Knowledge Representation and Reasoning with an Extended Dynamic Uncertain Causality Graph under the Pythagorean Uncertain Linguistic Environment

Abstract

A dynamic uncertain causality graph (DUCG) is a probabilistic graphical model for knowledge representation and reasoning, which has been widely used in many areas, such as probabilistic safety assessment, medical diagnosis, and fault diagnosis. However, the convention DUCG model fails to model experts’ knowledge precisely because knowledge parameters were crisp numbers or fuzzy numbers. In reality, domain experts tend to use linguistic terms to express their judgements due to professional limitations and information deficiency. To overcome the shortcomings of DUCGs, this article proposes a new type of DUCG model by integrating Pythagorean uncertain linguistic sets (PULSs) and the evaluation based on the distance from average solution (EDAS) method. In particular, experts express knowledge parameters in the form of the PULSs, which can depict the uncertainty and vagueness of expert knowledge. Furthermore, this model gathers the evaluations of experts on knowledge parameters and handles conflicting opinions among them. Moreover, a reasoning algorithm based on the EDAS method is proposed to improve the reliability and intelligence of expert systems. Lastly, an industrial example concerning the root cause analysis of abnormal aluminum electrolysis cell condition is provided to demonstrate the proposed DUCG model.

1. Introduction

A dynamic uncertain causality graph (DUCG) is a probabilistic graphical model for knowledge representation and probabilistic reasoning [1]. As a graphical and mathematical methodology, the DUCG aims to graphically represent complex uncertain causalities between variables and perform probabilistic reasoning [2]. A DUCG could be a directed graph comprising a number of nodes (variables) and directional arrows (causalities). Normally, the causalities and knowledge parameters associated with a DUCG are determined based on the knowledge and experience of field experts [3]. The knowledge reasoning process of a DUCG can be divided into three steps [4,5,6,7]: (1) simplification of DUCG based on evidence to decrease its scale; (2) extending causality chain expression of the consequence events composing of independent events; and (3) calculating the probabilities of consequence events according to these expressions. In addition, a complicated DUCG can be represented by a set of uncomplicated sub-DUCGs in the construction of a knowledge base [8]. Due to its outstanding capability in depicting uncertain causalities and performing efficient reasoning, the DUCG has been widely used in many fields, such as medical diagnosis and treatment [9,10], fault diagnosis of nuclear power plants [11], reliability analysis of dynamic reliability block diagram [12], shale-gas sweet-spot evaluation [13], and probabilistic safety assessment of a boiling water reactor [14].

Although the DUCGs have been utilized to solve various issues, the conventional DUCG model has been criticized for its shortcomings in knowledge representation and reasoning (KRR) [15]. For example, the knowledge parameters acquired from historical statistics or experts were restricted to be crisp values [16]. However, due to information scarcity, complex causalities between events, and inconsistent knowledge of specialists, the evaluations obtained from experts are usually uncertain and fuzzy [17,18,19]. Hence, to handle the vague knowledge, different uncertainty theories such as intuitionistic fuzzy sets [20], cloud model theory [21], and picture fuzzy sets [22] have been combined with DUCGs to improve their KRR ability. In practice, experts tend to use linguistic terms to describe their qualitative evaluations when quantitative information is not available or the cost of obtaining quantitative information is too high [23,24,25]. To more accurately and objectively describe the linguistic assessment information, Pythagorean uncertain linguistic sets (PULSs) [26] were proposed recently. The desirable advantage of PULSs is that they can describe two aspects of an object: an uncertain linguistic variable and a Pythagorean fuzzy number (PFN). Since its introduction, the PULS has received widespread attention in academia and has been applied in many practical fields [27,28]. Therefore, it is desirable to apply the PULSs to character the fuzziness and uncertainty expert knowledge in DUCGs.

In addition, obtaining accurate knowledge parameters play an important role in the KRR of DUCGs. However, previous research often directly gave the values of knowledge parameters or the representation of events [20,21,22]. Few studies have considered obtaining the knowledge parameters of DUCG from experts’ evaluations [22]. In addition, due to the different backgrounds of domain experts, experts often have diverse opinions about the knowledge parameters of a DUCG. However, existing studies fail to handle the conflicting opinions from experts on knowledge parameters. When experts have different opinions on an evaluation index, Yao et al. [13] adopted the weighting method to compute probabilities in DUCG; but they did not consider conflicting and inconsistent opinions among experts. Hence, developing effective methods to handle conflict opinions of experts to obtain knowledge parameters more accurately is vital for knowledge acquisition using DUCGs.

Motivated by the above analyses, this article aims to develop a new type of DUCGs, called PUL-DUCG, to enhance the ability of DUCGs in KRR. Firstly, the PULSs are used to express the vagueness and ambiguous evaluations obtained from a domain expert on knowledge parameters. Secondly, with the purpose of obtaining more accurate knowledge parameters, a correction algorithm for dealing with group conflict is employed to deal with cognitive nonconformity in the knowledge acquisition process. Moreover, a causal inference algorithm based on the improved evaluation based on the distance from average solution (EDAS) method is introduced to find out the most probable cause of an abnormal event. Lastly, a practical example is provided to illustrate the application and effectiveness of the established PUL-DUCG model.

The rest part of the paper is structured as follows: In the next section, we review the relevant work of DUCGs. Section 3 introduces the basic concepts of PULSs and DUCGs. The proposed PUL-DUCG model based on PULSs and the EDAS method is detailed in Section 4. A real case is performed to show and validate the feasibility of the new PUL-DUCG model in Section 5. Finally, we present conclusions in Section 6 and provide further study recommendations.

2. Literature Review

In recent years, some extended DUCGs have been proposed to compensate for the shortcomings of existing methods. For example, Zhang et al. [9] extended the DUCG to include the representation and inference algorithm for non-causal classification relationships. Zhang et al. [29] proposed an extended DUCG methodology to intuitively represent complex medical knowledge and perform effective clinical diagnose. Yao et al. [13] provided a DUCG model based on multiple conditional events and weighted graphs for shale-gas sweet-spot evaluation. Qiu and Zhang [30] presented a multi-valued DUCG method to calculate the joint probability distribution of the directed cycle graph with local data and domain causal knowledge. Jiao et al. [3] reported an artificial intelligence diagnostic model based on DUCG for improving the efficiency of differential dyspnea diagnosis. Dong and Zhang [11] proposed a cubic DUCG methodology characterized by causal connections and dynamic negative feedback loops for temporal process modeling and diagnostic logic inference. Bu et al. [31] presented a hybrid DUCG to reduce the misdiagnosis caused by outpatient triage error and help triage nurses improve their triage accuracy.

