Crane Operation Hazard Evaluation Based on Z-Number and CPT

Crane operation hazard evaluation plays an important role in theo prevention of accidents and casualties. Therefore, formulating effective measures to ensure the safety of cranes is vital. However, most studies on crane safety evaluation overlook the risk attitudes of decision makers. To overcome the limitations of the current research, this study proposes a novel risk prioritization method based on the Z-number and cumulative prospect theory (CPT) for crane operation hazard evaluation. First, considering the uncertainty and unreliability of decision makers’ assessments, the Z-number is utilized to capture the decision information with regard to crane operation hazards. Second, to consider decision makers’ psychological behavior in crane operation hazard evaluation, CPT is used in the Z-environment to rank the identified hazards. Third, grade discrimination rules and cumulative prospect values are integrated to classify the danger level of a hazard. Finally, a case study involving a crane operation hazard evaluation based on the Z-number and CPT is employed to verify the practicability of the proposed model. Sensitivity analyses and comparative studies are conducted to validate the proposed method.

SECTION I. Introduction

A crane is an essential piece of equipment used for transferring heavy loads and plays a crucial role in industrial operations. Unfortunately, accidents caused by lifting machinery and equipment occur more frequently than those caused by other special equipment. In China, where the number of cranes reached 2.7924 million as of the end of 2022, there were 25 crane accidents in 2022, resulting in 23 fatalities, according to the General Administration of Quality Supervision, Inspection and Quarantine. Unsafe behavior of people is the main cause of crane accidents. In response to this challenge, there has been a growing focus on improving the safety of crane operations over the past two decades [1], [2], [3]. Therefore, it is necessary to analyze crane safety and risk to ensure healthy and sustainable economic development, and crane hazard assessment has become a global issue in the field of safe production.

Research has highlighted the role of linguistic information in the assessment of crane operational risks [4]. Various techniques for uncertain linguistic representations have been applied to safety assessments of cranes. These techniques include trapezoidal fuzzy sets, triangular fuzzy sets, and Z-numbers. Zadeh proposed the idea of Z-number fuzzy theory for evaluating the dependability of fuzzy linguistic variables [5]. This reliability is manifested in the representation of information regarding the decision attribute values. Compared with triangular and trapezoidal fuzzy numbers, the Z-number more effectively captures the vagueness and uncertainty inherent in linguistic information, making it a more advantageous and practical analytical tool for the development of novel uncertain multicriteria decision-making methodologies. The Z-number has been applied to risk assessment in various fields [6], [7], [8], [9]. Consequently, it is preferable to incorporate the Z-number into crane operation hazard evaluation.

However, in the actual risk evaluation of crane operation, decision makers often exhibit bounded rational behavior when facing risks, displaying risk aversion in the case of potential gains and risk-seeking behavior in the case of potential losses [10], [11]. Fortunately, cumulative prospect theory (CPT) provides a better understanding of the psychological behaviors involved in decision making when dealing with gains and losses. It has been applied to crane risk assessments [4], [12]. However, the integration of prospect theory and Z-numbers into the study of crane operation hazard evaluation is limited. This integration provides a deeper understanding of human psychology in decision making and aids the development of more effective strategies for crane safety and risk management. It is possible to improve existing risk-assessment methods and develop more targeted safety training programs for crane operators.

In addition, for crane hazard assessment, the hazards of crane operations are identified. The hazards are then classified according to the risk grade to determine which risk factor has the most significant impact on crane safety. The proposed solution is advantageous in terms of effective crane safety management. However, most of the existing literature merely offers the risk ranking for hazards, rather than their risk grades.

Therefore, considering the bounded rationality of decision makers and the advantages of the Z-number in dealing with decision information, this study proposes a method based on CPT and the Z-number for assessing the operational hazards of cranes. The objective of this research was to develop a crane operation hazard evaluation framework that integrates the Z-number and CPT to aid risk management. To eliminate situations where decision makers have bounded rationality in the risk evaluation of crane operation hazards, CPT integrated with Z-number is adopted to simulate decision makers’ risk preferences. In view of the determination of risk grade with regard to hazard in crane operations, the integration of grade discrimination rules and cumulative prospect values is employed to classify the danger levels of hazards. Thus, the results of crane operation hazard assessment can be used to formulate effective measures for reducing risks. The contributions of this study are outlined as follows:

1. CPT is integrated with Z-numbers to consider the influence of decision makers’ psychological behavior.

2. The proposed method can provide not just the risk ranking for hazards but also their specific risk grades, which is convenient for formulating a plan to prevent casualties and economic losses.

3. An effective tool is introduced for crane practitioners and enterprise managers to cope with the hazards of crane operations.

The remainder of this paper is organized as follows. Section II provides a brief review of the literature. Section III introduces the basic concept of the Z-number. Additionally, CPT is presented. In Section IV, the model for crane operation hazard assessment is presented. Section V offers a case study that illustrates the implementation and effectiveness of the proposed model.. Finally, Section VI presents conclusions.

SECTION II. Literature Review on Crane Safety Evaluation

Research on crane safety evaluation models is mainly categorized into three primary perspectives [13]: probabilistic approaches, machine learning, and fuzzy mathematics, which encompass human factors, crane structure, management, and working environment risk factors, among others.

The first perspective is the probabilistic approach, which involves analyzing the likelihood of different safety hazards and their potential outcomes. For example, to evaluate the probability of a top event and the reliability of overhead cranes, Mohammadi et al. [14]. combined fault tree analysis, fuzzy set theory, Bayesian networks, and Markov chains. Wu et al. [15] presented a quantitative approach for crane accidents using Bayesian networks and an N-K model. Zheng et al. [16] established a fault diagnosis system for a crane spreader based on a fault tree and a Bayesian network, which was based on 13 years of historical fault data collected from their online service. It is crucial to determine the probability of basic events when using a probabilistic approach to analyze crane accidents.

The second perspective involves machine learning, which utilizes artificial intelligence and data analytics to identify safety risks and trends on the basis of historical data and real-time monitoring. This approach allows the development of predictive models to prevent potential accidents. Recently, intelligent assessment methods based on machine learning [17], [18] have been used for crane safety assessments. For instance, Guo et al. [19] developed a fault identification model for low-speed crane hub bearings based on wideband mode decomposition and a backpropagation neural network. However, machine-learning methods require numerous learning samples and are primarily used for fault diagnosis and identification of lifting mechanical parts.

