming or some other analytical method (such as AHP) applied, the team determine all possible FMs, and investigate the consequence of component failing through a specified FM. Several risk factors of FMs are taken into account: the probability of occurrence (O), the severity of the failure (S) and the probability of not detecting the failure (D). Certain values are given by DMs to evaluate the weight of risk factors and the performance of FM. After these preparations, the Risk Priority Number (RPN), which is calculated by multiplying the values of risk factors is introduced to determine the risk prioritization of all FMs. However, limitations and shortcomings are obvious and summarized as follows: (i) It is hard for DMs to give crisp numbers to determine the value of risk factors when FMs are evaluated in a complex system. (ii) Experts evaluate the relative performance of criteria by using their experience and professional knowledge, which presents the challenge of scientificity and objectivity in weight determination. (iii) The RPN calculating function is always being doubted for lacking complete scientific explanation about its derivation. Therefore, it is necessary to improve the traditional FMEA method, expand the application of FMEA in fuzzy environment, and make it more scientific and practical.

As FMEA can be treated as a multi-criteria decision making (MCDM) problem, many researchers use MCDM techniques corporated with fuzzy theory to determine the risk priority of FMs [6]. Carpitella et al. [7] established an FTOPSIS-based approach to prioritize FMs in street cleaning vehicles. Mohsen and Fereshteh [8] proposed an extended VIKOR method with Z-number to evaluate the risk of potential FMs in Geothermal Power Plant. Hajiagha et al. [9] solved a real-world problem of ranking causes of delay in Tehran metro system by VIKOR method, where the uncertainty is captured by Fuzzy Belief Structures. Wang et al. [10] proposed a new FMEA model that integrates COPRAS and ANP methods to rank the risk of hospital service FMs under interval-valued intuitionistic fuzzy context. Huang et al. [11] employed an improved TODIM method with linguistic distribution assessments to determine the risk priority of failure modes in a grinding wheel system. Certa et al. [12] proposed an ELECTRE Tri-based approach for the classification of FMs. The literature review reveals the following research gaps. First, although there are many kinds of research on FMEA under the fuzzy environment, these studies only consider one aspect of uncertain information, and seldom consider the hesitant information and probability information provided by DMs at the same time. Second, though the outranking method is an efficient MCDM technique to determine which FM is preferable, incomparable or indifferent [13], there is a limited study to extend the outranking method on FMEA under complex fuzzy environment.

In practical FMEA problem, due to its complexity and uncertainty, DMs like to express their opinions with linguistic terms. However, due to lack of sufficient relevant knowledge or different professional background, DMs may not able to give accurate opinions but hesitate between different linguistic terms. Rodríguez et al. [14, 15] proposed the concept of hesitant fuzzy linguistic term sets (HFLTSs), which is able to capture the hesitate information of DMs. However, the HFLTSs unable to describe the probability information in the process of opinion aggregation, and the probabilistic linguistic term set is suitable to solve this problem. In addition, compared with other outranking methods, the ELECTRE II method [16] is one of the best known and most widely used ranking method, it ranks the alternative from the best option to the worst option, which is especially suitable for sorting the risk of FMs.

Based on the above discussion, in this paper, we propose a novel FMEA model, extending the ELECTRE II method under a probabilistic linguistic environment to analyses the risk prioritization in FMEA problem. The innovations and contributions of this paper are summarized as follows:

(1) We employe PLTSs, which can avoid information loss or distortion, depict both the hesitate information and probability information given by DMs in FMEA problem. It is more flexible for DMs to express their opinions in linguistic terms.

(2) To better use the PLTSs in the decision-making process, we defined some important measures for PLTSs, and a method to obtain the combined weight based on entropy weight of PLTSs is also been proposed.

(3) We establish a score-deviation based PLTS-ELECTRE II model to study FMEA as a multi-criteria group decision-making problem under a complex fuzzy environment.

(4) We successfully apply this model in a nuclear reheat valve system and the effectiveness of the proposed method is verified by sensitivity analysis and comparative analysis.

The remainder of this paper is organized as follows: In section 2, some basic definitions and operation laws related to PLTS are introduced. New measures for PLTS are defined in section 3, containing new ordered relationships, hybrid PLTS-Hellinger distance measure and entropy weight for PLTSs. In section 4, a novel risk priority model of extended ELECTRE II method based on PLTS for FMEA is proposed. In section 5, a case study is presented. Section 6 presents a sensitivity analysis and comparative analysis. And section 7 makes a summary.

2 Preliminaries

In this section, some basic concepts and definitions are introduced. They will be utilized in the later analysis.

2.1 Linguistic term set and hesitant fuzzy linguistic term set

Xu [17, 18] proposed a finite and totally ordered discrete linguistic term set (LTS) S ={ s_t|t = - τ, . . . -1, 0, 1, . . . τ }, where the s_t expresses a possible value for a linguistic variable, such as “very low”, “high”. Usually, it is required that: (1) The set is ordered: s_α > s_β if and only if α > β; (2) there is a negation operator: neg (s_α) = s_-α.

Definition 1. [19] let S = {s_t|t = - τ, . . . -1, 0, 1, . . . τ} be a linguistic term set, a hesitant fuzzy linguistic term set (HFLTS), H_S, is an ordered finite subset of the consecutive linguistic terms of S.

Definition 2. [20] Let x_i ∈ X (i = 1, 2, . . . N) be fixed and S ={ s_t|t = - τ, . . . -1, 0, 1, . . . τ } be an LTS, an HFLTS on X, H_S, is in mathematical terms of H_S ={ < x_i, h_S (x_i) > |x_i ∈ X }, where h_S (x_i) is a set of some values in S. h_S (x_i) is expressed as h_S (x_i) ={ s_{φ
_l} (x_i) |s_{φ
_l} (x_i) ∈ S, l = 1, 2, . . . , # L }, # L is the number of linguistic terms in h_S (x_i). h_S (x_i) denotes the possible degree of the linguistic variable x_i to S. For convenience, h_S (x_i) is called the hesitant fuzzy linguistic element (HFLE).

For the convenience of operation, we need to transform the linguistic term to a definite semantic value [21]. A simple and commonly used transformation method is to divide the evaluation scale of the linguistic information on average.

Definition 3. [22] Let S = {s_t|t = - τ, . . . -1, 0, 1, . . . τ} be an LTS, h_S ={ s_t|t ∈ [- τ, τ] } be an HFLE, and h_γ ={ γ|γ ∈ [0, 1] } be a hesitant fuzzy element (HFE). Then the linguistic variable s_t that expresses the equivalent information to the membership degree γ is obtained with the following function g: $g : [- τ, τ] \to [0, 1], g (s_{t}) = \frac{t}{2 τ} + \frac{1}{2} = γ$ (1)

$\begin{matrix} g : [- τ, τ] \to [0, 1], g (h_{S}) \\ = {g (s_{t}) = \frac{t}{2 τ} + \frac{1}{2} | t \in [- τ, τ]} = h_{γ} \end{matrix}$ (2)

In addition, the membership degree γ that expresses the equivalent information to the linguistic variable s_t is obtained with the following function g^-1: $g^{- 1} : [0, 1] \to [- τ, τ], g^{- 1} (γ) = s_{(2 γ - 1) τ} = s_{t}$ (3)

$\begin{matrix} g^{- 1} : [0, 1] \to [- τ, τ], g^{- 1} (h_{γ}) \\ = {g^{- 1} (γ) = s_{(2 γ - 1) τ} | γ \in [0, 1]} = h_{S} \end{matrix}$ (4)

Definition 4. [23] Let the α and β be any two arbitrary HFEs, Hamming distance measures for α and β is defined as: $d_{1} (α, β) = \frac{1}{l} \sum_{i = 1}^{l} | α_{ρ (i)} - β_{ρ (i)} |$ (5) where l is the number of elements in these two HFEs, If l_α ≠ l_β, we extend the shorter one by adding the same value several times until the number of elements in α and β is equivalent. α_ρ(i) (i = 1, 2, . . . , n) is the i - th smallest value in α and β_ρ(i) (i = 1, 2, . . . , n) is the i - th smallest value in β.

2.2 Probabilistic linguistic term set

Pang [24] defined PLTS by adding probabilistic information to HFLTS.