On the one hand, many uncertainty theories have been incorporated into DUCGs to represent the experience and knowledge of experts. For instance, Li et al. [22] proposed a DUCG based on picture fuzzy sets for uncertain knowledge representation and reasoning in root cause diagnosis. Li and Yue [20] introduced a DUCG model for root cause analysis, in which the intuitionistic fuzzy sets were used for describing the uncertain event. Li et al. [21] developed a cloud reasoning dynamic DUCG model, in which the cloud model theory was employed to handle the fuzziness and randomness of uncertain information simultaneously. Combining a fuzzy decision tree with the DUCG, Zhao et al. [32] proposed a simplified DUCG model for fault diagnosis in nuclear power plants.

Furthermore, many researchers have introduced different knowledge inference algorithms for DUCGs. Nie and Zhang [33] suggested an inference algorithm of DUCG based on conditional stochastic simulation for complex cases with many state-unknown intermediate variables. Zhou and Zhang [14] integrated DUCG with event trees to perform probabilistic safety assessments by considering the problems of dependencies and circular loops. In [20,21], the technique of order preference similarity to the ideal solution (TOPSIS) method was utilized to implement fuzzy knowledge inference in DUCG. In [22], an enhanced knowledge reasoning algorithm based on the picture fuzzy operators was developed to resolve causal inference problems. Dong et al. [34] provided a methodology for modeling and reasoning about complex faults with negative feedback with cubic DUCG. Hao et al. [35] proposed a diagnostic modeling and reasoning system using DUCG for the intelligent diagnosis of jaundice, considering the causal interactions among diseases and symptoms.

Based on the above literature review, it can be seen that many upgraded DUCGs have been developed for KRR. Nevertheless, there is no research that combines PULS s with the EDAS method in DUCG. Additionally, most studies lack the capacity to deal with the conflict judgements of domain experts on knowledge parameters. Hence, this article aims to present a new type of DUCG that combine PULSs and the EDAS method for KRR. In addition, a preference modifying-based method is used to detect and eliminate the conflict among experts to obtain accurate knowledge parameters in the knowledge acquisition process, which is an important issue that received little attention in the literature.

3. Preliminaries

3.1. Pythagorean Uncertain Linguistic Sets

The PULSs were proposed by Liu et al. [26] based on uncertain linguistic term sets [36] and Pythagorean fuzzy sets [37] to character the fuzziness and uncertainty in practical situations.

Definition 1.

[38] Let$S=\left\{s_0,s_1,\dots ,s_g\right\}$be a finite and ordered linguistic term set, in which s_i indicates a possible value for a linguistic variable and g is an even number. In general, the linguistic term set has the following properties:

(1)

The set is ordered:$s_i>s_j$, if$i>j$;

(2)

Negation operator:$\mathit{neg}\left(s_i\right)=s_{g-i}$;

(3)

Max operator: max$\left(s_i,s_j\right)=s_i$, if$i\ge j$;

(4)

Min operator: min$\left(s_i,s_j\right)=s_i$, if$i\le j$.

To preserve all given linguistic information, the discrete linguistic term set S was extended by Xu [39] to a continuous form $\tilde{S}=\left\{s_i|i\in \left[0,\xi \right]\right\}$, in which $\xi \left(\xi >g\right)$ is a sufficiently large positive integer.

Definition 2.

[40] Let$\tilde{s}=\left[s_{\theta },s_{\tau }\right]$, $\theta \le \tau$and$s_{\theta },s_{\tau }\in \tilde{S}$, where$s_{\theta }$is the lower limit of$\tilde{s}$, and$s_{\tau }$is the upper limit of$\tilde{s}$, so$\tilde{s}$is called an uncertain linguistic variable.

Definition 3.

[26] Let X be a nonempty set of the universe, then a PULS A in X is expressed by

$$A=\left\{\langle x_i\mid \left(\tilde{s}\left(x_i\right),\left({\mu }_p\left(x_i\right),v_p\left(x_i\right)\right)\right)\rangle \mid x_i\in X\right\}$$

(1)

where$\tilde{s}\left(x_i\right)=\left[s_{\theta \left(x_i\right)},s_{\tau \left(x_i\right)}\right]$is an uncertain linguistic variable,${\mu }_p\left(x\right)$and$v_p\left(x\right)$represent the degrees of membership and nonmembership of the element$x_i\in X$to$\tilde{s}\left(x_i\right)$, respectively;${\mu }_p\left(x\right)\in [0,1]$,$v_p\left(x\right)\in [0,1]$, and for every$x_i\in X$, we have$0\le u_p^2(x_i)+v_p^2(x_i)\le 1$. The degree of indeterminacy can be computed by${\pi }_p\left(x_i\right)=\sqrt{1-{\left(u_p\left(x_i\right)\right)}^2-{\left(v_p\left(x_i\right)\right)}^2}$. For convenience,$\tilde{\alpha }=\langle \left[s_{\theta (\alpha )},s_{\tau (\alpha )}\right],\left(\mu \left(\alpha \right),v\left(\alpha \right)\right)\rangle$is named as a Pythagorean uncertain linguistic variable (PULV).

Definition 4.

[41] For any two PULVs $\tilde{\alpha }=\langle \left[s_{\theta (\alpha )},s_{\tau (\alpha )}\right],\left(\mu \left(\alpha \right),v\left(\alpha \right)\right)\rangle$and $\tilde{\beta }=\langle \left[s_{\theta (\beta )},s_{\tau (\beta )}\right],\left(\mu (\beta ),v\left(\beta \right)\right)\rangle$, the basic operations of $\tilde{\alpha }$and $\tilde{\beta }$are defined as follows:

(1)

$\tilde{\alpha }\oplus \tilde{\beta }=\langle \left[s_{\theta \left(\alpha \right)+\theta \left(\beta \right)},s_{\tau \left(\alpha \right)+\tau \left(\beta \right)}\right],\left(\sqrt{{\mu }^2\left(\alpha \right)+{\mu }^2\left(\beta \right)-{\mu }^2\left(\alpha \right){\mu }^2\left(\beta \right)},v\left(\alpha \right)v\left(\beta \right)\right)\rangle ;$

(2)