The third perspective is fuzzy mathematics, which involves dealing with the uncertainty and imprecision in safety assessments. In the context of safety assessment, fuzzy mathematics provides a framework for addressing ambiguous or imprecisely defined data and concepts. For example, extension theory [20], variable fuzzy theory [21], [22], set pair analysis [23], and cloud models [13] have been applied for crane safety evaluation. The basic process of this method is to calculate the membership of the evaluation index value relative to the evaluation standard interval and to determine the comprehensive safety level of the crane combined with the index weight. However, sometimes the evaluation index and the evaluation standard are both fuzzy numbers, which is inconvenient for calculating the membership degree. In view of this problem, the crane safety evaluation method for multicriteria decision making (MCDM) has several advantages. For example, as highlighted in previous studies, the MCDM method combined with fuzzy theory has been widely used in crane safety assessments. Li et al. [24] adopted fuzzy confidence theory and grey theory to calculate the grey correlation degrees associated with the failure modes of an overhead crane for metallurgic plants and provided the risk ranking of all fault models. Mandal et al. [25] studied human errors in overhead crane operations, where VIKOR was adopted to rank the priority order, and a trapezoidal fuzzy number (TrFN) was used to describe the weights of the risk factors. Aiming at the risk management problem of quay crane heightening, Zhao et al. [26] established a quay crane failure mode and effect analysis model based on entropy and Grey relational analysis (GRA), in which GRA was applied to rank the risk priority and objective weights were determined by entropy. Das et al. [27] developed a hazard prioritization model for crane operations based on the Z-number-integrated weighted VIKOR technique. Yu et al. [28] proposed a novel safety evaluation method for quayside container cranes using the BWM and Pythagorean hesitant fuzzy VIKOR, providing a valuable tool for port management.

According to the above literature review, MCDM methods integrated with fuzzy theory have made important contributions to improving crane safety assessment; nevertheless, the psychological behavior of decision makers has not been considered. To address this issue, Li and Zhao [12] proposed a model for crane safety assessment that integrates CPT and entropy. Li [4] proposed a model for prioritizing risks associated with human errors in crane operations based on CPT. Thus, in the present study, CPT was utilized to consider the bounded rationality of decision makers in crane hazard assessment. Moreover, there has been a notable absence of the integration of CPT with the Z-number in research on evaluating crane operation hazards. Therefore, it would be beneficial to develop a composite framework that integrates these two methodologies.

SECTION III. Methodology

A. Z-Number

The Z-number, introduced by Zadeh [29], represents both probability and uncertain information. The advantage of the Z-number is that it considers the confidence level of the relevant variable values [30]. It is expressed as Z$=(A$ , B). The first component A is a fuzzy subset of the domain X of the variable Z and is represented as $A=$ {(x, $\mu _{A}(x)\vert x\in X)$ }, where $\mu _{A}(x)$ denotes the membership function of x. Let A be a TrFN expressed as $A=(a_{1}$ , $a_{2}$ , $a_{3}$ , $a_{4}$ ), and its membership function $\mu _{A}(x)$ can be expressed as follows [31]: $$\begin{align*} \mu _{A} (x)= \begin{cases} \displaystyle 0 & {x\le a_{1}} \\ \displaystyle \frac {x-a_{1}}{a_{2} -a_{1}}& a_{1} \le x\le a_{2} \\ \displaystyle 1& a_{2} \le x\le a_{3} \\ \displaystyle \frac {a_{4} -x}{a_{4} -a_{3}}& a_{3} \le x\le a_{4} \\ \displaystyle 0& x\ge a_{4} \end{cases} \tag {1}\end{align*}$$

The second component B is a fuzzy subset of the unit interval indicating the reliability of A. Whereas A shows a restriction on the value of the variable, B reflects certainty or other similar concepts, such as sureness, confidence, the strength of truth, or probability. For instance, a Z-number of ((5, 6, 7, 8), (0.3, 0.5, 0.7)) indicates that the value of the variable (A) is defined by the (5, 6, 7, 8) TrFN, and the associated reliability (B) is described by the (0.3, 0.5, 0.7) triangular fuzzy number (TFN).

In the classical fuzzy MCDM process, the Z-number is usually converted into a regular fuzzy number by combining the first component (A) with the second component (B). Let B be a TFN expressed as $B=(b_{1}$ , $b_{2}$ , $b_{3}$ ), and its membership function $\mu _{B}(x)$ can be expressed as follows [32]: $$\begin{align*} \mu _{B} (x)= \begin{cases} \displaystyle 0 & {x\le b_{1}} \\ \displaystyle \frac {x-b_{1}}{b_{2} -b_{1}}& b_{1} \le x\le b_{2} \\ \displaystyle \frac {b_{3} -x}{b_{3} -b_{2}}& b_{2} \le x\le b_{3} \\ \displaystyle 0& x\ge b_{3} \end{cases} \tag {2}\end{align*}$$

The procedures for transforming Z-numbers into regular fuzzy numbers are as follows [33], [34] and [35].

• Step 1:

  The B component (a TFN) is transformed into a crisp number, and then the weighted Z-number ($Z^{\alpha }$ ) is calculated. $$\begin{align*} \alpha & =\frac {\int {x\mu _{B} (x)dx}}{\int {\mu _{B} (x)dx} }=\frac {1}{3}[(b_{3} -b_{1})+(b_{2} -b_{1})]+b_{1} \tag {3}\\ Z^{\alpha }& =(x,\mu _{Z^{\alpha }} (x)\vert \mu _{Z^{\alpha }} (x)=\alpha \mu _{A} (x),x\in X) \tag {4}\end{align*}$$ Here, $\alpha$ is the confidence coefficient of the decision maker; a higher value indicates that the decision is made more surely.

• Step 2:

  ${Z} ~^{\alpha }$ is converted into a regular fuzzy number $\tilde {Z}$ by using the following equation. $$\begin{align*} \tilde {Z}& =(x,\mu _{Z^{\alpha }} (x)\vert x\in X)=(x,\mu _{\tilde {Z}} (x)\vert \mu _{\tilde {Z}} (x) \\ & =\mu _{Z^{\alpha }} (x/\sqrt \alpha )=\sqrt \alpha \mu _{A} (x),x\in X) \tag {5}\end{align*}$$

If the TrFN A is ($a_{1}$ , $a_{2}$ , $a_{3}$ , $a_{4}$ ), $\tilde {Z}$ is expressed as $$\begin{equation*} \tilde {Z}=(\sqrt \alpha a_{1},\sqrt \alpha a_{2},\sqrt \alpha a_{3},\sqrt \alpha a_{4}) \tag {6}\end{equation*}$$

B. Cumulative Prospect Theory

Kahneman and Tversky [36] developed CPT, which can reflect decision makers’ psychological behavior under risk and uncertainty. It has been successfully applied to fuzzy decision-making issues [37], taking into account the decision maker’s risk attitude. According to CPT, the integrated prospect value is computed as follows. $$\begin{equation*} _{ }V_{i} =\sum \limits _{j=1}^{m} {v_{ij}^{+} \pi ^{+}} (\omega _{j})+\sum \limits _{j=1}^{m} {v_{ij}^{-} \pi ^{-}} (\omega _{j}) \tag {7}\end{equation*}$$ Here, $v_{ij}^{+}$ and $v_{ij}^{-}$ represent the negative and positive prospect values, respectively. $\pi ^{+}(\omega _{j})$ and $\pi ^{-}(\omega _{j})$ represent decision weight functions. The mathematical expressions of $v_{ij}^{+}$ and $v_{ij}^{-}$ are as follows. $$\begin{align*} v_{ij} = \begin{cases} \displaystyle v_{ij}^{+} =(\Delta x_{i})^{\alpha },& \Delta x\ge 0 \\ \displaystyle v_{ij}^{-} =-\theta (-\Delta x_{i})^{\beta },& \Delta x\lt 0 \end{cases} \tag {8}\end{align*}$$ Here, $\triangle x_{i}$ represents the value of gain or loss between $x_{i}$ and the reference point $x_{o}$ . $\alpha$ and $\beta$ are the exponent parameters related to gains and losses. According to Kahneman and Tversky’ study, these parameters are set as $\alpha =\beta =0.88$ , $\theta =2.25$ [38].