Definition 5. Let S ={ s_t|t = - τ, . . . -1, 0, 1, . . . τ } be an LTS, a PLTS is defined as: $\begin{matrix} L (p) = & {L^{(k)} (p^{(k)}) | L^{(k)} \in S, p^{(k)} ⩾ 0, \\ k = 1, 2, . . ., # L (p), \sum_{k = 1}^{# L (p)} p^{(k)} ⩽ 1} \end{matrix}$ where L^(k) (p^(k)) is the linguistic term s_{t
^k} associated with the probability p^(k), named as probabilistic linguistic element (PLE). # L (p) is the number of all different linguistic terms in L (p).

If the probabilities of all possible values in a PLTS can’t be explicitly provided, then these values are considered to have the same probability. In this case, PLTS degenerate to HFLE [25]. We assume that each value in HFLE has the same probability, then generate HFLE to PLTS by: $p^{(k)} = 1 / # L$ (6)

In this case, the sum of probabilities of all values in a PLTS equals 1.

In real-world problems, probabilistic information exists everywhere, it contains important meaning and can’t be ignored. In the example above, h_s (x₁) ={ s_-2, s₀, s₁ } means a DM is not sure to give his/her assessment by which linguistic term from these three elements. And L₁ (p) ={ s_-2 (1/3) , s₀ (1/3) , s₁ (1/3) } collects the very probability of each value being selected by this DM as his/her assessment. Compared to HFLE, PLTS not only collects various aspects of probabilistic information and show it in an explicit style but also keeps the original linguistic information without any distortion.

Aggregation operator for PLTSs is provided in Ref [24]:

Definition 6. Let $L_{i} (p) = {L_{i}^{(k)} (p_{i}^{(k)}) | k = 1, 2, . . ., # L_{i} (p)} (i = 1, 2, . . ., n)$ be n PLTSs, where $L_{i}^{(k)}$ and $p_{i}^{(k)}$ are the k - th linguistic term and its probability respectively in L_i (p). Then:

$\begin{matrix} {PLWA}_{w} (L_{1} (P), L_{2} (P), . . ., L_{n} (P)) \\ = w_{1} L_{1} (p) \oplus w_{2} L_{2} (p) \oplus . . . \oplus w_{n} L_{n} (p) \\ = ⋃_{L_{1}^{(k)} \in L_{1} (p)} {w_{1} p_{1}^{(k)} L_{1}^{(k)}} \oplus ⋃_{L_{2}^{(k)} \in L_{2} (p)} {w_{2} p_{2}^{(k)} L_{2}^{(k)}} \\ \oplus . . . \oplus ⋃_{L_{n}^{(k)} \in L_{n} (p)} {w_{n} p_{n}^{(k)} L_{n}^{(k)}} \end{matrix}$ (7) is called the probabilistic linguistic weighted averaging (PLWA) operator, where w = (w₁, w₂, . . . , w_n) ^T is the weight vector of L_i (p) (i = 1, 2, . . . , n), w_i ⩾ 0, i = 1, 2, . . . , n and $\sum_{i = 1}^{n} w_{i} = 1$ .

2.3 Hellinger distance measure

Hellinger distance is used to measure the difference between two continuous or discrete probability distributions. Compared with other probability distance measures, such as Kullback-Leibler (KL) divergence, the advantages of Hellinger distance are that the HD (f, g) satisfies the property of symmetry and bounded, that is 0 ⩽ HD (f, g) ⩽1.

Definition 7. [26, 27] Let P and Q be two probability density functions. The Hellinger distance between P and Q in the discrete case is given by $HD (P, Q) = \sqrt{\frac{1}{2} \sum_{i = 1}^{n} {(\sqrt{p (x_{i})} - \sqrt{q (x_{i})})}^{2}}$ (8)

3 Some new measures for PLTSs

Since the concept of PLTS is defined, a lot of work has been done to study its characteristics [28 –35]. Gou [36] redefined some more logical operational laws for PLTSs based on equivalent transformation functions. Wu [37] presented new operations of PLTSs based on the adjusted PLTSs and the semantics of linguistic terms. Bai [38] proposed a possibility degree formula for PLTSs comparison. Liu [39] proposed the general operational laws for PLTSs by Archimedean t-conorm and t-norm and Linguistic scale functions. To better use the PLTSs in the FMEA process, we have improved the previous work and put forward our own method. We use the linguistic transformation function g to redefine the score and deviation degree of PLTS, making the comparison result more accurate and clear. By merging the Hamming distance and Hellinger distance, we define a hybrid distance for PLTSs, which calculate the difference of probability information and hesitation information separately. A method to obtain combined weight based on entropy weight of PLTSs is also proposed. The adjusted weight will make the ranking result more accurate.

3.1 A new ordered relationship for PLTSs

Pang [24] defined the score and deviation degree of L (p) with linguistic variables, we extend his definition by introducing function g. In this way, the calculation result of the original function is quantified and a crisp number is obtained.

Definition 8. Let L (p) ={ L^(k) (p^(k)) |k = 1, 2, . . . , # L (p) } be a PLTS, by using the function g, the score of L (p) is calculated as: $E (L (p)) = \sum_{k = 1}^{# L (p)} g (L^{(k)}) p^{(k)} / \sum_{k = 1}^{# L (p)} p^{(k)}$ (9)

Definition 9. Let L (p) ={ L^(k) (p^(k)) |k = 1, 2, . . . , # L (p) } be a PLTS, the score of L (p) is E (L (p)). Then the deviation degree of L (p) is:

$\begin{matrix} σ (L (p)) = & {(\sum_{k = 1}^{# L (p)} {(p^{(k)} (g (L^{(k)}) - E (L (p))))}^{2})}^{1 / 2} / \\ \sum_{k = 1}^{# L (p)} p^{(k)} \end{matrix}$ (10)

Given two PLTSs L₁ (p) and L₂ (p):

If E (L₁ (p)) > E (L₂ (p)), then L₁ (p)> L₂ (p) ;

If E (L₁ (p)) < E (L₂ (p)), then L₁ (p)< L₂ (p) ;

If E (L₁ (p)) = E (L₂ (p)), then

If σ (L₁ (p)) > σ (L₂ (p)), then L₁ (p)< L₂ (p) ;

If σ (L₁ (p)) < σ (L₂ (p)), then L₁ (p)> L₂ (p) ;

If σ (L₁ (p)) = σ (L₂ (p)), then L₁ (p) = L₂ (p).

3.2 The hybrid distance for PLTSs

A PLTS is composed of several linguistic terms and their respective probability. Therefore, the distance between any two PLTSs should contain both hesitant information and probabilistic information. In the following section, we will introduce two hybrid distance measure for any two PLTSs.

Let S ={ s_t|t = - τ, . . . -1, 0, 1, . . . τ }, L^a (p) and L^b (p) be any two arbitrary PLTSs. The hybrid Hamming-Hellinger distance between L^a (p) and L^b (p) is defined as:

$\begin{array}{l} d_{H H} (L^{a} (p), L^{b} (p)) \\ = θ \frac{1}{# l} \sum_{i = 1}^{# l} |g (s_{ρ (i)}^{a}) - g (s_{ρ (i)}^{b})| \\ + (1 - θ) \sqrt{\frac{1}{2} \sum_{t = - τ}^{τ} {(\sqrt{s_{t}^{a} (p)} - \sqrt{s_{t}^{b} (p)})}^{2}} \end{array}$ (11) where θ ∈ [0, 1], s_t (p) means the probability value of each linguistic variable, $\sum_{t = - τ}^{τ} s_{t} (p) = 1$ , and if $\sum_{t = - τ}^{τ} s_{t} (p) < 1$ , we normalize each probability value by: $s_{t} (p^{'}) = s_{t} (p) / \sum_{t = - τ}^{τ} s_{t} (p)$ (12)

The distance satisfies:

0 ⩽ d (L^a (p) , L^b (p)) ⩽1

d (L^a (p) , L^b (p)) =0 if and only if L^a (p) = L^b (p)

d (L^a (p) , L^b (p)) = d (L^b (p) , L^a (p))

Proof.