$\tilde{\alpha }\otimes \tilde{\beta }=\langle \left[s_{\theta \left(\alpha \right)\times \tau \left(\beta \right)}\right],\left(\mu \left(\alpha \right)\mu \left(\beta \right),\sqrt{v^2\left(\alpha \right)+v^2\left(\beta \right)-v^2\left(\alpha \right)v^2\left(\beta \right)}\right)\rangle ;$

(3)

$\lambda \tilde{\alpha }=\langle \left[s_{\lambda \times \theta \left(\alpha \right)},s_{\lambda \times \tau \left(\beta \right)}\right],\left(\sqrt{1-{\left(1-{\mu }^2\left(\alpha \right)\right)}^{\lambda }},{\left(v\left(\alpha \right)\right)}^{\lambda }\right)\rangle ;$

(4)

${\tilde{\alpha }}^{\lambda }=\langle \left[s_{{\left(\theta \left(\alpha \right)\right)}^{\lambda }},s_{{\left(\tau (\overline{\beta })\right)}^{\lambda }}\right],\left({\left(\mu \left(\alpha \right)\right)}^{\lambda },\sqrt{1-{\left(1-v^2\left(\alpha \right)\right)}^{\lambda }}\right)\rangle .$

Definition 5.

[42] Let $\tilde{\alpha }=\langle \left[s_{\theta (\alpha )},s_{\tau (\alpha )}\right],\left(\mu \left(\alpha \right),v\left(\alpha \right)\right)\rangle$be a PULV, then the expected value $E\left(\tilde{\alpha }\right)$and the accuracy function $H\left(\tilde{\alpha }\right)$of$\alpha$are defined as follows:

$$E\left(\tilde{\alpha }\right)=s\left(\left(\theta \left(\alpha \right)+\tau \left(\alpha \right)\right)\times \left({\mu }^2\left(\alpha \right)+1-v^2\left(\alpha \right)\right)\right)/4,$$

(2)

$$H\left(\tilde{\alpha }\right)=s_{\left(\left({\mu }^2\left(\alpha \right)+v^2\left(\alpha \right)\right)\times \left(\theta \left(\alpha \right)+\tau \left(\alpha \right)\right)\right)/2}.$$

(3)

Definition 6.

[26] For any two PULVs $\tilde{\alpha }=\langle \left[s_{\theta (\alpha )},s_{\tau (\alpha )}\right],\left(\mu \left(\alpha \right),v\left(\alpha \right)\right)\rangle$and $\tilde{\beta }=\langle \left[s_{\theta (\beta )},s_{\tau (\beta )}\right],\left(\mu (\beta ),v\left(\beta \right)\right)\rangle$, then:

(1)

If$E\left(\tilde{\alpha }\right)>E\left(\tilde{\beta }\right)$, then$\tilde{\alpha }>\tilde{\beta }$;

(2)

If$E\left(\tilde{\alpha }\right)=E\left(\tilde{\beta }\right)$, then:

(a)

If$H\left(\tilde{\alpha }\right)>H\left(\tilde{\beta }\right)$, then$\tilde{\alpha }>\tilde{\beta }$;

(b)

If$H\left(\tilde{\alpha }\right)>H\left(\tilde{\beta }\right)$, then$\tilde{\alpha }=\tilde{\beta }$.

Definition 7.

[43,44] For any two PULVs$\tilde{\alpha }=\langle \left[s_{\theta (\alpha )},s_{\tau (\alpha )}\right],\left(\mu \left(\alpha \right),v\left(\alpha \right)\right)\rangle$and$\tilde{\beta }=\langle \left[s_{\theta (\beta )},s_{\tau (\beta )}\right],\left(\mu (\beta ),v\left(\beta \right)\right)\rangle$, the Hamming distance between$\tilde{\alpha }$and$\tilde{\beta }$can be computed by

$$d\left(\tilde{\alpha },\tilde{\beta }\right)=\frac{1}{4g}\left(\begin{array}{l}\left|\mu {\left(\alpha \right)}^2\theta \left(\alpha \right)-\mu {\left(\beta \right)}^2\theta \left(\beta \right)\right|+\left|v{\left(\alpha \right)}^2\theta \left(\alpha \right)-v{\left(\beta \right)}^2\theta \left(\beta \right)\right|\hfill \\ +\left|\pi {\left(\alpha \right)}^2\theta \left(\alpha \right)-\pi {\left(\beta \right)}^2\theta \left(\beta \right)\right|+\left|\mu {\left(\alpha \right)}^2\tau \left(\alpha \right)-\mu {\left(\beta \right)}^2\tau \left(\beta \right)\right|\hfill \\ +\left|v{\left(\alpha \right)}^2\tau \left(\alpha \right)-v{\left(\beta \right)}^2\tau \left(\beta \right)\right|+\left|\pi {\left(\alpha \right)}^2\tau \left(\alpha \right)-\pi {\left(\beta \right)}^2\tau \left(\beta \right)\right|\hfill \end{array}\right)$$

(4)

Definition 8.

[42] Let${\tilde{\alpha }}_i=\langle \left[s_{\theta \left({\alpha }_i\right)},s_{\tau \left({\alpha }_i\right)}\right],\left(\mu \left({\alpha }_i\right),v\left({\alpha }_i\right)\right)\rangle \left(i=1,2,\dots ,n\right)$be a collection of PULVs. Then, the Pythagorean uncertain linguistic weighted averaging (PULWA) operator is defined as:

$$\begin{array}{l}PULWA\left({\tilde{\alpha }}_1,{\tilde{\alpha }}_2,\dots ,{\tilde{\alpha }}_n\right)=\underset{i=1}{\overset{n}{\oplus }}{w}_i{\tilde{\alpha }}_i\\ =\langle \left[s_{{\displaystyle \sum _{i=1}^n{w}_i}\theta \left({\alpha }_i\right)},s_{{\displaystyle \sum _{i=1}^n{w}_i}\tau \left({\alpha }_i\right)}\right],\left(\sqrt{1-{\displaystyle \prod _{i=1}^n{\left(1-{\mu }^2\left({\alpha }_i\right)\right)}^{w_i}}},{\displaystyle \prod _{i=1}^n{v}^{w_i}}\left({\alpha }_i\right)\right)\rangle ,\end{array}$$

(5)

where$w={(w_1,w_2,\dots ,w_n)}^T$is the weight vector of${\tilde{\alpha }}_i\left(i=1,2,\dots ,n\right)$, with$w_i\ge 0$and${\displaystyle \sum _{i=1}^n{w}_i}=1$.