The cumulative prospect weights are computed as follows. $$\begin{align*} \pi (\omega _{j})= \begin{cases} \displaystyle \pi ^{+}(\omega _{j})=\frac {\omega _{j}^{\gamma ^{+}}}{[\omega _{j} ^{\gamma ^{+}}+(1-\omega _{j})^{\gamma ^{+}}]^{\frac {1}{^{\gamma ^{+}}}}} \\ \displaystyle \pi ^{-}(\omega _{j})=\frac {\omega _{j}^{\gamma ^{-}}}{[\omega _{j} ^{\gamma ^{-}}+(1-\omega _{j})^{\gamma ^{-}}]^{\frac {1}{^{\gamma ^{-}}}}} \end{cases} \tag {9}\end{align*}$$ Here, $\gamma ^{+}$ and $\gamma ^{-}$ are risk attitude parameters, and $\omega _{j}$ is the combined weight of the risk factors. The parameters $\gamma ^{+}$ and $\gamma ^{-}$ are set as follows: $\gamma ^{+} =0.61$ , $\gamma ^{-}=0.69$ [36].

SECTION IV. Proposed Model for Crane Operation Hazard Evaluation

This study proposes a model for evaluating crane operation hazards, which is based on the Z-number and CPT. Hazard evaluation is regarded as an MCDM problem, and the risk factors for each hazard are considered as evaluation indices. Consequently, the procedure of the proposed model is treated as a ranking process of hazards in crane operations, where expert preferences and risk attitudes are both considered. The steps of the proposed model are depicted in Fig. 1 and explained in detail below.

FIGURE 1.

Flowchart of the proposed model.

A. Collect Experts’ Evaluation Information

1) Determine Rating of Risk Factors for Each Hazard

It is assumed that there are i hazards in crane operations, which are represented by ${}\text{H}_{i}$ ($i=1$ , 2, …, n); risk factors (O, S, and D) represented by $C_{j}(j=1$ , 2, 3); and k assessment experts represented by $E_{k}$ ($k=1$ , …, K). Let $Z_{ij}^{(k)}$ be the evaluation value for ${}\text{H}_{i}$ for index $C_{j}$ associated with $E_{k}$ . The assessment value of each hazard with regard to the risk factors is expressed as $$\begin{equation*} Z_{ij}^{k} =(A_{ij}^{k},B_{ij}^{k}) \tag {10}\end{equation*}$$

The linguistic variables O, S, and D are shown in Table 1 [35], along with the corresponding Z-numbers.

TABLE 1 Linguistic Variables O, S, and D

Using (3)–​(6), $Z_{ij}^{(k)}$ is transformed into the traditional trapezoidal fuzzy number $e_{ij}^{(k)}$ , which is expressed as $$\begin{equation*} e_{ij}^{(k)} =\left ({{e_{ij(1)}^{(k)},e_{ij(2)}^{(k)},e_{ij(3)}^{(k)},e_{ij(4)}^{(k)}}}\right ) \tag {11}\end{equation*}$$

2) Integrate Evaluation Information of Operation Hazard

Each risk factor’s integrated score is expressed as follows. $$\begin{equation*} \textrm {Agg}(e_{ij})=\textrm {Agg}\left ({{e_{ij}^{(1)},e_{ij}^{(2)},\cdots, e_{ij}^{(k)}}}\right ) \tag {12}\end{equation*}$$ Here, Agg($^{\mathbf {.}}$ ) represents the aggregation function, and $$\begin{align*} e_{ij1} & =\min \limits _{k} \left ({{e_{ij(1)}^{(k)}}}\right ) \tag {13}\\ e_{ij2} & =\frac {1}{K}\sum \limits _{k=1}^{K} {\left ({{e_{ij(2)}^{(k)}}}\right )} \tag {14}\\ e_{ij3} & =\frac {1}{K}\sum \limits _{k=1}^{K} {\left ({{e_{ij\left ({{ 3 }}\right )}^{(k)}}}\right )} \tag {15}\\ e_{ij4} & =\max \limits _{k} \left ({{e_{ij(4)}^{(k)}}}\right ). \tag {16}\end{align*}$$

The centroid defuzzification of trapezoidal fuzzy numbers is given by the following equation. $$\begin{align*} e_{ij} & =\frac {1}{3}\left [{{\vphantom {\left. {{-\frac {\left \{{{e_{ij\left ({{ 3 }}\right )} \cdot e_{ij\left ({{ 4 }}\right )} -e_{ij\left ({{ 1 }}\right )} \cdot e_{ij\left ({{ 2 }}\right )}}}\right \}}{\left \{{{\left ({{e_{ij\left ({{ 3 }}\right )} +e_{ij\left ({{ 4 }}\right )}}}\right )-\left ({{e_{ij\left ({{ 1 }}\right )} +e_{ij\left ({{ 2 }}\right )}}}\right )}}\right \}} }}\right ]} e_{ij\left ({{ 1 }}\right )} +e_{ij\left ({{ 2 }}\right )} +e_{ij\left ({{ 3 }}\right )} +e_{ij\left ({{ 4 }}\right )} }}\right. \\ & \qquad \quad \left. {{-\frac {\left \{{{e_{ij\left ({{ 3 }}\right )} \cdot e_{ij\left ({{ 4 }}\right )} -e_{ij\left ({{ 1 }}\right )} \cdot e_{ij\left ({{ 2 }}\right )}}}\right \}}{\left \{{{\left ({{e_{ij\left ({{ 3 }}\right )} +e_{ij\left ({{ 4 }}\right )}}}\right )-\left ({{e_{ij\left ({{ 1 }}\right )} +e_{ij\left ({{ 2 }}\right )}}}\right )}}\right \}} }}\right ] \tag {17}\end{align*}$$

B. Determine Combination Weight

Let $w_{\mathbf {1}}$ represent the subjective weight vector and $w_{\mathbf {2}}$ represent the objective weight vector for O, S, and D. The combination weight is calculated as $$\begin{equation*} {\boldsymbol { w}}=\xi {\boldsymbol { w}}_{1} +(1-\xi ){\boldsymbol { w}}_{2} ;\,\xi \in [{0, 1}] \tag {18}\end{equation*}$$ where $\xi$ represents combination coefficient.