Since $g (s_{ρ (i)}^{a}), g (s_{ρ (i)}^{b}) \in [0, 1]$ , then $0 ⩽ \frac{1}{# l} \sum_{i = 1}^{# l} | g (s_{ρ (i)}^{a}) - g (s_{ρ (i)}^{b}) | ⩽ 1$ , thus $0 ⩽ θ \frac{1}{# l} \sum_{i = 1}^{# l} | g (s_{ρ (i)}^{a}) - g (s_{ρ (i)}^{b}) | ⩽ θ$ . From the definition of Hellinger distance, we know that $0 ⩽ \sqrt{\frac{1}{2} \sum_{t = - τ}^{τ} {(\sqrt{s_{t}^{a} (p)} - \sqrt{s_{t}^{b} (p)})}^{2}} ⩽ 1$ , thus $0 ⩽ (1 - θ) \sqrt{\frac{1}{2} \sum_{t = - τ}^{τ} {(\sqrt{s_{t}^{a} (p)} - \sqrt{s_{t}^{b} (p)})}^{2}} ⩽ 1 - θ$ . Add these two parts together, therefore 0 ⩽ d_HH (L^a (p) , L^b (p)) ⩽1.

If d (L^a (p) , L^b (p)) =0, then $| g (s_{ρ (i)}^{a}) - g (s_{ρ (i)}^{b}) |=0$ , $\sqrt{s_{t}^{a} (p)} - \sqrt{s_{t}^{b} (p)} = 0$ , thus $g (s_{ρ (i)}^{a}) = g (s_{ρ (i)}^{b})$ , $s_{t}^{a} (p) = s_{t}^{b} (p)$ , therefore L^a (p) = L^b (p).

Since $| g (s_{ρ (i)}^{a}) - g (s_{ρ (i)}^{b}) | = | g (s_{ρ (i)}^{b}) - g (s_{ρ (i)}^{a}) |$ , ${(\sqrt{s_{t}^{a} (p)} - \sqrt{s_{t}^{b} (p)})}^{2} = {(\sqrt{s_{t}^{b} (p)} - \sqrt{s_{t}^{a} (p)})}^{2}$ , therefore d (L^a (p) , L^b (p)) = d (L^b (p) , L^a (p)).

The parameter θ indicates the relative importance of hesitant information and probabilistic information in calculating the distance of PLTSs. # l = max(l^a, l^b), l^a and l^b represent the number of elements contained by L^a (p) and L^b (p) respectively. The shorter one should be extended by adding the same value several times until it has the same length as the longer one. Here, we set the adding value to be the one that has the highest probability in the shorter PLTS. g function is applied to translate the linguistic variables to membership function. s_ρ(i) (i = 1, 2, . . . # l) is the i - th smallest linguistic term value in L (p). And for the linguistic terms that only exist in one PLTS, we set their probabilities equal 0 in the other PLTS.

Example 1. Set θ = 0.5 τ = 3, the distance between L₁ (p) ={ s_-2 (0.4) , s₀ (0.2) , s₁ (0.4) } and L₂ (p) ={ s_-1 (0.4) , s₀ (0.6) } is calculated as: $\begin{matrix} d_{HH} & = \frac{0.5}{3} (| \frac{1}{6} - \frac{2}{6} | + | \frac{3}{6} - \frac{3}{6} | + | \frac{4}{6} - \frac{3}{6} |) + 0.5 \sqrt{\frac{{(\sqrt{0.4})}^{2} + {(- \sqrt{0.4})}^{2} + {(\sqrt{0.2} - \sqrt{0.6})}^{2} + {(\sqrt{0.4})}^{2}}{2}} \\ = 0.4598 \end{matrix}$

Note if θ = 1, the measure degenerates to the distance measure of HFLEs, which ignores the probabilistic information; if θ = 0, the measure merely considers the probability distribution information of linguistic terms, which ignores the hesitant information.

3.3 The entropy weight for PLTSs

Entropy [40, 41] is a measure of uncertainty in the information. In this section, it was used to determine the objective weight of evaluation factors where the assessments are given by PLTSs.

Let S ={ s_t|t = - τ, . . . -1, 0, 1, . . . τ }, and X = [{ L^(k) (p^(k)) _ij|k = 1, 2, . . . , # L (p) }] _m×n (i = 1, 2 . . . m, j = 1, 2 . . . n), where m is the number of alternatives, and n is the number of criteria. Then normalize the original data by:

$y_{ij} = \frac{{max}_{j} (E (L^{(k)} (p^{(k)})_{ij})) - E (L^{(k)} (p^{(k)})_{ij})}{{max}_{j} (E (L^{(k)} (p^{(k)})_{ij})) - {min}_{j} (E (L^{(k)} (p^{(k)})_{ij}))}$ (13)

When the j - th criterion is a benefit criterion, or

$y_{ij} = \frac{E (L^{(k)} (p^{(k)})_{ij}) - {min}_{j} (E (L^{(k)} (p^{(k)})_{ij}))}{{max}_{j} (E (L^{(k)} (p^{(k)})_{ij})) - {min}_{j} (E (L^{(k)} (p^{(k)})_{ij}))}$ (14)

When j - th criterion is a cost criterion.

Then the entropy measure for X is formed as: $E_{j} = - \frac{1}{ln m} \sum_{i = 1}^{m} P_{ij} ln P_{ij}$ (15)

Where $P_{ij} = y_{ij} / \sum_{i = 1}^{m} y_{ij}$ (16)

(ln 0 replaced by 0.)

The deviation degree and weight is calculated as: $d_{j} = 1 - E_{j}$ (17) $w_{j}^{o} = d_{j} / \sum_{j = 1}^{n} d_{j}$ (18)

Suppose the subjective weight is $w_{s}^{j} (j = 1, 2..., n)$ , then the combined weight is obtained: $w_{j}^{c} = w_{j}^{o} w_{j}^{s} / \sum_{j = 1}^{n} w_{j}^{o} w_{j}^{s}$ (19)

4 A new approach for FMEA

In this part, an extended ELECTRE II method is proposed for ranking FMs. In Ref. [42] a hesitant fuzzy ELECTRE II is proposed to solve an MCDM problem under a hesitant fuzzy environment by means of score function and deviation function. Based on this, an extended ELECTRE II method was used to deal with FMEA problems, which considering both subjective weight and objective weight of risk factors. The assessments are taken the form of HFLTSs and the probabilistic information is also considered. Figure 1 shows the flow chart of the proposed approach.

Fig. 1

Flow chart of the proposed approach.

Let FM = { A₁, A₂, . . . . , A_u } (u ⩾ 2) be a finite set of u FMs, RF = (r₁, r₂, . . . r_v) be a finite set of v risk factors, TM = (TM₁, TM₂, . . . , TM_l) be a team consisted of l experts, each TM is assigned a weight vector λ_i, where λ_i > 0 and $\sum_{i = 1}^{l} λ_{i} = 1$ , to represent the importance of TM_l in the risk evaluation process. Notably, the weight vector of TM is designated directly by management or indirectly by using existing methods such as AHP.

Suppose the evaluation matrix M_k for all the FMs and the evaluation vector $W_{k}^{s}$ for risk factors given by TM_k is described as follows: $M_{k} = (\begin{matrix} m_{11}^{k} & m_{12}^{k} & \dots & m_{1 n}^{k} \\ m_{21}^{k} & m_{22}^{k} & \dots & m_{2 n}^{k} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ m_{m 1}^{k} & m_{m 2}^{k} & \dots & m_{mn}^{k} \end{matrix}), k = 1, 2, . . ., l$ (20) $W_{k}^{s} = {(w_{1}^{k}, w_{2}^{k}, . . ., w_{v}^{k})}^{T}, k = 1, 2, . . ., l$ (21) where $m_{ij}^{l} = {< x_{ij}, h_{S} (x_{ij}) >}$ is the evaluation value for the A_i with respect to r_j. And $W_{k}^{s}$ is the subjective evaluation vector for risk factors, where $w_{j}^{l} = {< y_{jl}, h_{U} (y_{jl}) >}$ is the importance degree of r_j. S ={ s_t|t = - τ, . . . -1, 0, 1, . . . τ } and U ={ u_t|t = - τ, . . . -1, 0, 1, . . . τ } are LTSs.

The relative attitude weight vector ω = (ω_C, ω_C′, ω_C″, ω_D, ω_D′, ω_D″, ω_{J
⁼}) of different types of hesitant fuzzy concordance, discordance and indifferent sets which depends on the DMs attitude is identified.