3.2. The DUCG Model

The DUCG is a probabilistic graphical model originally proposed by Zhang [45] for KRR, which intuitively represents uncertain causalities between variables and performs probabilistic reasoning. A typical DUCG is shown in Figure 1 composed of three sorts of variables [7,46]:

(1)

The root variable B, which can be drawn as a square, represents the root cause;

(2)

The intermediate variable X, which can be drawn as a circle, represents the consequence and can also be a cause variable;

(3)

The variable F, which can be written as $F_{n;i}$ in text, represents a connection event variable and is represented by a certain probability between 0 and 1.

In Figure 1, a one-way arrow from variable B₁ to variable X₂ represents that B₁ is the cause of X₂. Two-way arrows between variables X₂ and X₃ represent mutual relationships between them. $F_{2;1}$ indicates the possibility of variable B₁ causing variable X₂.

Definition 9.

[16] Given a variable$V_i$,$V_i\in \left\{B_i, X_i\right\}$, the DUCG is characterized as:

$$X_n={\displaystyle \sum _i{F}_{n;i}}{V}_i$$

(6)

$$F_{n;i}=\left(r_{n;i}/r_n\right)A_{n;i}$$

(7)

where$r_{n;i}$ is the causal relationship intensity between child event$X_n$ and parent event$V_i$, with$r_{n;i}\ge 0$ and$r_n\equiv {\displaystyle \sum _i{r}_{n;i}}$;$A_{n;i}$is the virtual event representing the causality between the child event$X_n$ and the parent event$V_i$. The connection variable$F_{n;i}$ can be decomposed into the weighted part$\left(r_{n;i}/r_n\right)$ and the virtual event$A_{n;i}$.

Suppose $B_i=\left(B_{i1},B_{i2},\cdots ,B_{ik}\right)$, $X_i=\left(X_{i1},X_{i2},\cdots ,X_{ik}\right)$ and $B_i,X_i\in R^{k\times 1}$, then the DUCG can be further described as:

$$X_{nk}={\displaystyle \sum _i{\displaystyle \sum _j{F}_{nk;ij}}}{V}_{ij}={\displaystyle \sum _i\left(r_{n;i}/r_n\right)}{\displaystyle \sum _j{A}_{nk;ij}}{V}_{ij}$$

(8)

where $V_{ij}$ represents a parent event (state j of parent variable i) of child event $X_{nk}$ (state k of child variable n). $F_{nk;ij}$, the element of the connection matrix $F_{n;i}$, represents the causality between $X_{nk}$ and $V_{ij}$; $A_{nk;ij}$, the element of the virtual event $A_{n;i}$, denotes the virtual event representing the causality between $V_{ij}$ and $X_{nk}$; $r_{n;i}$ denotes the causal relationship intensity between $X_{nk}$ and $V_{ij}$.

Definition 10.

[16] Suppose that$a_{nk;ij}=\mathrm{Pr}\left\{A_{nk;ij}\right\}$and$x_{nk}=\mathrm{Pr}\left\{X_{nk}\right\}$, in which Pr means the probability of the event, and$a_{nk;ij}$represents the occurrence probability between$V_{ij }$and$X_{nk}$, which satisfies${\displaystyle \sum _k{a}_{nk;ij}}=1$, then

$$\begin{array}{l}{x}_{nk}=\mathrm{Pr}\left\{X_{nk}\right\}={\displaystyle \sum _i\left(r_{n;i}/r_n\right)}{\displaystyle \sum _j{a}_{nk;ij}}\mathrm{Pr}\left\{V_{ij}\right\}\\ ={\displaystyle \sum _i\left(r_{n;i}/r_n\right)}{\displaystyle \sum _j{a}_{nk;ij}}{v}_{ij}\\ ={\displaystyle \sum _i\left(r_{n;i}/r_n\right) }{a}_{nk;ij}{v}_{ij}\end{array}$$

(9)

where the knowledge parameters$a_{nk;ij}$and$r_{n;i}$are usually obtained from domain specialists.

Note that any condition of a variable cannot bring about any condition of the same variable [16]. As indicated by the standard of logical inconsistency, an event itself cannot be both root and consequence at the same time; therefore, it breaks the coordinated cyclic cases in DUCG.

4. The Proposed PUL-DUCG Model

4.1. Definition of the PUL-DUCG

In this section, a new type of DUCG based on PULSs and the EDAS method [47] is presented for KRR.

The events are represented by some graphical symbols in the classical DUCG. A square is used to represent the B-type variable, which could only be an independent cause. A circle is used to represent the X-type variable, which can be divided into two types: $X_c$ and $X_r$. The $X_c$-type variable can represent a consequence; the $X_r$-type variable can represent both a consequence and a cause. A certain state j of variable V_i (V_i can be a B- and $X_r$-type), referred to as $V_{ij}$, represents a parent event. A certain state k of variable $X_n$($X_n$ can be $X_c$ and $X_r$ type), referred to as $X_{nk}$, represents a child event. The connection event variable $F_{nk;ij}=\left(r_{n;i}/r_n\right)A_{nk;ij}$, denoted by the arrow $V_{ij}$→$X_{nk}$, represents the cause-and-effect relationship between $V_{ij}$ and $X_{nk}$. The causal mechanism of $V_{ij}$ independently causing $X_{nk}$ is quantified using the virtual event $A_{nk;ij}$, with all knowledge parameters given in a virtual event matrix $A={\left[a_{nk;ij}\right]}_{p\times q}$ for p child events $X_{nk}$ and q states of child events. A typical PUL-DUCG is established as shown in Figure 2. Similar to the normal DUCG model, the variables of the PUL-DUCG model can be expressed as

$$X_n={\displaystyle \sum _i{F}_{n;i}}{V}_i={\displaystyle \sum _i\left(r_{n;i}/r_n\right)}{A}_{n;i}{V}_i.$$

(10)

When there are multiple states for a variable, the model can be expressed as

$$X_{nk}={\displaystyle \sum _i{\displaystyle \sum _j{F}_{nk;ij}}}{V}_{ij}={\displaystyle \sum _i\left(r_{n;i}/r_n\right)}{\displaystyle \sum _j{A}_{nk;ij}}{V}_{ij}.$$

(11)