C. Obtain Decision Results Based on CPT

1) Normalize Decision-Making Matrix

To better reflect the gains and losses in prospect theory, it is necessary to normalize the decision-making matrix $R=(r_{ij})$ . The normalized values of the benefit- and cost-related indices are calculated as follows:

For the cost index, $e_{ij}$ can be normalized as follows. $$\begin{align*} r_{i j}=\frac {\max \limits _{j}\left ({{e_{i j}}}\right )-e_{i j}}{\max \limits _{i}\left ({{e_{i j}}}\right )-\min \limits _{i}\left ({{e_{i j}}}\right )},\quad i=1,2, \cdots, n; ~j=1,2,3. \tag {19}\end{align*}$$

For the beneficial index, $e_{ij}$ can be normalized as follows. $$\begin{align*} r_{i j}=\frac {e_{i j}-\min \limits _{j}\left ({{e_{i j}}}\right )}{\max \limits _{j}\left ({{e_{i j}}}\right )-\min \limits _{j}\left ({{e_{i j}}}\right )},\quad i=1,2, \ldots, n; ~j=1,2,3. \tag {20}\end{align*}$$ Here, $\max \limits _{j} (e_{ij} )$ denotes the maximum defuzzification of trapezoidal fuzzy numbers among hazards for the $i^{\mathrm {th}}$ risk factor, and $\min \limits _{j} (e_{ij} )$ denotes the minimum defuzzification of trapezoidal fuzzy numbers among hazards for the $i^{\mathrm {th}}$ risk factor.

The normalized decision-making matrix R’ is expressed as $$\begin{align*} {\boldsymbol { R'}}& =\left [{{r_{ij}}}\right ]=\left [{{\begin{array}{cccccccccccccccccccc} {r_{11}} & \quad {r_{12}} & \quad \cdots & \quad {r_{1j}} \\ {r_{21}} & \quad {r_{22}} & \quad \cdots & \quad {r_{2j}} \\ \vdots & \quad \vdots & \quad \vdots & \quad \vdots \\ {r_{n1}} & \quad {r_{n2}} & \quad \cdots & \quad {r_{nj}} \end{array}}}\right ] \\ & \quad i=1,2,\cdots, n;~j=1,2,\cdots, m \tag {21}\end{align*}$$

In this study, O, S, and D are view as cost indices [39].

2) Determine Positive Ideal and Negative Ideal Solutions

When utilizing prospect theory as a framework for decision-making, decision-makers typically evaluate the potential gains and losses associated with various alternatives in relation to a chosen reference point. This study viewed positive and negative ideal solutions as reference points. The positive ideal solution (PIS) is denoted as S$^{+}=$ ($r_{1}^{+}$ , $r_{2}^{+}$ , …, r$_{m}^{+}$ ), and the negative ideal solution (NIS) is denoted as S$^{-}=$ (${r} _{1}^{-}$ , r$_{2}^{-}$ , …, r$_{m}^{-}$ ).

3) Determine Positive and Negative Prospect Value Matrices

According to the matrix R, the PIS is expressed as S$^{+}=(r_{1}^{+}$ , r$_{2}^{+}$ , …, r$_{m}^{+}$ ), and the NIS is expressed as S$^{-}= (r_{1}^{-}$ , r$_{2}^{-}$ , …, r$_{m}^{-}$ ). The PIS and NIS can be viewed as reference points. Using (8), PIS matrix is obtained as follows. $$\begin{align*} \boldsymbol {V}^{+}=(v_{ij}^{+})_{n\times m} =\left [{{\begin{array}{cccccccccccccccccccc} {v_{11}^{+}} & \quad {v_{12}^{+}} & \quad \cdots & \quad {v_{1m}^{+}} \\ {v_{21}^{+}} & \quad {v_{22}^{+}} & \quad \cdots & \quad {v_{2m}^{+}} \\ \vdots & \quad \vdots & \quad \vdots & \quad \vdots \\ {v_{n1}^{+}} & \quad {v_{n2}^{+}} & \quad \cdots & \quad {v_{nm}^{+}} \end{array}}}\right ]_{ } \tag {22}\end{align*}$$ Here, $i=1$ , 2, …, n; $j=1$ , 2, …, m; $v_{ij}^{+}=(r_{ij}-r_{j}^{-})^{0.88}$ .

The NIS matrix is established as follows. $$\begin{align*} \boldsymbol {V}^{-}=(v_{ij}^{-})_{n\times m} =\left [{{\begin{array}{cccccccccccccccccccc} {v_{11}^{-}} & \quad {v_{12}^{-}} & \quad \cdots & \quad {v_{1m}^{-}} \\ {v_{21}^{-}} & \quad {v_{22}^{-}} & \quad \cdots & \quad {v_{2m}^{-}} \\ \vdots & \quad \vdots & \quad \vdots & \quad \vdots \\ {v_{n1}^{-}} & \quad {v_{n2}^{-}} & \quad \cdots & \quad {v_{nm}^{-}} \end{array}}}\right ]_{ } \tag {23}\end{align*}$$ Here, $i=1$ , 2, …, n; $j=1$ , 2, …, m; $v_{ij}^{-}=-2.25$ ($r_{j}^{+}-r_{ij})^{0.88}$ .

4) Determine Risk Rank

The integrated prospect value $V_{i}$ for ${}\text{H}_{i}$ is calculated using (7). Hazard prioritization is determined by ranking ${}\text{V}_{i}$ .

5) Determine Risk Grade

Based on the cumulative prospect value $V_{i}$ , the discrimination range of risk grade for each hazard can be determined. Subsequently, the risk grade of the hazard associated with crane operations can be assigned.

SECTION V. Case Study

This section presents an example of hazard prioritization, using EOT crane operations [27] as a case study to illustrate the application of the proposed model. Additionally, the effectiveness of the proposed method is verified through sensitivity and comparative analyses. A detailed analysis of this application example is presented below.

A. Collect Experts’ Evaluation Information

1) Determine Rating of Risk Factors Associated with Each Hazard

Cranes are commonly utilized in the construction industry, ports, railroad transportation, and various manufacturing sectors for lifting and moving heavy loads. They have become an integral part of industrial enterprises and can cause accidents. The identification of hazards is a crucial aspect of the proposed model. In this study, an expert team comprising three members, i.e., $DM_{k}$ ($k=1$ , 2, 3), was established to evaluate the potential risk associated with each operation hazards. Experts identified thirteen operational risks labeled as H1 to H13. The identified hazards are presented in Table 2 [27].

TABLE 2 Hazard Descriptions [27]

In this study, the two tuple linguistic terms (Z-number) mentioned in Section III-A were adopted by the same team of three experts to determine the scores for O, S, and D. The fuzzy linguistic assessments of the 13 hazards with regard to O, S, and D are listed in Table 3 [27]. Next, the fuzzy linguistic assessment was transformed into a TrFN. For example, the Z-number for hazard H1 for S given by DM1 is Z$_{\mathrm {H1}}=$ (G, H) = [(7,8,8,9), (0.7,0.9,1)]. Subsequently, the obtained reliability weight was applied to the first component of the Z-number. First, $\alpha$ was calculated as 0.8667 using (3). Then, the obtained value of $\alpha$ was used in (6), yielding Z$_{\mathrm {H1}}=$ (7,8,8,9; 0.8667). Finally, the Z-number was converted into a TrFN as follows: $$\begin{align*} \text {Z}_{\mathrm {H1}}& =(7\times \sqrt {0.8667},8\times \sqrt {0.8667},8\times \sqrt {0.8667}, 9 \\ & \quad \times \sqrt {0.8667} )=(6.6166,7.4476,7.4476,8.3785).\end{align*}$$

TABLE 3 Experts’ Assessment of Each Hazard [27]

Similarly, the experts’ assessment of each hazard in the Z-number environment was converted into a TrFN (see Table 4).