In algorithmic form, the application of score-deviation based PLTS-ELECTRE II involves the following steps:

Step 1. Aggregate DMs assessments to construct the group probabilistic linguistic distribution decision matrix M′ = [L (p) _ij] _m×n and the group probabilistic linguistic distribution weighting vector of risk factors W^s by Eq. (6)-(7).

Step 2. Obtain the subjective weights and the combined weights. Calculate the subjective weights of risk factors by Eq. (9), and normalize the result. Calculate the objective weight of criteria $w_{j}^{o} (j = 1, 2 . . ., n)$ by using Eq. (13)-(18) (Note that all the risk factors are benefit factors), then obtain the combined weight $W^{c} = (w_{1}^{c}, w_{2}^{c}, . . . w_{n}^{c})$ of the criteria by using Eq. (19).

Step 3. Establish the extended PLTSs-ELECTRE II model.

i. Calculate the score degree E (L (P) _ij) and deviation degree σ (L (P) _ij) of the evaluation of each alternative with respect to each different criterion by using Eq. (9)-(10).

ii. Construct the probabilistic hesitant fuzzy concordance, discordance and indifferent set by the following rules:

(1) The probabilistic hesitant fuzzy strong concordance set J_{C
_kl}:

$\begin{matrix} J_{C_{kl}} = & {j | E (L (P)_{kj}) > E (L (P)_{lj}) and \\ σ (L (P)_{kj}) < σ (L (P)_{lj})} \end{matrix}$ (22) (2) The probabilistic hesitant fuzzy medium concordance set $J_{C_{kl}^{'}}$ :

$\begin{matrix} J_{C_{kl}^{'}} = & {j | E (L (P)_{kj}) > E (L (P)_{lj}) and \\ σ (L (P)_{kj}) ⩾ σ (L (P)_{lj})} \end{matrix}$ (23) (3) The probabilistic hesitant fuzzy weak concordance set $J_{C_{kl}^{″}}$ :

$\begin{matrix} J_{C_{kl}^{″}} = & {j | E (L (P)_{kj}) = E (L (P)_{lj}) and \\ σ (L (P)_{kj}) < σ (L (P)_{lj})} \end{matrix}$ (24) where J_{C
_kl}, $J_{C_{kl}^{'}}$ and $J_{C_{kl}^{″}}$ denote the concordance coalition of the attributes and support the assertion that A_k is at least as risky as A_l (denoted by A_kSA_L).

(4) The probabilistic hesitant fuzzy strong discordance set J_{D
_kl}:

$\begin{matrix} J_{D_{kl}} = & {j | E (L (P)_{kj}) < E (L (P)_{lj}) and \\ σ (L (P)_{kj}) > σ (L (P)_{lj})} \end{matrix}$ (25) (5) The probabilistic hesitant fuzzy medium discordance set $J_{D_{kl}^{'}}$ :

$\begin{matrix} J_{D_{kl}^{'}} = & {j | E (L (P)_{kj}) < E (L (P)_{lj}) and \\ σ (L (P)_{kj}) ⩽ σ (L (P)_{lj})} \end{matrix}$ (26) (6) The probabilistic hesitant fuzzy weak discordance set $J_{D_{kl}^{″}}$ :

$\begin{matrix} J_{D_{kl}^{″}} = & {j | E (L (P)_{kj}) = E (L (P)_{lj}) and \\ σ (L (P)_{kj}) > σ (L (P)_{lj})} \end{matrix}$ (27) where J_{D
_kl}, $J_{D_{kl}^{'}}$ and $J_{D_{kl}^{″}}$ denotes the discordance coalition and against the assertion that A_k is at least as risky as A_l.

(7) The probabilistic hesitant fuzzy indifferent set $J_{kl}^{=}$ :

$\begin{matrix} J_{kl}^{=} = & {j | E (L (P)_{kj}) = E (L (P)_{lj}) and \\ σ (L (P)_{kj}) = σ (L (P)_{lj})} \end{matrix}$ (28)

Where $J_{kl}^{=}$ means A_k is indifferent with A_l.

iii. Calculate the probabilistic hesitant fuzzy concordance index c_kl (k, l = 1, 2, . . , m ; k ≠ l) as:

$\begin{matrix} c_{kl} = \frac{ω_{C} \times \sum_{j \in J_{C_{kl}}} w_{j}^{c} + ω_{C^{'}} \times \sum_{j \in J_{C_{kl}^{'}}} w_{j}^{c} + ω_{C^{″}} \times \sum_{j \in J_{C_{kl}^{″}}} w_{j}^{c} + ω_{J_{kl}^{=}} \times \sum_{j \in J_{kl}^{=}} w_{j}^{c}}{\sum_{j = 1}^{n} w_{j}^{c}} \\ = ω_{C} \times \sum_{j \in J_{C_{kl}}} w_{j}^{c} + ω_{C^{'}} \times \sum_{j \in J_{C_{kl}^{'}}} w_{j}^{c} + ω_{C^{″}} \times \sum_{j \in J_{C_{kl}^{″}}} w_{j}^{c} + ω_{J_{kl}^{=}} \times \sum_{j \in J_{kl}^{=}} w_{j}^{c} \end{matrix}$ (29)

iv. Construct the concordance matrix by the indices c_kl as: $C = [\begin{matrix} - & \dots & c_{1 l} & \dots & c_{1 (m - 1)} & c_{1 m} \\ ⋮ & ⋱ & ⋮ & ⋱ & ⋮ & ⋮ \\ c_{k 1} & \dots & c_{kl} & \dots & c_{k (m - 1)} & c_{km} \\ ⋮ & ⋱ & ⋮ & ⋱ & ⋮ & ⋮ \\ c_{m 1} & \dots & c_{ml} & \dots & c_{m (m - 1)} & - \end{matrix}]$ (30)

v. Calculate the weighted distance between any two alternatives with respect to each criterion by using Eq. (11).

vi. Calculate the probabilistic hesitant fuzzy discordance index d_kl (k, l = 1, 2, . . , m ; k ≠ l) as:

$\begin{matrix} d_{kl} \\ = \frac{{max}_{j \in J_{D_{kl}} \cup J_{D_{kl}^{'}} \cup J_{D_{kl}^{″}}} {\begin{matrix} ω_{D} \times d (w_{j}^{c} L (P)_{kj}, w_{j}^{c} L (P)_{lj}), \\ ω_{D^{'}} \times d (w_{j}^{c} L (P)_{kj}, w_{j}^{c} L (P)_{lj}), \\ ω_{D^{″}} \times d (w_{j}^{c} L (P)_{kj}, w_{j}^{c} L (P)_{lj}) \end{matrix}}}{{max}_{j \in J} d (w_{j}^{c} L (P)_{kj}, w_{j}^{c} L (P)_{lj})} \end{matrix}$ (31)

vii. Construct the discordance matrix by the indices d_kl as: $D = [\begin{matrix} - & \dots & d_{1 l} & \dots & d_{1 (m - 1)} & d_{1 m} \\ ⋮ & ⋱ & ⋮ & ⋱ & ⋮ & ⋮ \\ d_{k 1} & \dots & d_{kl} & \dots & d_{k (m - 1)} & d_{km} \\ ⋮ & ⋱ & ⋮ & ⋱ & ⋮ & ⋮ \\ d_{m 1} & \dots & d_{ml} & \dots & d_{m (m - 1)} & - \end{matrix}]$ (32)

Step 4. Construct outranking relations:

Let c^*, c⁰, c^- be three decreasing levels of concordance satisfying 0 < c^- < c⁰ < c^* < 1, d⁰, d^* be two increasing levels of discordance satisfying 0 < d⁰ < d^* < 1. Defined two types of outranking relationships: a strong relationship A_kS^FA_L and a weak relationship A_kS^fA_L. Comparing to A_kS^fA_L, A_kS^FA_L represents that the judgment of A_k is at least as risky as A_l that yield a more refined and stricter ranking procedure.

The relationship A_kS^FA_L is defined if any or both of the following set of conditions (a) and (b) hold: $\begin{matrix} (a) {\begin{matrix} C (A_{k}, A_{l}) ⩾ c^{*} \\ D (A_{k}, A_{l}) ⩽ d^{*} \\ C (A_{k}, A_{l}) ⩾ C (A_{l}, A_{k}) \end{matrix} \\ (b) {\begin{matrix} C (A_{k}, A_{l}) ⩾ c^{0} \\ D (A_{k}, A_{l}) ⩽ d^{0} \\ C (A_{k}, A_{l}) ⩾ C (A_{l}, A_{k}) \end{matrix} \end{matrix}$ (33)

The relationship A_kS^fA_L is defined when the conditions (c) hold: $(c) {\begin{matrix} C (A_{k}, A_{l}) ⩾ c^{-} \\ D (A_{k}, A_{l}) ⩽ d^{*} \\ C (A_{k}, A_{l}) ⩾ C (A_{l}, A_{k}) \end{matrix}$ (34)

Step 5. Draw the strong and weak outranking graphs according to the outranking relations.