4.2. Knowledge Acquisition and Representation

In this stage, the PULSs are used to acquire the knowledge parameters of PUL-DUCG (e.g., r_n;I, b_ij and $a_{nk;ij}$). Here, we use the knowledge parameter b_ij as an example to illustrate the knowledge acquisition process. Suppose that the parameter b_ij is evaluated by l experts $\left\{e_1,e_2,\dots ,e_l\right\}$. Each expert is assigned a weight ${\lambda }_k$ with ${\displaystyle \sum _{k=1}^l{\lambda }_k=1}$ because of their different backgrounds and experience. Let ${\tilde{B}}^k={\left[{\tilde{b}}_{ij}{}^k\right]}_{p\times q}$ be the knowledge assessment matrix of the kth expert, where ${\tilde{b}}_{ij}^k=\langle \left[s_{{\theta }_{ij}}^k,s_{{\tau }_{ij}}^k\right],\left({\mu }_{ij}^k,{\nu }_{ij}^k\right)\rangle$ is the linguistic evaluation for the jth state of the root event B_i. Next, the proposed knowledge acquisition is explained in detail.

• Step 1: Construct the collective knowledge assessment matrix $\tilde{B}$.

By using the PULWA operator, all individual knowledge assessment matrixes ${\tilde{B}}^k\left(k=1,2,\dots ,l\right)$ can be aggregated to obtain the collective knowledge assessment matrix $\tilde{B}={\left[{\tilde{b}}_{ij}\right]}_{p\times q}$, where

$${\tilde{b}}_{ij}=PULWA\left({\tilde{b}}_{ij}^1,{\tilde{b}}_{ij}^2,\dots ,{\tilde{b}}_{ij}^l\right)=\underset{k=1}{\overset{l}{\oplus }}{\lambda }_k{\tilde{b}}_{ij}^k.$$

(12)

• Step 2: Obtain the first-order cut-set expression of the target event.

By using Equation (8), the expression of any child event is the combination of parent events and connection events. Then, the expression could be simplified by removing unrelated events from the expression based on the DUCG.

For example, as shown in Figure 2, the child events X₂ and X₃ are expressed as:

$$X_2=F_{2;1}{B}_1+F_{2;3}{X}_3$$

(13)

$$X_3=F_{3;4}{B}_4+F_{3;2}{X}_2$$

(14)

Using Equations (13) and (14), the expression X₂ is expanded as:

$$X_2=F_{2;1}{B}_1+F_{2;3}\left(F_{3;4}{B}_4+F_{3;2}{X}_2\right)=F_{2;1}{B}_1+F_{2;3}{F}_{3;4}{B}_4+F_{2;3}{F}_{3;2}{X}_2$$

(15)

A duplicated variable X₂ appears on the right-hand side of the equation, so X₂ can be removed. When X₂ is deleted, it will disappear as a parent event of X₃. Finally, X₃ has only one parent event B₄. Then, $r_3=r_{3;4}+r_{3;2}$ changes into $r_3^{\left\{2\right\}}=r_{3;4}$, where the superscript index {2} indicates event X₂. Finally, the first-order cut-set expression of the target event X₂ can be expressed as:

$$\begin{array}{l}{X}_2=F_{2;1}{B}_1+F_{2;3}{F}_{3;4}^{\left\{2\right\}}{B}_4\\ =\left(r_{2;1}/r_2\right)A_{2;1}{B}_1+\left(r_{2;3}/r_2\right)A_{2;3}\left(r_{3;4}/r_3^{\left\{2\right\}}\right)A_{3;4}{B}_4\\ =\left(r_{2;1}/r_2\right)A_{2;1}{B}_1+\left(r_{2;3}/r_2\right)A_{2;3}{A}_{3;4}{B}_4\end{array}$$

(16)

4.3. Reasoning Algorithm

In this section, we use the EDAS method to conduct knowledge reasoning in the proposed PUL-DUCG model, which can locate the maximal probability event. The detailed reasoning steps of the PUL-DUCG are expressed below:

• Step 3: Compute the occurrence probabilities of target child events.

In this study, the occurrence probability of a target child event is computed by

$$\mathrm{Pr}\left\{H_{kj}\mid E\right\}=\mathrm{Pr}\left\{H_{kj}E\right\}/\mathrm{Pr}\left\{E\right\}$$

(17)

where H_kj and E represent B- type events and X-type events, respectively.

Suppose that we are interested in the events of $B_{1,2}\mid X_{2,2}{X}_{3,2}$ and $B_{4,2}\mid X_{2,2}{X}_{3,2}$ in Figure 3. Namely, we want to know which is more likely to cause the event of $X_{2,2}{X}_{3,2}$ between $B_{1,2}$ and $B_{4,2}$. Thus, we can compute the probabilities of $B_{1,2}\mid X_{2,2}{X}_{3,2}$ and $B_{4,2}\mid X_{2,2}{X}_{3,2}$ by

$$\mathrm{Pr}\left\{B_{1,2}\mid X_{2,2}{X}_{3,2}\right\}=\mathrm{Pr}\left\{B_{1,2}{X}_{2,2}{X}_{3,2}\right\}/\mathrm{Pr}\left\{X_{2,2}{X}_{3,2}\right\}$$

(18)

• Step4: Obtain the average probability vector $\tilde{P}$.

The average probability for each target child event ${\tilde{p}}_j$ can be derived by

$${\tilde{p}}_j=\frac{{\displaystyle \sum _{i=1}^m{\tilde{p}}_{ij}}}{m}, j=1,2,\dots ,q$$

(19)

As a result, the average probability vector is established as $\tilde{P}=\left[{\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_q\right]$.

• Step5: Determine the positive distance matrix $D^{+}$ and the negative distance matrix $D^{-}$ of probabilities to the average probability vector $\tilde{P}$.

The positive distance matrix $D^{+}={\left[d_{ij}^{+}\right]}_{p\times q}$ can be obtained by

$$d_{ij}^{+}=\{\begin{array}{l}\frac{d\left({\tilde{p}}_{ij},{\tilde{p}}_j\right)}{E\left({\tilde{p}}_j\right)} if {\tilde{p}}_{ij}>{\tilde{p}}_j, \\ 0 if {\tilde{p}}_{ij}\le {\tilde{p}}_j.\end{array}$$

(20)

The negative distance matrix $D^{-}={\left[d_{ij}^{-}\right]}_{p\times q}$ can be derived by

$$d_{ij}^{-}=\{\begin{array}{l}\frac{d\left({\tilde{p}}_{ij},{\tilde{p}}_j\right)}{E\left({\tilde{p}}_j\right)} if {\tilde{p}}_{ij}\ge {\tilde{p}}_j, \\ 0 if {\tilde{p}}_{ij}<{\tilde{p}}_j.\end{array}$$

(21)

• Step 6: Calculate the weighted sum of the probability assessment positive distance and negative distance for each target child event.