TABLE 4 Values of Risk Factors After Converting into TrFN

2) Integrate Evaluation Information of Operation Hazard

Thereafter, the TrFNs (Table 4) were aggregated using (11)–​(16). The aggregated fuzzy values of the operation hazard were defuzzified into crisp values using (17). For example, the aggregated rating and crisp value for H1 for risk factor S obtained using the rating given by the three decision makers was computed as follows. $$\begin{align*} e_{ij(1)}^{(k)} & =\min \limits _{k} \left ({{e_{ij(1)}^{(k)}}}\right )=\min \limits _{k} (6.5166, 0, 6.5166)=0, \\ e_{ij(2)}^{(k)} & =\frac {1}{K}\sum \limits _{k=1}^{K} {\left ({{e_{ij(2)}^{(k)}}}\right )} =\frac {1}{3}(7.4476\!+\!0\!+\!7.4476)=4.9651, \\ e_{ij(3)}^{(k)} & =\frac {1}{K}\sum \limits _{k=1}^{K} {\left ({{e_{ij\left ({{ 3 }}\right )}^{(k)}}}\right )}= \frac {1}{3}(7.4476+0.3651+7.4476) \\ & =5.0868, \\[-5pt] e_{ij(4)}^{(k)} & =\max \limits _{k} \left ({{e_{ij(4)}^{(k)}}}\right )=\max \limits _{k} \left ({{8.3785, 0.7303, 8.3785}}\right ) \\[-5pt]& =8.3785, \\[-5pt] e_{ij} & =\frac {1}{3}\left [{{\vphantom {\left. {{-\frac {\left \{{{e_{ij\left ({{ 3 }}\right )} \cdot e_{ij\left ({{ 4 }}\right )} -e_{ij\left ({{ 1 }}\right )} \cdot e_{ij\left ({{ 2 }}\right )}}}\right \}}{\left \{{{\left ({{e_{ij\left ({{ 3 }}\right )} +e_{ij\left ({{ 4 }}\right )}}}\right )-\left ({{e_{ij\left ({{ 1 }}\right )} +e_{ij\left ({{ 2 }}\right )}}}\right )}}\right \}} }}\right ]} e_{ij\left ({{ 1 }}\right )} +e_{ij\left ({{ 2 }}\right )} +e_{ij\left ({{ 3 }}\right )} +e_{ij\left ({{ 4 }}\right )} }}\right. \\ & \quad \qquad \left. {{-\frac {\left \{{{e_{ij\left ({{ 3 }}\right )} \cdot e_{ij\left ({{ 4 }}\right )} -e_{ij\left ({{ 1 }}\right )} \cdot e_{ij\left ({{ 2 }}\right )}}}\right \}}{\left \{{{\left ({{e_{ij\left ({{ 3 }}\right )} +e_{ij\left ({{ 4 }}\right )}}}\right )-\left ({{e_{ij\left ({{ 1 }}\right )} +e_{ij\left ({{ 2 }}\right )}}}\right )}}\right \}} }}\right ] \\[3pt] & =\frac {1}{3}\left [{{\vphantom {\left. {{-\frac {\left \{{{5.0868\times 8.3785-0\times 4.9651}}\right \}}{\left \{{{\left ({{5.0868+8.3785}}\right )-\left ({{0+4.9651}}\right )}}\right \}} }}\right ]} 0+4.9651+5.0868+8.3785}}\right. \\ & \qquad \quad \left. {{-\frac {\left \{{{5.0868\times 8.3785-0\times 4.9651}}\right \}}{\left \{{{\left ({{5.0868+8.3785}}\right )-\left ({{0+4.9651}}\right )}}\right \}} }}\right ] \\ & =4.4721\end{align*}$$

Likewise, the aggregated ratings and crisp values of the risk factors (S, O, and D) for the three decision makers (DM1, DM2, and DM3) with regard to the 12 hazards were computed. To rank operational hazards and compare them with prioritization standards (denoted by H-VP-VH, H-P-VH, H-MP-VH, H-M-VH, H-MG-VH, and H-G-VH), the Z-numbers for the fuzzy rates of hazards (Table 1) were transformed into crisp values using the same method. The results of the integration and defuzzification are presented in Table 5.

TABLE 5 Results for Integration and Defuzzification

B. Determine Combination Weights

The subjective, objective, and combined weights of each risk factor were adopted and are presented in Table 6.

TABLE 6 Weights of Risk Factors [27]

C. Obtain Decision Results Based on CPT

1) Normalize Decision-Making Matrix

In Table 5, the decision matrix for hazards is presented as R. The O, S, and D were considered as cost indices. In the next step, the decision matrix of the hazards was normalized using (19). For example, the normalized rating for H1 for risk factor S is given as follows. $$\begin{equation*} r_{ij} =\frac {9.0672-4.4721}{9.0672-0.7647}=0.5535\end{equation*}$$