Step 6. Rank all the FMs.

The ranking procedures consist of a direct ranking ν′ (x), a reverse ranking ν″ (x) and a median order $\bar{ν} (x) = (ν^{'} + ν^{″}) / 2$ . The ranking of FMs is according to the value of $\bar{ν} (x)$ . The higher the value is, the greater the risk an FM is. More detail refers to the Ref. [43].

5 Case study

In this part, a case study of a reheat valve system in a nuclear steam turbine comes from [44] is adopted to demonstrate how to make use of the proposed FMEA approach.

5.1 Description

Operating under complex conditions, the nuclear reheat valve system is essential to the reliability of the whole nuclear power plant. The valve must be fully opened and closed timely when transporting radioactive medium with high temperatures and pressure. Any incorrect operation may cause serious consequences. To identify known and potential risk issues, FMEA was carried out to enhance the safety and reliability of processes in the nuclear reheat valve system. When implementing FMEA, it is necessary to form a team, which is composed of several individuals with multi-disciplinary and multi-functional backgrounds. Then, divide the valve system into various components. Through brainstorming, the possible failure modes of the analysis components are listed, and the influence and detection methods of these failure modes are also been discussed. After this preparation, eight FMs of nuclear reheat valve system were obtained, shown in Table 1.

Table 1
The causes, effects, and detection methods of nuclear reheat valve system FMs

No. FM Causes of FMs Effects of FMs Detection methods

A₁ Valve’s closing time is too long or no action Unreasonable spring selection Over speeding of steam turbine rotor and parts breakdown Valve closing test

A₂ Valve cannot be closed tightly Small bushing clearance and shaft bending Blade corrosion of steam turbine Valve leak test

A₃ Large leak if valve shaft Compaction force of sealing filler is not enough Waste of chemical water and thermal loss Inspection after packing removal

A₄ Valve fluctuations Hydraulic cylinder leaks Valve cannot open and close normally, and unsafe operation Visual inspection of cylinder pressure gauge

A₅ Valve jam when opening and closing Large deformation of valve shaft or body due to process and material defects Valve cannot open and close affecting normal operation of turbine Valve operation test

A₆ Valve shaft fracture Fatigue fracture under alternating stress Tripping of turbine unit Metallographic tests on the fracture gap

A₇ Malfunction of valve shaft support bearing Low strength of bearing material and long-term wear and tear Abnormal operation of valve system Disassemble inspection

A₈ Excessive noise or abnormal noise of valve system System vibration due to unreasonable components clearance selection Make the user feel uncomfortable, and reduce service life of components Change operating condition or frequency measurement of valve system

No.	FM	Causes of FMs	Effects of FMs	Detection methods
A₁	Valve’s closing time is too long or no action	Unreasonable spring selection	Over speeding of steam turbine rotor and parts breakdown	Valve closing test
A₂	Valve cannot be closed tightly	Small bushing clearance and shaft bending	Blade corrosion of steam turbine	Valve leak test
A₃	Large leak if valve shaft	Compaction force of sealing filler is not enough	Waste of chemical water and thermal loss	Inspection after packing removal
A₄	Valve fluctuations	Hydraulic cylinder leaks	Valve cannot open and close normally, and unsafe operation	Visual inspection of cylinder pressure gauge
A₅	Valve jam when opening and closing	Large deformation of valve shaft or body due to process and material defects	Valve cannot open and close affecting normal operation of turbine	Valve operation test
A₆	Valve shaft fracture	Fatigue fracture under alternating stress	Tripping of turbine unit	Metallographic tests on the fracture gap
A₇	Malfunction of valve shaft support bearing	Low strength of bearing material and long-term wear and tear	Abnormal operation of valve system	Disassemble inspection
A₈	Excessive noise or abnormal noise of valve system	System vibration due to unreasonable components clearance selection	Make the user feel uncomfortable, and reduce service life of components	Change operating condition or frequency measurement of valve system

Note: part of the FMs is listed in Table 1 due to space limitations and privacy regulations.

In order to analyses the risk priority of these eight FMs, a cross-functional team consisting of 4 experts is established, take S, O, and D as risk factors, and use seven-grade linguistic rating variables s ∈ S to evaluate each FM with respect to each risk factor, shown in Table 2. Meanwhile, five-grade linguistic weighting variables u ∈ U to assess the importance of each risk factor, as shown in Table 3. Each expert is assigned an importance weight λ = (0.23, 0.26, 0.25, 0.26), and experts assign the relative attitude weights of strong, medium and weak probabilistic hesitant fuzzy concordance, discordance, and indifferent sets as: ω = (ω_C, ω_C′, ω_C″, ω_D, ω_D′, ω_D″, ω_{J
⁼}) = (1, 0.9, 0.8, 1, 0.9, 0.8, 0.7).

Table 2

Evaluation of the FMs

TMs	FMs	Risk factors			TMs	FMs	Risk factors
		S	O	D			S	O	D
TM₁	A₁	{H, VH}	{ML, M}	{H}	TM₃	A₁	{VH}	{M}	{MH, H}
	A₂	{L}	{ML, M}	{ML, M}		A₂	{VL, L}	{M, MH}	{M}
	A₃	{L , ML}	{MH}	{ML}		A₃	{M}	{M}	{M}
	A₄	{MH}	{L , ML}	{ML}		A₄	{MH}	{ML, M}	{ML, M}
	A₅	{H, VH}	{ML}	{ML}		A₅	{MH}	{ML}	{L , ML}
	A₆	{VH}	{VL}	{L , ML}		A₆	{VH}	{L}	{ML, M}
	A₇	{MH}	{M}	{L , ML}		A₇	{H, VH}	{MH}	{ML}
	A₈	{M, MH}	{M, MH}	{ML}		A₈	{MH}	{M, MH}	{ML}
TM₂	A₁	{H, VH}	{ML, M}	{H}	TM₄	A₁	{H, VH}	{L}	{M}
	A₂	{L , ML}	{ML, M}	{M}		A₂	{ML}	{ML}	{ML}
	A₃	{ML, M}	{MH, H}	{ML, M}		A₃	{L , ML}	{M}	{ML, M}
	A₄	{MH}	{ML, M}	{L}		A₄	{M}	{ML, M}	{ML, M}
	A₅	{VH}	{ML}	{ML, M}		A₅	{H}	{M}	{ML}
	A₆	{VH}	{L}	{L}		A₆	{VH}	{L , ML}	{M}
	A₇	{MH, H}	{M}	{L , ML}		A₇	{H, VH}	{M}	{ML}
	A₈	{M}	{MH, H}	{M}		A₈	{VH}	{MH}	{ML, M}

Table 3

Assessments on risk factors weights

Risk factors	TMs
	TM₁	TM₂	TM₃	TM₄
S	{M, I}	{I, VI}	{I, VI}	{VI}
O	{UI, M}	{M, I}	{M, I}	{I}
D	{M}	{M, I}	{M}	{I}

The two linguistic term sets are described as: $\begin{matrix} S = {s_{- 3} = Verylow (VL), s_{- 2} = Low (L), \\ s_{- 1} = Moderately low (ML), s_{0} = Moderate (M), \\ s_{1} = Moderately high (MH), s_{2} = High (H), \\ s_{3} = Very high (VH)} \end{matrix}$

$\begin{matrix} U = {u_{- 2} = Very unimportant (VU), \\ u_{- 1} = Unimportant (UI), u_{0} = Medium (M), \\ u_{1} = Important (I), u_{2} = Very important (VI)} \end{matrix}$

5.2 The ranking progress

In the following section, we use the proposed approach to construct and exploit the outrank relationships of all the identified FMs.

Step 1. Translate the hesitant fuzzy linguistic evaluations into PLTSs. Take TM₁’s evaluation of A₁ with respect to risk factor S as an example, it is translated as: {s₂ (0.5) , s₃ (0.5)}. And TM₁’s evaluations about risk factors S is translated as: {u₀ (0.5) , u₁ (0.5)}.