The weighted sum of the probability assessment positive distance (SP_i) is calculated by

$$S{P}_i={\displaystyle \sum _{j=1}^q{w}_j{d}_{ij}^{+}}.$$

(22)

In a similar way, the weighted sum of the probability assessment negative distance (SN_i) is obtained by

$$S{N}_i={\displaystyle \sum _{j=1}^q{w}_j{d}_{ij}^{-}}.$$

(23)

• Step 7: Normalize the values of SP_i and SN_i for each event.

The normalized values of SP_i and SN_i for each event can be obtained by

$$\overline{S{P}_i}=\frac{S{P}_i}{\underset{i}{\max }S{P}_i},$$

(24)

$$\overline{S{N}_i}=1-\frac{S{N}_i}{\underset{i}{\max }S{N}_i}.$$

(25)

• Step 8: Obtain the ranking order of target child events.

The appraisal score for the ith event $A{S}_i$ can be determined by

$$A{S}_i=\frac{\left(\overline{S{P}_i}+\overline{S{N}_i}\right)}{2}, i=1,2,\dots ,p$$

(26)

where $0\le A{S}_i\le 1$.

The greater the appraisal score $A{S}_i$, the higher occurrence probability of the target child events. Therefore, all the child events can be ranked based on the descending order of their appraisal score $A{S}_i\left(i=1,2,\dots ,p\right)$.

5. Case Study

In this section, an example about the root cause analysis of the abnormal aluminum electrolysis cell condition [48] is used to demonstrate the feasibility and practicability of the proposed PUL-DUCG model.

5.1. Problem Description

The aluminum electrolysis cell is the most important equipment in aluminum electrolysis. Due to the interference of complex physical and chemical reactions under high temperatures and corrosive conditions, the cell conditions in aluminum electrolysis are quite sophisticated. These conditions can be judged based on the superheat degree [49]. Therefore, the root cause analysis of abnormal aluminum electrolysis cell condition can be transformed into the root cause analysis of an abnormal superheat degree. The superheat degree is mainly determined by technical parameters. If technical parameters are abnormal, it will result in an abnormal superheat degree. Thus, it is a vital significance to recognize the abnormal technical parameter, which leads to an abnormal superheat degree. By consulting experienced technicians and experts, the technical parameters aluminum height (AH), electrolyte level (EL), molecular ratio (MR), heat insulation capacity (IP), electrolyte temperature (ET), voltage fluctuation (VF) and crystallization temperature (CT) are chosen for assessing the abnormal superheat degree. These technical parameters, together with their corresponding roles, are shown in

Table 1. Selected technical parameters.

Technical Parameter	Role Analysis
Aluminum height (AH)	AH can stabilize battery voltage.
Electrolyte level (EL)	EL maintains cell energy stability.
Molecular ratio (MR)	MR will influence the dissolution of alumina in electrolyte.
Heat insulation capacity (IP)	Heat insulation performance of aluminum reduction cell.
Electrolyte temperature (ET)	The appropriate temperature to ensure the normal operation of aluminum batteries.
Voltage fluctuation (VF)	VF is influenced by AH.
Crystallization temperature (CT)	The primary temperature of crystallization.

[48]. Subsequently, the proposed PUL-DUCG is established for root cause analysis of the abnormal superheat degree, as shown in Figure 3.

Five experts were invited to evaluate the knowledge parameters of the PUL-DUCG model using the linguistic term set S = $\{s_0$ = Very low, s₁ = Low, s₂ = Slightly low, s₃ = Normal, s₄ = Slightly high, s₅ = Normal, s₆ = Very high}. These experts from different departments or organizations include an electrical engineer, a chemical production engineer, a nonferrous metal engineer and two professors of aluminum electrolysis. The weights assigned to the five experts are assumed as ${\lambda }_1=0.21,{\lambda }_2=0.19,{\lambda }_3=0.18,{\lambda }_4=0.20,{\lambda }_5=0.22$ in line with their different background knowledge and experience. For example, the experts’ evaluations on parameter b are listed in

Table 2. The knowledge assessment matrix on parameter b_ij by the experts.

	Experts	States
		High	Normal	Low
AH	E1	<[s4, s5], (0.18,0.82)>	<[s2, s3], (0.72,0.28)>	<[s1, s2], (0.00,1.00)>
	E2	<[s5, s6], (0.18,0.82)>	<[s3, s4], (0.65,0.57)>	<[s1, s2], (0.02,0.99)>
	E3	<[s4, s5], (0.18,0.82)>	<[s3, s4], (0.71,0.37)>	<[s2, s3], (0.01,0.98)>
	E4	<[s4, s5], (0.18,0.82)>	<[s3, s4], (0.61,0.39)>	<[s2, s3], (0.00,1.00)>
	E5	<[s4, s5], (0.18,0.82)>	<[s3, s4], (0.71,0.37)>	<[s2, s3], (0.00,1.00)>
EL	E1	<[s3, s4], (0.10,0.70)>	<[s2, s3], (0.75,0.20)>	<[s1, s2], (0.00,1.00)>
	E2	<[s5, s6], (0.27,0.82)>	<[s2, s3], (0.18,0.82)>	<[s2, s3], (0.03,0.89)>
	E3	<[s5, s6], (0.18,0.82)>	<[s1, s2], (0.18,0.82)>	<[s2, s3], (0.01,0.98)>
	E4	<[s5, s6], (0.10,0.80)>	<[s2, s3], (0.18,0.82)>	<[s2, s3], (0.03,0.89)>
	E5	<[s5, s6], (0.18,0.82)>	<[s1, s2], (0.18,0.82)>	<[s2, s3], (0.01,0.98)>
MR	E1	<[s3, s4], (0.20,0.20)>	<[s5, s6], (0.85,0.15)>	<[s1, s2], (0.01,0.98)>
	E2	<[s3, s4], (0.13,0.18)>	<[s3, s4], (0.56,0.31)>	<[s2, s3], (0.01,0.98)>
	E3	<[s4, s5], (0.25,0.28)>	<[s3, s4], (0.67,0.35)>	<[s1, s2], (0.18,0.82)>
	E4	<[s3, s4], (0.30,0.30)>	<[s3, s4], (0.56,0.31)>	<[s2, s3], (0.01,0.98)>
	E5	<[s3, s4], (0.11,0.11)>	<[s2, s3], (0.67,0.35)>	<[s2, s3], (0.11,0.72)>

.