Normalized hazard values for the remaining risk factors were computed similarly. The normalized fuzzy decision matrix for the hazards is expressed as R’. R and R’ are given as follows. $$\begin{align*} \begin{array}{l} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \textrm {Hazard} \\[-1pt] {\boldsymbol { R}}=\left [{{\begin{array}{cccccccccccccccccccc} {4.4721} & \quad {4.2880} & \quad {7.1959} \\[-1pt] {3.5595} & \quad {2.7006} & \quad {5.7835} \\[-1pt] {7.1959} & \quad {4.7661} & \quad {7.9131} \\[-1pt] {4.7296} & \quad {2.5311} & \quad {6.2572} \\[-1pt] {5.7227} & \quad {3.8176} & \quad {6.0282} \\[-1pt] {5.4193} & \quad {3.8637} & \quad {4.8067} \\[-1pt] {4.9650} & \quad {4.4774} & \quad {6.6837} \\[-1pt] {7.1555} & \quad {6.5166} & \quad {7.9131} \\[-1pt] {5.9699} & \quad {4.8579} & \quad {7.6507} \\[-1pt] {5.6862} & \quad {3.7055} & \quad {5.9792} \\[-1pt] {8.3821} & \quad {6.3265} & \quad {8.3174} \\[-1pt] {6.1459} & \quad {6.7429} & \quad {8.1901} \\[-1pt] {5.6697} & \quad {5.0331} & \quad {5.7835} \\[-1pt] {0.7647} & \quad {0.7647} & \quad {0.7647} \\[-1pt] {1.9664} & \quad {1.9664} & \quad {1.9664} \\[-1pt] {3.4412} & \quad {3.4412} & \quad {3.4412} \\[-1pt] {4.9160} & \quad {4.9160} & \quad {4.9160} \\[-1pt] {6.3907} & \quad {6.3907} & \quad {6.3907} \\[-1pt] {7.8655} & \quad {7.8655} & \quad {7.8655} \\[-1pt] {9.0672} & \quad {9.0672} & \quad {9.0672} \\[-1pt] \end{array}}}\right ]{\begin{array}{cccccccccccccccccccc} {\textrm {H1}} \\[-1pt] {\textrm {H2}} \\[-1pt] {\textrm {H3}} \\[-1pt] {\textrm {H4}} \\[-1pt] {\textrm {H5}} \\[-1pt] {\textrm {H6}} \\[-1pt] {\textrm {H7}} \\[-1pt] {\textrm {H8}} \\[-1pt] {\textrm {H9}} \\[-1pt] {\textrm {H10}} \\[-1pt] {\textrm {H11}} \\[-1pt] {\textrm {H12}} \\[-1pt] {\textrm {H13}} \\[-1pt] {\textrm {H-VP-VH}} \\[-1pt] {\textrm {H-P-VH}} \\[-1pt] {\textrm {H-MP-VH}} \\[-1pt] {\textrm {H-M-VH}} \\[-1pt] {\textrm {H-MG-VH}} \\[-1pt] {\textrm {H-G-VH}} \\[-1pt] {\textrm {H-VG-VH}} \\[-1pt] \end{array}} \\[-1pt] \end{array} \\ \begin{array}{l} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \textrm {Hazard} \\[-1pt] {\boldsymbol { R'}}=\left [{{\begin{array}{cccccccccccccccccccc} {\textrm {0.5535}} & \quad {\textrm {0.5756}} & \quad {\textrm {0.2254}} \\[-1pt] {\textrm {0.6634}} & \quad {\textrm {0.7668}} & \quad {\textrm {0.3955}} \\[-1pt] {\textrm {0.2254}} & \quad {\textrm {0.5181}} & \quad {\textrm {0.1390}} \\[-1pt] {\textrm {0.5224}} & \quad {\textrm {0.7872}} & \quad {\textrm {0.3385}} \\[-1pt] {\textrm {0.4028}} & \quad {\textrm {0.6323}} & \quad {\textrm {0.3660}} \\[-1pt] {\textrm {0.4394}} & \quad {\textrm {0.6267}} & \quad {\textrm {0.5132}} \\[-1pt] {\textrm {0.4941}} & \quad {\textrm {0.5528}} & \quad {\textrm {0.2871}} \\[-1pt] {\textrm {0.2303}} & \quad {\textrm {0.3072}} & \quad {\textrm {0.1390}} \\[-1pt] {\textrm {0.3731}} & \quad {\textrm {0.5070}} & \quad {\textrm {0.1706}} \\[-1pt] {\textrm {0.4072}} & \quad {\textrm {0.6458}} & \quad {\textrm {0.3719}} \\[-1pt] {\textrm {0.0825}} & \quad {\textrm {0.3301}} & \quad {\textrm {0.0903}} \\[-1pt] {\textrm {0.3519}} & \quad {\textrm {0.2800}} & \quad {\textrm {0.1056}} \\[-1pt] {\textrm {0.4092}} & \quad {\textrm {0.4859}} & \quad {\textrm {0.3955}} \\[-1pt] {\textrm {1.0000}} & \quad {\textrm {1.0000}} & \quad {\textrm {1.0000}} \\[-1pt] {\textrm {0.8553}} & \quad {\textrm {0.8553}} & \quad {\textrm {0.8553}} \\[-1pt] {\textrm {0.6776}} & \quad {\textrm {0.6776}} & \quad {\textrm {0.6776}} \\[-1pt] {\textrm {0.5000}} & \quad {\textrm {0.5000}} & \quad {\textrm {0.5000}} \\[-1pt] {\textrm {0.3224}} & \quad {\textrm {0.3224}} & \quad {\textrm {0.3224}} \\[-1pt] {\textrm {0.1447}} & \quad {\textrm {0.1447}} & \quad {\textrm {0.1447}} \\[-1pt] {0} & \quad {0} & \quad {0} \\[-1pt] \end{array}}}\right ]{\begin{array}{cccccccccccccccccccc} {\textrm {H1}} \\[-1pt] {\textrm {H2}} \\[-1pt] {\textrm {H3}} \\[-1pt] {\textrm {H4}} \\[-1pt] {\textrm {H5}} \\[-1pt] {\textrm {H6}} \\[-1pt] {\textrm {H7}} \\[-1pt] {\textrm {H8}} \\[-1pt] {\textrm {H9}} \\[-1pt] {\textrm {H10}} \\[-1pt] {\textrm {H11}} \\[-1pt] {\textrm {H12}} \\[-1pt] {\textrm {H13}} \\[-1pt] {\textrm {H-VP-VH}} \\[-1pt] {\textrm {H-P-VH}} \\[-1pt] {\textrm {H-MP-VH}} \\[-1pt] {\textrm {H-M-VH}} \\[-1pt] {\textrm {H-MG-VH}} \\[-1pt] {\textrm {H-G-VH}} \\[-1pt] {\textrm {H-VG-VH}} \\[-1pt] \end{array}} \\[-1pt] \end{array}\end{align*}$$

2) Determine Positive Ideal and Negative Ideal Solution

From matrix R’, the PIS and NIS are determined as follows: $$\begin{equation*} {\boldsymbol { S}}^{\mathbf {+}}= (1, 1, 1),\quad {\boldsymbol { S}}^{-}= (0, 0, 0).\end{equation*}$$