Step 2. Construct group probability linguistic decision matrix M′ and the weighting vector of risk factors W^S. The results are shown below:

$M = (\begin{matrix} {s_{2} (0.375), s_{3} (0.625)} & {s_{- 2} (0.26), s_{- 1} (0.245), s_{0} (0.495)} & {s_{0} (0.26), s_{1} (0.125), s_{2} (0.615)} \\ {s_{- 3} (0.125), s_{- 2} (0.485), s_{- 1} (0.39)} & {s_{- 1} (0.505), s_{0} (0.37), s_{1} (0.125)} & {s_{- 1} (0.375), s_{0} (0.625)} \\ {s_{- 2} (0.245), s_{- 1} (0.375), s_{0} (0.38)} & {s_{0} (0.51), s_{1} (0.36), s_{2} (0.13)} & {s_{- 1} (0.49), s_{0} (0.51)} \\ {s_{0} (0.26), s_{1} (0.74)} & {s_{- 2} (0.115), s_{- 1} (0.5), s_{0} (0.385)} & {s_{- 2} (0.26), s_{- 1} (0.485), s_{0} (0.255)} \\ {s_{1} (0.25), s_{2} (0.375), s_{3} (0.375)} & {s_{- 1} (0.74), s_{0} (0.26)} & {s_{- 2} (0.125), s_{- 1} (0.745), s_{0} (0.13)} \\ {s_{3} (1)} & {s_{- 3} (0.23), s_{- 2} (0.64), s_{- 1} (0.13)} & {s_{- 2} (0.375), s_{- 1} (0.24), s_{0} (0.385)} \\ {s_{1} (0.36), s_{2} (0.385), s_{3} (0.255)} & {s_{0} (0.75), s_{1} (0.25)} & {s_{- 2} (0.245), s_{- 1} (0.755)} \\ {s_{0} (0.26) s_{1} (0.25), s_{2} (0.115), s_{3} (0.375)} & {s_{0} (0.24), s_{1} (0.63), s_{2} (0.13)} & {s_{- 1} (0.61), s_{0} (0.39)} \end{matrix})$ $W^{s} = (\begin{matrix} {u_{0} (0.115), u_{1} (0.37), u_{2} (0 . 515)} & {u_{- 1} (0.115), u_{0} (0.37), u_{1} (0.515)} & {u_{0} (0.61), u_{1} (0.39)} \end{matrix})$

Step 3. Calculate the subjective weight of risk factors, the normalized result is: w = (0 . 4151, 0 . 2930, 0 . 2918) ^T. Calculate the objective weight and the combined weight of risk factors, the results are obtained as: W^o = (0.4689, 0 . 3597, 0 . 1714) and W^c = (0.5560, 0 . 3011, 0 . 1429).

Step 4. Calculate the score degree E (L (P) _ij) and deviation degree σ (L (P) _ij) of the evaluation of each alternative with respect to each different criterion.

Step 5. Construct the probabilistic hesitant fuzzy strong, medium, weak concordance, the probabilistic hesitant fuzzy strong, medium, weak discordance and the probabilistic hesitant fuzzy indifferent set following the rules (21) - (27). $\begin{matrix} J_{C} = [\begin{matrix} - & 1 & 1 & - & 1 & 3 & 1 & 1 \\ 2 & - & 3 & 3 & - & 3 & - & 3 \\ 2 & - & - & 3 & - & 3 & - & - \\ 2 & 1 & 1 & - & - & - & - & - \\ 2 & - & - & 3 & - & - & 3 & 1 \\ 1 & 1 & 1 & 1 & 1 & - & 1 & 1 \\ 2 & 2 & 1 & 2 & 2 & - & - & 1 \\ 2 & 2 & 2 & 23 & 2 & 3 & - & - \end{matrix}] \end{matrix}$ $\begin{matrix} J_{C^{'}} = [\begin{matrix} - & 3 & 3 & 13 & 3 & 2 & 3 & 3 \\ - & - & - & 2 & 2, 3 & 2 & 3 & - \\ - & 12 & - & 2 & 2 . 3 & 2 & 23 & 3 \\ - & - & - & - & 2 & 2 & 3 & - \\ - & 1 & 1 & 1 & - & 2 & 1 & - \\ - & - & - & 3 & 3 & - & 3 & - \\ - & 1 & - & 1 & - & 2 & - & - \\ - & 1 & 1 & 1 & 3 & 2 & 23 & - \end{matrix}] \end{matrix}$ $\begin{matrix} J_{D} = [\begin{matrix} - & 2 & 2 & 2 & 2 & 1 & 2 & 2 \\ 1 & - & - & 1 & - & 1 & 2 & 2 \\ 1 & 3 & - & 1 & - & 1 & 1 & 2 \\ - & 3 & 3 & - & 3 & 1 & 2 & 2, 3 \\ 1 & - & - & - & - & 1 & 2 & 2 \\ 3 & 3 & 3 & - & - & - & - & 3 \\ 1 & - & - & - & 3 & 1 & - & - \\ 1 & 3 & - & - & 1 & 1 & 1 & - \end{matrix}] \\ J_{D^{'}} = [\begin{matrix} - & - & - & - & - & - & - & - \\ 3 & - & 1, 2 & - & 1 & - & 1 & 1 \\ 3 & - & - & - & 1 & - & - & 1 \\ 1, 3 & 2 & 2 & - & 1 & 3 & 1 & 1 \\ 3 & 2, 3 & 2, 3 & 2 & - & 3 & - & 3 \\ 2 & 2 & 2 & 2 & 2 & - & 2 & 2 \\ 3 & 3 & 2, 3 & 3 & 1 & 3 & - & 2, 3 \\ 3 & - & 3 & - & - & - & - & - \end{matrix}] \end{matrix}$ $J_{C^{″}} = J_{D^{″}} = J^{=} = {[-]}_{8 \times 8}$

Step 6. Calculate probabilistic hesitant fuzzy concordance index c_kl (k, l = 1, 2, . . , m ; k ≠ l), and use the indices to construct the concordance matrix: $C = {(c_{kl})}_{8 \times 8} = [\begin{matrix} - & 0.6845 & 0.6845 & 0.6289 & 0.6845 & 0.4138 & 0.6845 & 0.6845 \\ 0.3011 & - & 0.1428 & 0.4138 & 0.3995 & 0.4138 & 0.1285 & 0.1428 \\ 0.3011 & 0.7714 & - & 0.4138 & 0.3995 & 0.4138 & 0.3995 & 0.1285 \\ 0.3011 & 0.5560 & 0.5560 & - & 0.2710 & 0.2710 & 0.1285 & 0 \\ 0.3011 & 0.5004 & 0.5004 & 0.6432 & - & 0.2710 & 0.6432 & 0.5560 \\ 0.5560 & 0.5560 & 0.5560 & 0.6845 & 0.6845 & - & 0.6845 & 0.5560 \\ 0.3011 & 0.8015 & 0.5560 & 0.8015 & 0.3011 & 0.2710 & - & 0.5560 \\ 0.3011 & 0.8015 & 0.8015 & 0.9443 & 0.4296 & 0.4138 & 0.3995 & - \end{matrix}]$

Step 7. Calculate the weighted distance between any two alternatives with respect to each criterion by using the hybrid Hamming-Hellinger distance, where θ = 0.5, Table 4 shows the weighted distances respect to risk factor S.