5.2. Implementation and Results

First, the knowledge parameters of the PUL-DUCG are determined according to the proposed knowledge acquisition method. Then, the reasoning algorithm of the proposed PUL-DUCG model is implemented to find out the root cause event with the maximum probability. We use the knowledge parameter a as an example to explain the knowledge acquisition process.

• Step 1: By applying Equation (12), the collective knowledge assessment matrix $\tilde{B}$ is derived as shown in

  Table 3. The collective knowledge assessment matrix $\tilde{B}$.

  States	High	Normal	Low
  AH	<[s4, s5], (0.18,0.82)>	<[s2, s3], (0.72,0.28)>	<[s1, s2], (0.00,1.00)>
  EL	<[s5, s6], (0.18,0.82)>	<[s3, s4], (0.65,0.57)>	<[s1, s2], (0.02,0.99)>
  MR	<[s4, s5], (0.18,0.82)>	<[s3, s4], (0.71,0.37)>	<[s1, s2], (0.01,0.98)>

  .
• Step2: The expression of target child events X_nk under different states could be simplified as

  $$\begin{array}{cc}\hfill X_{4k}& =F_{4k;1}{B}_1=\left(r_{4;1}/r_4\right)a_{4k;1}{B}_1\hfill \\ \hfill X_{5k}& =F_{5k;1}{B}_1+F_{5k;2}{B}_2=\left(r_{5;1}/r_5\right)a_{5k;1}{B}_1+\left(r_{5;2}/r_5\right)a_{5k;2}{B}_2\hfill \\ \hfill X_{6k}& =F_{6k;1}{B}_1+F_{6k;2}{B}_2=\left(r_{6;1}/r_6\right)a_{6k;1}{B}_1+\left(r_{6;2}/r_6\right)a_{6k;2}{B}_2\hfill \\ \hfill X_{7k}& =F_{7k;3}{B}_3=\left(r_{7;3}/r_7\right)a_{7k;3}{B}_3\hfill \end{array}$$
• Step 3: Via Equation (17), the occurrence probabilities of target child events X_nk under different states are calculated as listed in

  Table 4. The occurrence probabilities of target child events.

  Target Child Events	Occurrence Probabilities
  $B_{11}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	<[s1, s2], (0.18,0.82)>
  $B_{12}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	<[s1, s2], (0.70,0.23)>
  $B_{13}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	<[s5, s6], (0.00,1.00)>
  $B_{21}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	<[s3, s4], (0.70,0.10)>
  $B_{22}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	<[s5, s6], (0.70,0.18)>
  $B_{23}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	<[s1, s2], (0.00,1.00)>
  $B_{31}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	<[s5, s6], (0.88,0.01)>
  $B_{32}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	<[s3, s4], (0.79,0.08)>
  $B_{33}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	<[s1, s2], (0.00,1.00)>

  .

For example, when events X₄₂, X₅₂, X₆₁ and X₇₃ occur, we need to acquire the different states of $B_1$, $B_2$ and $B_3$, namely the fuzzy representation of the posterior probability of events $B_{1k}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$, $B_{2k}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$ and $B_{3k}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$, where k = 1, 2, 3.

• Step 4: Via Equation (19), the average probability vector $\tilde{P}$ is established as

  $$\begin{array}{l}\tilde{P}=\{\langle \left[s_{3.118}, s_{4.113}\right],\left(0.696,0.695\right)\rangle , \langle \left[s_{3.050}, s_{4.048}\right],\left(0.675,0.664\right)\rangle ,\\ \langle \left[s_{3.048}, s_{3.975}\right],\left(0.682,0.688\right)\rangle \}.\end{array}$$
• Step 5: With Equations (20) and (21), the positive distance matrix $D^{+}={\left[d_{ij}^{+}\right]}_{3\times 3}$ and the negative distance matrix $D^{-}={\left[d_{ij}^{-}\right]}_{3\times 3}$ of target child events to the average probability vector $\tilde{P}$ are determined as

  $$D^{+}=\left[\begin{array}{ccc}0.248& 0.324& 0.332\\ 0.076& 0.165& 0.142\\ 0.030& 0.056& 0.067\end{array}\right], D^{-}=\left[\begin{array}{ccc}0.248& 0.324& 0.332\\ 0.076& 0.165& 0.142\\ 0.030& 0.056& 0.067\end{array}\right].$$
• Step 6: By using Equations (22)–(23), the weighted sums of SP and SN for each target child event SP_i (i = 1, 2, ..., 9) and SN_i (i = 1, 2, ..., 9) are determined as expressed in

  Table 5. The computation results by the EDAS method.

  Target Child Events	$S{P}_i$	$S{N}_i$	${\overline{SP}}_i$	${\overline{SN}}_i$	$A{S}_i$	Ranking
  $B_{11}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	0.000	0.021	0.000	0.925	0.462	6
  $B_{12}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	0.122	0.000	0.411	1.000	0.705	4
  $B_{13}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	0.000	0.282	0.000	0.000	0.000	9
  $B_{21}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	0.190	0.000	0.640	1.000	0.820	2
  $B_{22}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	0.000	0.074	0.000	0.734	0.367	7
  $B_{23}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	0.027	0.025	0.090	0.909	0.500	5
  $B_{31}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	0.297	0.000	1.000	1.000	1.000	1
  $B_{32}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	0.129	0.000	0.4360	1.000	0.718	3
  $B_{33}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$	0.000	0.173	0.0000	0.384	0.192	8

  .
• Step 7: Via Equations (24) and (25), the normalized values of $\overline{S{P}_i}$(i = 1, 2, ..., 9) and $\overline{S{N}_i}$(i = 1, 2,..., 9) for each target child event are computed and shown in Table 5.
• Step 8: Applying Equation (26), the appraisal scores for nine target child events are displayed in Table 5. According to the descending order of the appraisal scores $A{S}_i\left(i=1,2,\dots ,9\right)$, the ranking of nine target child events is determined as shown in Table 5.