3) Determine Positive and Negative Prospect Value Matrices

Using (22) and (23), the positive and negative prospect value matrices were obtained. The results were as follows. $$\begin{align*} & \begin{array}{l} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \textrm {Hazard} \\[1pt] {\boldsymbol { V}}^{+}=\left [{{\begin{array}{cccccccccccccccccccc} {\textrm {0.5942}} & \quad {\textrm {0.6151}} & \quad {\textrm {0.2695}} \\[1pt] {\textrm {0.6969}} & \quad {\textrm {0.7917}} & \quad {\textrm {0.4421}} \\[1pt] {\textrm {0.2695}} & \quad {\textrm {0.5606}} & \quad {\textrm {0.1762}} \\[1pt] {\textrm {0.5648}} & \quad {\textrm {0.8102}} & \quad {\textrm {0.3854}} \\[1pt] {\textrm {0.4493}} & \quad {\textrm {0.6680}} & \quad {\textrm {0.4129}} \\[1pt] {\textrm {0.4849}} & \quad {\textrm {0.6629}} & \quad {\textrm {0.5559}} \\[1pt] {\textrm {0.5377}} & \quad {\textrm {0.5936}} & \quad {\textrm {0.3335}} \\[1pt] {\textrm {0.2746}} & \quad {\textrm {0.3539}} & \quad {\textrm {0.1762}} \\[1pt] {\textrm {0.4199}} & \quad {\textrm {0.5500}} & \quad {\textrm {0.2109}} \\[1pt] {\textrm {0.4536}} & \quad {\textrm {0.6806}} & \quad {\textrm {0.4188}} \\[1pt] {\textrm {0.1113}} & \quad {\textrm {0.3771}} & \quad {\textrm {0.1205}} \\[1pt] {\textrm {0.3988}} & \quad {\textrm {0.3262}} & \quad {\textrm {0.1384}} \\[1pt] {\textrm {0.4555}} & \quad {\textrm {0.5299}} & \quad {\textrm {0.4421}} \\[1pt] {\textrm {1.0000}} & \quad {\textrm {1.0000}} & \quad {\textrm {1.0000}} \\[1pt] {\textrm {0.8715}} & \quad {\textrm {0.8715}} & \quad {\textrm {0.8715}} \\[1pt] {\textrm {0.7100}} & \quad {\textrm {0.7100}} & \quad {\textrm {0.7100}} \\[1pt] {\textrm {0.5434}} & \quad {\textrm {0.5434}} & \quad {\textrm {0.5434}} \\[1pt] {\textrm {0.3693}} & \quad {\textrm {0.3693}} & \quad {\textrm {0.3693}} \\[1pt] {\textrm {0.1825}} & \quad {\textrm {0.1825}} & \quad {\textrm {0.1825}} \\[1pt] {\textrm {0.0000}} & \quad {\textrm {0.0000}} & \quad {\textrm {0.0000}} \\[1pt] \end{array}}}\right ]{\begin{array}{cccccccccccccccccccc} {\textrm {H1}} \\[1pt] {\textrm {H2}} \\[1pt] {\textrm {H3}} \\[1pt] {\textrm {H4}} \\[1pt] {\textrm {H5}} \\[1pt] {\textrm {H6}} \\[1pt] {\textrm {H7}} \\[1pt] {\textrm {H8}} \\[1pt] {\textrm {H9}} \\[1pt] {\textrm {H10}} \\[1pt] {\textrm {H11}} \\[1pt] {\textrm {H12}} \\[1pt] {\textrm {H13}} \\[1pt] {\textrm {H-VP-VH}} \\[1pt] {\textrm {H-P-VH}} \\[1pt] {\textrm {H-MP-VH}} \\[1pt] {\textrm {H-M-VH}} \\[1pt] {\textrm {H-MG-VH}} \\[1pt] {\textrm {H-G-VH}} \\[1pt] {\textrm {H-VG-VH}} \\[1pt] \end{array}} \\[1pt] \end{array} \\ & \begin{array}{l} \qquad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \textrm {Hazard} \\ {\boldsymbol { V}}^{-}=\left [{{\begin{array}{cccccccccccccccccccc} {-\textrm {1.1068}} & \quad {-\textrm {1.0583}} & \quad {-\textrm {1.7971}~} \\ {-\textrm {0.8631}} & \quad {-\textrm {0.6248}} & \quad {-\textrm {1.4448}~} \\ {-\textrm {1.7971}} & \quad {-\textrm {1.1837}} & \quad {-\textrm {1.9723}~} \\ {-\textrm {1.1742}} & \quad {-\textrm {0.5764}} & \quad {-\textrm {1.5641}~} \\ {-\textrm {1.4294}} & \quad {-\textrm {0.9329}} & \quad {-\textrm {1.5066}~} \\ {-\textrm {1.3521}} & \quad {-\textrm {0.9453}} & \quad {-\textrm {1.1942}~} \\ {-\textrm {1.2353}} & \quad {-\textrm {1.1082}} & \quad {-\textrm {1.6705}~} \\ {-\textrm {1.7872}} & \quad {-\textrm {1.6290}} & \quad {-\textrm {1.9723}~} \\ {-\textrm {1.4919}} & \quad {-\textrm {1.2075}} & \quad {-\textrm {1.9085}~} \\ {-\textrm {1.4201}} & \quad {-\textrm {0.9027}} & \quad {-\textrm {1.4943}~} \\ {-\textrm {2.0858}} & \quad {-\textrm {1.5815}} & \quad {-\textrm {2.0702}~} \\ {-\textrm {1.5362}} & \quad {-\textrm {1.6852}} & \quad {-\textrm {2.0394}~} \\ {-\textrm {1.4159}} & \quad {-\textrm {1.2529}} & \quad {-\textrm {1.4448}~} \\ {\textrm {0.0000}} & \quad {\textrm {0.0000}} & \quad {\textrm {0.0000}} \\ {-\textrm {0.4107}} & \quad {-\textrm {0.4107}} & \quad {-\textrm {0.4107}~} \\ {-\textrm {0.8309}} & \quad {-\textrm {0.8309}} & \quad {-\textrm {0.8309}~} \\ {-\textrm {1.2226}} & \quad {-\textrm {1.2226}} & \quad {-\textrm {1.2226}~} \\ {-\textrm {1.5976}} & \quad {-\textrm {1.5976}} & \quad {-\textrm {1.5976}~} \\ {-\textrm {1.9608}} & \quad {-\textrm {1.9608}} & \quad {-\textrm {1.9608}~} \\ {-\textrm {2.2500}} & \quad {-\textrm {2.2500}} & \quad {-\textrm {2.2500}~} \end{array}}}\right ]{\begin{array}{cccccccccccccccccccc} {\textrm {H1}} \\ {\textrm {H2}} \\ {\textrm {H3}} \\ {\textrm {H4}} \\ {\textrm {H5}} \\ {\textrm {H6}} \\ {\textrm {H7}} \\ {\textrm {H8}} \\ {\textrm {H9}} \\ {\textrm {H10}} \\ {\textrm {H11}} \\ {\textrm {H12}} \\ {\textrm {H13}} \\ {\textrm {H-VP-VH}} \\ {\textrm {H-P-VH}} \\ {\textrm {H-MP-VH}} \\ {\textrm {H-M-VH}} \\ {\textrm {H-MG-VH}} \\ {\textrm {H-G-VH}} \\ {\textrm {H-VG-VH}} \end{array}} \end{array}\end{align*}$$

4) Determine Risk Rank

To rank the crane operation hazard priorities, the overall prospect values of each hazard were computed. First, we combined the risk attitude coefficients and obtained comprehensive weights (see Table 6) using (9). The cumulative prospect weights were computed as follows: $$\begin{align*} \pi ^{+}(\omega _{j})& =(0.347850, 0.326169, 0.333673), \\ \pi ^{-}(\omega _{j})& =(0.364142, 0.337239, 0.346544).\end{align*}$$

By integrating the prospect value matrices and cumulative prospect weights, the overall prospect values were calculated using (1). The results are presented in Table 7.

TABLE 7 Ranking Results Based on Different Methods

Comparing the cumulative prospect values (${}\text{V}_{i}$ , see Table 7) of each hazard, revealed the following risk priority ranking: H11 > H8 > H12 > H3 > H9 > H13 > H7 > H1 > H5 > H10 > H6 > H4 > H2. Thus, H11 has the highest risk priority and should be given the highest priority for future corrective actions, followed by H8, H12, H3, H9, H13, H7, H1, H5, H10, H6, H4, and H2.

5) Determining Risk Grade

Furthermore, the risk grade discrimination range for each hazard based on the cumulative prospect value ${}\text{V}_{i}$ (see Table 7) was determined as follows.

• Grade I:

  $\mathrm {V}(\mathrm {H}-\mathrm {VG}-\mathrm {VH})=-2.3578 \leq \mathrm {V}_{i}\lt \mathrm {V}(\mathrm {H}-\mathrm {G}-\mathrm {VH})= -1.8708$ .

• Grade II:

  $\mathrm {V}(\mathrm {H}-\mathrm {G}-\mathrm {VH})=-1.8708 \leq \mathrm {V}_{i}\lt \mathrm {V}(\mathrm {H}-\mathrm {MG}-\mathrm {VH})= -1.3020 \text {.}$

• Grade III:

  $\mathrm {V}(\mathrm {H}-\mathrm {MG}-\mathrm {VH})=-1.3020 \leq \mathrm {V}_{i}\lt \mathrm {V}(\mathrm {H}-\mathrm {M}-\mathrm {VH})= -0.7336$ .