Table 4
Weighted distances respect to risk factor S

L(P)₁₁ L(P)₂₁ L(P)₃₁ L(P)₄₁ L(P)₅₁ L(P)₆₁ L(P)₇₁ L(P)₈₁

L(P)₁₁ 0.4942 0.4479 0.3707 0.1352 0.1504 0.1615 0.2123

L(P)₂₁ 0.1916 0.4016 0.4633 0.5097 0.4633 0.4402

L(P)₃₁ 0.3074 0.4170 0.4633 0.4170 0.3344

L(P)₄₁ 0.2716 0.3938 0.2552 0.1895

L(P)₅₁ 0.2194 0.0289 0.1485

L(P)₆₁ 0.2419 0.2426

L(P)₇₁ 0.1412

L(P)₈₃

	L(P)₂₁	L(P)₃₁	L(P)₄₁	L(P)₅₁	L(P)₆₁	L(P)₇₁	L(P)₈₁
L(P)₁₁	0.4942	0.4479	0.3707	0.1352	0.1504	0.1615	0.2123
L(P)₂₁		0.1916	0.4016	0.4633	0.5097	0.4633	0.4402
L(P)₃₁			0.3074	0.4170	0.4633	0.4170	0.3344
L(P)₄₁				0.2716	0.3938	0.2552	0.1895
L(P)₅₁					0.2194	0.0289	0.1485
L(P)₆₁						0.2419	0.2426
L(P)₇₁							0.1412
L(P)₈₃

Step 8. Calculate probabilistic hesitant fuzzy discordance index, and use the indices to construct the discordance matrix: $D = {(d_{kl})}_{8 \times 8} = [\begin{matrix} - & 0.1 937 & 0.3491 & 0.0818 & 0.5786 & 1 & 0.7897 & 0.8106 \\ 1 & - & 0.9 & 1 & 0.9 & 1 & 0.9 & 0.9 \\ 1 & 0.0 307 & - & 1 & 0.9 & 1 & 1 & 0.9 \\ 0.9 & 0.1732 & 0.3 534 & - & 0.9 & 1 & 0.9 & 0 . 9276 \\ 1 & 0.1144 & 0.3133 & 0.1646 & - & 1 & 1 & 1 \\ 0.7296 & 0.3179 & 0.4 387 & 0.2 941 & 0.6844 & - & 0.7779 & 0.8378 \\ 1 & 0.1181 & 0.1289 & 0.1091 & 0.1890 & 1 & - & 0.5124 \\ 1 & 0.0 272 & 0.0 164 & 0 & 0.8621 & 1 & 1 & - \end{matrix}]$

Step 9. Construct outranking relations, let (c^-, c⁰, c^*) = (0 . 5, 0 . 6, 0 . 75, (d⁰, d^*) = (0 . 5, 0 . 9, the result displayed in Table 5.

Table 5

Outranking relations

	A₁	A₂	A₃	A₄	A₅	A₆	A₇	A₈
A₁		S^F	S^F	S^F	S^f		S^f	S^f
A₂
A₃		S^F
A₄		S^f	S^f
A₅		S^f	S^f	S^F
A₆	S^f	S^f	S^f	S^F	S^f		S^f	S^f
A₇		S^F	S^f	S^F				S^f
A₈		S^F	S^F	S^F

Step 10. Draw the strong and weak outranking graphs. See Fig. 2.

Fig. 2

Strong (a) and weak (b) outranking graphs for FMs.

Step 11. Rank all FMs. The result presented in Table 6:

Table 6

The ranking result of the FMs

	A₁	A₂	A₃	A₄	A₅	A₆	A₇	A₈
ν′	2	7	6	5	3	1	3	4
ν″	2	7	6	5	4	1	3	4
$\bar{ν}$	2	7	6	5	3.5	1	3	4

So, the final ranking of the eight FMs is: A₆ ≻ A₁ ≻ A₇ ≻ A₅ ≻ A₈ ≻ A₄ ≻ A₃ ≻ A₂.

Note that “≻” means the risk of A_i is greater than A_j, and “∼” means the risk of A_i is indifferent to A_j.

6 Discussion

In this section, sensitivity analysis and comparative analysis have been done to verify the validity of the proposed approach.

6.1 Sensitivity analysis

We perform a sensitivity analysis by changing parameter θ. θ indicates the different attitudes of DMs to the importance of probabilistic information and hesitant information. The hybrid Hamming-Hellinger weighted distances changed as θ varying from 0 to 1. Figure 3 shows the graphs of outranking relations when (c^-, c⁰, c^*) = (0 . 5, 0 . 6, 0 . 75, (d⁰, d^*) = (0 . 5, 0 . 9. The final ranking results are presented in Table 7.

Table 7
The final ranking results

θ Ranking results

0, 0.1,..., 0.7 A₆ ≻ A₁ ≻ A₇ ≻ A₅ ≻ A₈ ≻ A₄ ≻ A₃ ≻ A₂

0.8 A₆ ≻ A₁ ≻ A₇ ≻ A₅ ≻ A₈ ≻ A₄ ≻ A₃ ≻ A₂

0.9 A₆ ≻ A₁ ∼ A₇ ≻ A₅ ∼ A₈ ≻ A₄ ≻ A₃ ≻ A₂

1 A₆ ∼ A₁ ≻ A₇ ≻ A₅ ≻ A₈ ≻ A₄ ≻ A₃ ≻ A₂

θ	Ranking results
0, 0.1,..., 0.7	A₆ ≻ A₁ ≻ A₇ ≻ A₅ ≻ A₈ ≻ A₄ ≻ A₃ ≻ A₂
0.8	A₆ ≻ A₁ ≻ A₇ ≻ A₅ ≻ A₈ ≻ A₄ ≻ A₃ ≻ A₂
0.9	A₆ ≻ A₁ ∼ A₇ ≻ A₅ ∼ A₈ ≻ A₄ ≻ A₃ ≻ A₂
1	A₆ ∼ A₁ ≻ A₇ ≻ A₅ ≻ A₈ ≻ A₄ ≻ A₃ ≻ A₂

Fig. 3

Strong (a) and weak (b) outranking graphs with the different value of θ.

From Fig. 3, we note that when θ between 0 and 0.7, the strong and weak outranking relations stay the same. As θ keeps increasing to 0.8, the weak relation that “A₁S^fA₅” changes to strong relation that “A₁S^FA₅”. When θ = 0 . 9, The previously accepted relation of “A₁S^fA₇” is rejected. And the relation “A₁S^fA₆” is also rejected when θ = 1. In Table 7, we can see that the ranking order get slight changes when θ ⩾ 0 . 9. These changes are not critical enough to influence the ranking order of all the FMs.

6.2 Comparative analysis

The HF-ELECTRE II method introduced in Ref. [45] only considers the hesitant information and the subjective weight of the criteria. In order to illustrate the advancement of the PLTS-ELECTRE II method, we investigate the same numerical example used in Ref. [45]. Moreover, some probabilistic information is added to make a comparison between these two methods.

Suppose that the weight vector of the criteria is w = (0 . 2 0 . 2 0 . 15 0 . 1 0 . 1 0 . 15 0 . 1) ^T, three DMs with the same importance, relative attitude weights of strong, medium and weak probabilistic hesitant fuzzy concordance, discordance, and indifferent sets are assigned as: ω = (ω_C, ω_C′, ω_C″, ω_D, ω_D′, ω_D″, ω_{J
⁼}) = (1, 0.9, 0.8, 1, 0.9, 0.8, 0.7), θ is determined as 0.5, the probabilistic information is given and the decision matrix is presented in Table 8.