Accordingly, the ranking order of target child events within the circumstance of $X_{43}{X}_{51}{X}_{62}{X}_{71}$ is $B_{31}$ > $B_{21}$ > $B_{32}$ > $B_{12}$ > $B_{23}$ > $B_{11}$ > $B_{22}$ > $B_{33}$ > $B_{13}$. Thus, we can know that event $B_{31}\mid X_{43}{X}_{51}{X}_{62}{X}_{71}$ has the highest occurrence probability, indicating that $B_{31}$ is the most likely cause event in the case of $X_{43}{X}_{51}{X}_{62}{X}_{71}$. This result is consistent with the statistical data obtained from the actual production.

5.3. Comparisons and Discussions

To verify the effectiveness and advantages of the proposed PUL-DUCG model, a comparative analysis with the fuzzy-Bayesian network (FBN) [48], the DUCG [1], the intuitionistic fuzzy DUCG (IFDUCG) [20], and the cloud model DUCG (CDUCG) [21] is conducted in this section. The reasoning results of the listed methods are presented in

Table 6. The reasoning results according to the listed methods.

Group	Results Given by Model					Actual Result	Consistency
	FBN	DUCG	IFDUCG	CDUCG	PUL-DUCG
1	AH Low	SV Low	AH Low	AH Low	AH Low	AH Low	Y
2	AH Low	AH Low	AH Low	AH Low	AH Low	AH Normal	N
3	AH Low	AH Low	AH Low	AH Low	AH Low	AH Low	Y
4	AH Low	AH Low	AH Low	AH Low	AH Low	AH Low	Y
5	AH Low	AH Low	AH Low	AH Low	AH Low	AH Low	Y
6	AH Low	AH Low	AH Low	AH Low	AH Low	AH Low	Y
7	AH Low	AH Low	AH Low	AH Low	AH Low	AH Low	Y
8	AH Low	AH Low	AH Low	AH Low	AH Low	AH Low	Y
9	AH High	AH High	AH High	AH High	AH High	AH High	Y
10	AH High	AH High	AH High	AH High	AH High	AH High	Y
11	AH High	AH High	AH High	AH High	AH High	AH High	Y
12	AH High	AH High	AH High	AH High	AH High	AH High	Y
13	AH High	AH High	AH High	AH High	AH High	AH High	Y
14	AH High	AH High	AH High	AH High	AH High	AH High	Y
15	AH Normal	AH High	AH High	AH High	AH High	AH High	Y
16	AH High	AH High	AH High	AH High	AH High	AH High	Y
17	AH High	AH High	AH High	AH High	AH High	AH High	Y
18	AH Normal	AH High	AH High	AH High	AH High	AH High	Y
19	EL High	EL High	EL High	EL High	EL High	EL High	Y
20	EL High	EL High	EL High	EL High	EL High	EL High	Y
Accuracy	91.10%	93.30%	95.60%	95.60%	95.60%

.

First, the accuracy of PUL-DUCG was the same as the CDUCG, which was 95.6%. This confirms that the proposed PUL-DUCG model is effective for root cause analysis. Furthermore, this method has advantages over the compared methods. Since the DUCG model ignores uncertain information, it is likely to obtain inaccurate reasoning results in complex situations. Compared with the IFDUCG, the proposed model integrates the advantages of PULSs and linguistic variables. Thus, the PUL-DUCG has a more powerful representation ability than the IFDUCG model to handle the fuzziness and vulnerability information in practical situations. In addition, the TOPSIS method in [19] is used to find the root event with a maximum posteriori probability, whereas the proposed model determines the probability ranking of events by the EDAS method. The TOPSIS method did not consider the weights of events, which are not in line with the real situations. Moreover, the CDUCG and the IFDUCG have limitations in handling the inconsistent cognition of experts. Therefore, the knowledge parameters acquired by the proposed PUL-DUCG model are more credible because they are based on a preference modifying-based method.

From Table 6, it can be seen that the accuracy of the FBN method is lower than the DUCG. This is mainly because the FBN can only depict the quantitative evaluation information. Within the circumstance of a qualitative depiction, the FBN is unable to describe fuzzy information. In contrast, the PUL-DUCG model is proposed based on the PULSs, which can effectively describe both qualitative and quantitative information. Moreover, the FBN requires a lot of statistics and empirical information to express the precise conditional probability between events. In comparison, the PUL-DUCG permits inadequate information representation, which could significantly diminish the workload and trouble of setting up an information base.

From the above comparison analysis, it can be found that more accurate and reasonable knowledge acquisition and reasoning results can be determined by the proposed PUL-DUCG model. Compared with the existing DUCG methods, the advantages of the developed PUL-DUCG model are summarized as follows:

1.

The model used the membership degree and non-membership degree of PULSs to express the fuzzy knowledge of experts. The fuzziness of initial information will be retained, which can avoid information loss and distortion in the process of causal analysis. Hence, the proposed model is more capable of representing and reasoning uncertain information.

2.

Based on a correction algorithm for solving the contradictory opinions of specialists, the proposed model can manage the conflicts and inconsistencies among expert evaluations in knowledge parameters. Thus, it can determine the knowledge parameters more precisely when acquiring knowledge.

3.

By means of the modified EDAS method, the proposed model can obtain rational and distinguishing occurrence probabilities and select the maximal probability event. Therefore, it is more accurate and effective in the practical utilization of causal analysis problems.

6. Conclusions

In this paper, a new PUL-DUCG model based on PULSs and the EDAS was developed to enhance the KRR ability of conventional DUCGs. The PULSs are adopted to character the fuzziness and uncertainty of expert knowledge in knowledge representation. A modified EDAS method is utilized to determine the root cause event with the maximal probability. Moreover, this method comprehensively considered the conflicting opinions among experts, which can avoid the limitations of individual expert evaluations and obtain more accurate and reliable knowledge parameters. Finally, a practical aluminum electrolysis case was presented to illustrate the application and advantages of our proposed DUCG model. The results indicate that the PUL-DUCG model is a promising and effective modeling technique for knowledge representation, acquisition and reasoning.

However, the proposed PUL-DUCG model has some disadvantages which can be addressed in further studies. First, the complexity of the proposed algorithm will be increased when a larger number of experts are involved. Thus, further work is needed to solve the problems of conflicting opinions in a large group environment. Second, other knowledge parameters, such as the time factor, can be considered to improve the online analysis ability of DUCG. Another research direction is developing a computer-based application system for the PUL-DUCG, which can perform automatic diagnosis after inputting knowledge parameters obtained from experts.