• Grade IV:

  $\mathrm {V}(\mathrm {H}-\mathrm {M}-\mathrm {VH})=-0.7336 \leq \mathrm {V}_{i}\lt \mathrm {V}(\mathrm {H}-\mathrm {MP}-\mathrm {VH}) =-0.1552 \text {.}$

• Grade V:

  $\mathrm {V}(\mathrm {H}-\mathrm {MP}-\mathrm {VH})=-0.1552 \leq \mathrm {V}_{i}\lt \mathrm {V}(\mathrm {H} -\mathrm {P}-\mathrm {VH})=0.4478$ .

• Grade VI:

  $\mathrm {V}(\mathrm {H}-\mathrm {P}-\mathrm {VH})=0.4478 \leq \mathrm {V}_{i}\lt \mathrm {V}(\mathrm {H}-\mathrm {VP}-\mathrm {VH}) =1.0077$ .

From Fig. 2, Table 7, and the risk-grade discrimination range, the risk grade of each hazard can be determined. For example, the cumulative prospect value for H1 was –0.8855, which was between –0.7336 and –1.3020. This shows that the risk grade for H1 is classified III. Similarly, the other 12 risk grades with regard to crane operation hazards were determined, and the results are presented in the last column of Table 7.

FIGURE 2.

Range of risk grades.

6) Sensitivity Analysis

To verify the robustness of the proposed model and study the influence of the associated parameters on crane operation hazard risk prioritization, a sensitivity analysis was conducted. According to CPT, the risk preferences for determining the risk prioritization are reflected by the parameters $\theta$ , $\alpha$ , and $\beta$ in (8). The value of $\theta$ (risk aversion coefficient) varies from 1 to 10. Meanwhile, $\alpha$ and $\beta$ range from 0 to 1.

The trends in the prospect values of the 13 crane operation hazards computed for various sensitivity analysis scenarios under different risk preferences are presented in Figs. 3–​5. In Fig. 3, with an increase in a, the prospect values of the crane operation hazards decrease to varying degrees. In Fig. 4, as $\beta$ increases, the prospect values of crane operation hazards increase to varying degrees. As shown in Fig. 5, as $\theta$ fluctuates, the prospect values of crane operation hazards decrease to varying degrees. Clearly, the hazard rankings are not sensitive to the values of $\alpha$ , $\beta$ and $\theta$ , indicating that the hazard prioritization is not influenced in this case.

FIGURE 3.

Prospect values with respect to a.

FIGURE 4.

Prospect values with respect to $\beta$ .

FIGURE 5.

Prospect values with respect to $\theta$ .

The ranking results obtained for the dynamic scenarios are presented in Table 8. The results in column 7 of Table 7 are consistent with the calculation results in Table 8, indicating that the proposed framework is robust and effective.

TABLE 8 Results for Risk Rank and Grade in Dynamic Scenarios

7) Comparison Analysis

In order to validate the effectiveness of the proposed model, it was compared with Souvik Das’ method (Z-number and VIKOR). The results for the risk ranking based on the different methods are presented in Table 7. As shown, the importance ranking for Souvik Das’s method is H11 > H12 > H8 > H3 > H9 > H1 > H3 > H7 > H5 > H6 > H10 > H4 > H2. This study also ranked 13 hazards using Souvik Das’s method. The risk priorities were H11, H12, H8, H3, H9, H1, H3, H7, H5, H10, H6, H4, and H2 (see column 5 of Table 7). Comparing the cumulative prospect values ${}\text{V}_{i}$ (see column 6 of Table 7) of the hazards reveals that the risk priority ranking is H11 > H8 > H12 > H3 > H9 > H13 > H7 > H1 > H5 > H10 > H6 > H4 > H2. The ranking orders of the different methods are not the same, which can be explained as follows: Z-VIKOR ignores the risk attitudes and risk preferences of decision makers, whereas the proposed model Z-CPT considers the psychological behaviors of decision makers, such as loss aversion and reference dependence. Therefore, the risk priority ranking obtained by the proposed model, i.e., H11 > H8 > H12 > H3 > H9 > H13 > H7 > H1 > H5 > H10 > H6 > H4 > H2, is more reasonable than that obtained using the Z- VIKOR method [27].

H11 had the highest risk priority and should be given the highest priority for future corrective actions, and the ranking orders obtained using Z-VIKOR were approximately consistent with the outcomes of the Z-CPT model. Crane operation hazards H11, H8, H12, and H3, which were classified as grade II in the Z-CPT model, were ranked in the first four positions using the Souvik Das method. Crane operation hazards H4 and H2 belonging to grade IV in the Z-CPT model were ranked in the last two positions using the Souvik Das method. Crane operation hazards H9, H13, H7, H1, and H5 belonging to grade III in the Z-CPT model were ranked in the middle positions using the Souvik–Das method. These results indicate the practicality and effectiveness of the Z-CPT method. The Z-VIKOR method [27] solely offers a ranking of crane operation hazards, while the Z-CPT method is capable of categorizing these hazards. Hence, the Z-CPT model outperforms the Z-VIKOR model and provides theoretical support for implementing preventative measures against risk.

SECTION VI. Conclusion

The focus of this paper is on determining the severity of risks associated with crane operations using uncertain linguistic evaluations. To solve this problem, we propose a Z-CPT model that considers the psychological behavior of experts during the decision-making process.

First, Z-numbers are applied to capture the fuzzy information for S, O, and D. The Z-number represents how people usually express their decisions. The integration of subjective and objective weights is determined by a combined weighting method. Risk ranking and hazard grades are determined according to the overall prospect theory. A case study was conducted to demonstrate the efficiency of the proposed model. We found that the proposed method can provide more practical results. The stability of our method was verified through a sensitivity analysis. Our findings indicate that the proposed method can be employed for the risk evaluation of critical equipment in the railway and aerospace fields, among others.

The practical contributions of this study are as follows:

1. The proposed model can be adopted for analyzing risk by considering the psychological behavior of decision makers.

2. The proposed model not only provides the risk ranking of crane operation hazards but also classifies their risk grades.

3. The Z-CPT method is employed for hazard evaluation of crane operations. This method can provide theoretical support for controlling hazards, and the evaluation results can reduce costs due to misjudgment.

4. The proposed model can be employed to rank risks and determine risk grades for other critical equipment, such as railways, airplanes, and nuclear power facilities.

However, the proposed method has limitations. To facilitate the sensitivity analysis, we did not consider the combined effects of the parameters ($\alpha$ , $\beta$ , and $\theta$ ) on the crane operation hazard risk prioritization and grade. In addition, changes in the reference points were not considered. Moreover, the entropy weight variation with the addition of prioritization standards (denoted as H-VP-VH, H-P-VH, H-MP-VH, H-M-VH, H-MG-VH, and H-G-VH) in the decision matrix was not included in this study. Nevertheless, it is worth mentioning that converting Z-numbers into classical fuzzy numbers leads to a loss of original information, and more effective methods should be adopted to reduce the information loss of each hazard in future research.