Table 8
Probabilistic hesitant fuzzy decision matrix for the numerical example

A₁ A₂ A₃

C₁ {0 . 7 (1/3) , 0 . 8 (1/3) , 0 . 9 (1/3)} {0 . 7 (1/3) , 0 . 8 (2/3)} {0 . 7 (2/3) , 0 . 9 (1/3)}

C₂ {0 . 6 (1/3) , 0 . 7 (2/3)} {0 . 2 (1/3) , 0 . 3 (1/3) , 0 . 6 (1/3)} {0 . 6 (1/3) , 0 . 7 (1/3) , 0 . 8 (1/3)}

C₃ {0 . 2 (1/3) , 0 . 3 (2/3)} {0 . 2 (2/3) , 0 . 3 (1/3)} {0 . 1 (1/3) , 0 . 3 (1/3) , 0 . 4 (1/3)}

C₄ {0 . 5 (1/3) , 0 . 6 (1/3) , 0 . 7 (1/3)} {0 . 4 (2/3) , 0 . 5 (1/3)} {0 . 7 (1/3) , 0 . 8 (1/3) , 1 (1/3)}

C₅ {0 . 7 (1/3) , 0 . 8 (1/3) , 1 (1/3)} {0 . 5 (1/3) , 0 . 7 (2/3)} {0 . 5 (1/3) , 0 . 7 (2/3)}

C₆ {0 . 2 (2/3) , 0 . 3 (1/3)} {0 . 3 (1/3) , 0 . 4 (1/3) , 0 . 6 (1/3)} {0 . 3 (1/3) , 0 . 5 (2/3)}

C₇ {0 . 6 (1/3) , 0 . 9 (2/3)} {0 . 2 (1/3) , 0 . 3 (2/3)} {0 . 7 (1/3) , 0 . 8 (1/3) , 0 . 9 (1/3)}

A₄ A₅

C₁ {0 . 6 (1/3) , 0 . 8 (2/3)} {0 . 9 (1/3) , 1 (2/3)}

C₂ {0 . 3 (1/3) , 0 . 4 (1/3) , 0 . 5 (1/3)} {0 . 8 (1/3) , 0 . 9 (2/3)}

C₃ {0 . 3 (1/3) , 0 . 5 (2/3)} {0 . 2 (1/3) , 0 . 5 (1/3) , 0 . 6 (1/3)}

C₄ {0 . 5 (1/3) , 0 . 6 (1/3) , 0 . 7 (1/3)} {0 . 7 (1)}

C₅ {0 . 6 (1/3) , 0 . 8 (1/3) , 0 . 9 (1/3)} {0 . 8 (1/3) , 0 . 9 (1/3) , 1 (1/3)}

C₆ {0 . 3 (2/3) , 0 . 5 (1/3)} {0 . 4 (2/3) , 0 . 6 (1/3)}

C₇ {0 . 4 (1/3) , 0 . 5 (1/3) , 0 . 6 (1/3)} {0 . 7 (1/3) , 0 . 8 (1/3) , 0 . 9 (1/3)}

	A₁	A₂	A₃
C₁	{0 . 7 (1/3) , 0 . 8 (1/3) , 0 . 9 (1/3)}	{0 . 7 (1/3) , 0 . 8 (2/3)}	{0 . 7 (2/3) , 0 . 9 (1/3)}
C₂	{0 . 6 (1/3) , 0 . 7 (2/3)}	{0 . 2 (1/3) , 0 . 3 (1/3) , 0 . 6 (1/3)}	{0 . 6 (1/3) , 0 . 7 (1/3) , 0 . 8 (1/3)}
C₃	{0 . 2 (1/3) , 0 . 3 (2/3)}	{0 . 2 (2/3) , 0 . 3 (1/3)}	{0 . 1 (1/3) , 0 . 3 (1/3) , 0 . 4 (1/3)}
C₄	{0 . 5 (1/3) , 0 . 6 (1/3) , 0 . 7 (1/3)}	{0 . 4 (2/3) , 0 . 5 (1/3)}	{0 . 7 (1/3) , 0 . 8 (1/3) , 1 (1/3)}
C₅	{0 . 7 (1/3) , 0 . 8 (1/3) , 1 (1/3)}	{0 . 5 (1/3) , 0 . 7 (2/3)}	{0 . 5 (1/3) , 0 . 7 (2/3)}
C₆	{0 . 2 (2/3) , 0 . 3 (1/3)}	{0 . 3 (1/3) , 0 . 4 (1/3) , 0 . 6 (1/3)}	{0 . 3 (1/3) , 0 . 5 (2/3)}
C₇	{0 . 6 (1/3) , 0 . 9 (2/3)}	{0 . 2 (1/3) , 0 . 3 (2/3)}	{0 . 7 (1/3) , 0 . 8 (1/3) , 0 . 9 (1/3)}
	A₄	A₅
C₁	{0 . 6 (1/3) , 0 . 8 (2/3)}	{0 . 9 (1/3) , 1 (2/3)}
C₂	{0 . 3 (1/3) , 0 . 4 (1/3) , 0 . 5 (1/3)}	{0 . 8 (1/3) , 0 . 9 (2/3)}
C₃	{0 . 3 (1/3) , 0 . 5 (2/3)}	{0 . 2 (1/3) , 0 . 5 (1/3) , 0 . 6 (1/3)}
C₄	{0 . 5 (1/3) , 0 . 6 (1/3) , 0 . 7 (1/3)}	{0 . 7 (1)}
C₅	{0 . 6 (1/3) , 0 . 8 (1/3) , 0 . 9 (1/3)}	{0 . 8 (1/3) , 0 . 9 (1/3) , 1 (1/3)}
C₆	{0 . 3 (2/3) , 0 . 5 (1/3)}	{0 . 4 (2/3) , 0 . 6 (1/3)}
C₇	{0 . 4 (1/3) , 0 . 5 (1/3) , 0 . 6 (1/3)}	{0 . 7 (1/3) , 0 . 8 (1/3) , 0 . 9 (1/3)}

Rank these five alternatives in the framework of the proposed PLTS-ELECTRE II method, the calculation results are shown as bellows:

The objective weight: w^o = (0 . 0563 0 . 0807 0 . 2150 0 . 0723 0 . 0847 0 . 3506 0 . 1404) ^T

The combined weight: w^c = (0 . 0794 0 . 1137 0 . 2271 0 . 0509 0 . 0596 0 . 3704 0 . 0989) ^T.

When (c^-, c⁰, c^*) = (0 . 5, 0 . 6, 0 . 75, (d⁰, d^*) = (0 . 5 0 . 9, the outranking relations are presented in Table 9, and the outranking graphs are presented in Fig. 4:

Table 9

The outranking relations for the numerical example

	A₁	A₂	A₃	A₄	A₅
A₁
A₂
A₃		S^f		S^f
A₄	S^F
A₅	S^F	S^F	S^F	S^F

Fig. 4

Strong (a) and weak (b) outranking graphs for the numerical example.

And the final ranking is: A₅ ≻ A₃ ≻ A₄ ≻ A₁ ∼ A₂, this ranking order is different from A₅ ≻ A₃ ∼ A₁ ≻ A₄ ≻ A₂ obtained by HF-ELECTRE II method. The advantages of our proposed approach are listed as following: Firstly, we use linguistic terms to assess the alternatives, which are more human-centered. It renders the assessment more explicit. Secondly, we combine the subjective weight and objective weight together, so the weight of the criteria is considered more systematically. This causes the result more accurate. And finally, In PLTS-ELECTRE II method, the importance of probabilistic information is reflected. Obviously, the proposed method takes into account more comprehensive information. Therefore, the results obtained are more convincing than the previous method.

7 Conclusion

FMEA is a powerful tool to deal with real-world engineering and management problems. When it is treated as an MCDM problem, the fuzzy set theory and outranking MCDM techniques can be applied to improve its effectiveness. In this paper, a suitable fuzzy set-PLTS which combine the hesitant information and probabilistic information together is applied to describe FMEA problem under a complex hesitant environment. An entropy weight which can measure the uncertainty of PLTSs, further determine the weight of criteria is proposed. Based on these discussions, a new FMEA model of score-deviation based PLTS-ELECTRE II is developed and has been successfully applied in a nuclear reheat valve system. The effectiveness of the proposed method has been verified by sensitivity and comparative analysis.

In further, more complex fuzzy environments will be considered in the FMEA problem [46 –48]. In this paper, we suppose the risk factor is independent of each other. In fact, it does not conform to reality. Moreover, Linguistic term sets employed in this paper are suggested to be uniformly and symmetrically distributed, this also needs to be reconsidered by means of unbalanced linguistic term sets.

Disclosure statement

No potential conflict of interest was reported by the authors.

Footnotes

Acknowledgments

The authors are very grateful to the anonymous reviewers for their valuable comments and suggestions to help improve the overall quality of this paper. This work was supported by the National Natural Science Foundation of China under Grant No. 71871228.

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A novel risk assessment model based on failure mode and effect analysis and probabilistic linguistic ELECTRE II method

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Linguistic term set and hesitant fuzzy linguistic term set

3.1 A new ordered relationship for PLTSs

5.1 Description

6.1 Sensitivity analysis

Table 7 The final ranking results θ Ranking results 0, 0.1,..., 0.7 A6 ≻ A1 ≻ A7 ≻ A5 ≻ A8 ≻ A4 ≻ A3 ≻ A2 0.8 A6 ≻ A1 ≻ A7 ≻ A5 ≻ A8 ≻ A4 ≻ A3 ≻ A2 0.9 A6 ≻ A1 ∼ A7 ≻ A5 ∼ A8 ≻ A4 ≻ A3 ≻ A2 1 A6 ∼ A1 ≻ A7 ≻ A5 ≻ A8 ≻ A4 ≻ A3 ≻ A2

Disclosure statement

Footnotes

Acknowledgments

References