A hybrid fuzzy FTA-AHP method for risk decision-making in accident emergency response of work system

Abstract

This paper focuses on a risk decision making problem in high-risk work system. A special risk decision-making problem with dynamic and risky characteristics needs to be considered in work system which involves many complex factors in safety risks and control measurements. Furthermore, taking into account the inherent imprecision and uncertainty of the available data, a method based on Fuzzy Fault Tree Analysis (FFTA) is proposed to solve the problem. Unlike conventional fuzzy fault tree analysis, the fuzzy analytic hierarchy process is applied to eliminate ambiguity and subjectivity in determining the weights of criteria. Furthermore, a more simple and effective method for ranking fuzzy numbers is introduced to determine the relationships among the fuzzy results. Based on these methods, a hybrid approach is proposed for risk decision-making in accident emergency response of work system. Finally, a case study on the accident of crane hitting the high bent is given to demonstrate the proposed the validity and objectivity of the method. Compared with other available methods in the literature, the results reported here suggest that the hybrid method is more reasonable under the conditions of limited decision data and possible evolvement of emergency states.

Keywords

Emergency response risk decision-making work system fuzzy fault tree analysis analytic hierarchy process

1 Introduction

As high-risk work system processes are complex and dangerous, emergencies appeared frequently. Decision-makers need to make a reasonable emergency response to minimize the negative effects. However, researches on high-risk work systems at present mainly focused on analysis of safety risk factors and risk assessment [1 –3]. There are a few achievements on emergency decision in the process of high-risk work system. As safety risks and control measurements in high-risk work system involve many complex factors, the decision-making problems in emergency response have dynamic and risky characteristics. Thus a special risk decision-making problem which has characteristics including dynamic evolvement process of emergency, multiple states, and impact of emergency alternatives on the emergency states needs to be considered.

The existing studies have made significant contributions to decision analysis in emergency response. For example, the IF-PROMETHEE method proposed in [4] takes into account intuitionistic fuzzy preferences and intuitionistic fuzzy weights. It is more applicable in handling uncertain information. But it is difficult to meet the timeliness requirements in emergency management [5]. The HF-VIKOR proposed in [6] is effective in solving multi-criteria decision making problems with hesitant preference information and conflicting criteria. Although it is an effective tool for multiple criteria decision making (MCDM) problems, some errors would occur in multi-criteria optimization calculation [7]. The method based on cumulative prospect theory proposed in [8] is incorporated into decision analysis in emergency response so as to consider decision-maker’s psychological behavior. However, this method is not applicable to the emergency response problem with uncertain criterion values or uncertain reference points. In [9], the satisfaction based interactive hesitant fuzzy decision making method is introduced for hesitant fuzzy multi-criteria decision making with incomplete weight information. The IF-AHP method proposed in [10] improved the inconsistent intuitionistic preference relation without the participation of the decision maker. A procedure for group decision making with IFPRs based on the multiplicative consistency of IFPRs is proposed in [11]. In [12], the hesitant fuzzy linguistic method is applied in multi-criteria decision making due to its distinguished power and efficiency in representing uncertainty and vagueness within the process of decision making. In [13], a new distance-based multi-criteria group decision-making methodology is proposed to support multi-person emergency decision problems. In order to consider uncertainties arising from data, modeling and human judgment in a fuzzy environment, a fuzzy optimization method based on the concept of ideal and anti-ideal points is presented in [14]. In [15], the Markov decision process is used to treat the marine oil spill problem. Multiple criteria decision-making and decision tree analysis are combined to propose an integrated analytical framework for effective management of project risks in [16]. There are also other methods [17 –21], but they could not solve the decision-making problem with dynamic and risky characteristics in addition to Fault Tree Analysis (FTA).

FTA can be used in the analysis of risk decision-making problem with above characteristics [22]. It has been used [23 –26] as a powerful technique in risk analysis studies. However, because of ambiguity and imprecision of some basic events in conventional FTA, the failure rates of the system components are considered as crisp values which could not reflect real situation of system [27, 28]. Based on the fuzzy set theory, the fuzzy fault tree analysis (FFTA) may be a way to overcome these difficulties and limitations in FTA by treating the probabilities of basic events as triangular fuzzy numbers (TFNs). The introduction of fuzzy theory and technology has important theoretical significance and urgently need for practical engineering [29]. Moreover, there are two key issues that could not be ignored in FFTA.

On the one hand, when fuzzy arithmetic is incorporated into FTA, the final results are also fuzzy numbers. How to determine the relationships among them is a problem must be solved. Many investigators have proposed fuzzy ranking methods to compare fuzzy numbers. For example, WABL method is used to convert a fuzzy set to a point-wise and crisp value in [30]. The left and right fuzzy ranking method proposed is used to convert fuzzy number into fuzzy possibility score in [31]. A new method for ranking intuitionistic fuzzy values (IFVs) is proposed to solve multi-attribute decision making problem by using similarity measure and accuracy degree of IFVs in [32]. However, converting a fuzzy number into a real number would lead to a loss of necessary information for decision-making [33]. There are other ideas which can be used to avoid this defect. In order to rank IFVs, scholars have proposed different procedures which have been made in-depth comparison in [34]. Compared with them, the method introduced in [35] has strengths in some aspects. The only condition we need is that some special points of the fuzzy number, contains the interval points of the support and the middle points which membership degree are 1. It is easy to obtain the condition. Compared with other exist ranking methods, we do not need to compute membership functions of the fuzzy number or know some other relative information about them which are hard to get. Therefore, this method is more effective and simple. In this article, this method is applied to solve the problem mentioned above.

On the other hand, emergency decision problem in work system is MCDM problem because it involves many complex factors. How to determine the weights of criteria more objectively is an un-ignored problem for multiple criteria decision making problems. At present, various methods, Such as TOPSIS method [36], Dempster–Shafer evidence theory [37], the analytic hierarchy process (AHP) [38] have been applied to solve this problem. Among these methods, AHP firstly developed by Saaty is one of the most effective decision-making tools for selecting best alternatives according to a set of multi-criteria [38]. An advantage of AHP over other MADM methods is that it is designed to incorporate tangible as well as nontangible factors. This advantage is especially considerable where the subjective judgments constitute an important part of the decision process [39].

However, when expert preferences are affected by uncertainty and imprecision, it is not very reasonable to use definite and precise numbers to represent lin-guistic judgments [40]. Based on fuzzy set theory, which was introduced in [41], numerous researchers have incorporated fuzzy theory into AHP. The first FAHP method which is presented in [42] com-pares fuzzy ratios which are described by triangular fuzzy numbers. Then the method was extended to trapezoidal fuzzy judgments and hierarchical analysis in [43]. Here, Chang’s extent analysis to handle FAHP described in [44] is applied to convert fuzzy weights to exact numbers because of its better time complexity.

Based on above analysis, we established a fuzzy fault tree model which is in line with the trend of the evolution of the accident by analyzing event and properties. According to the fuzzy probabilities of basic events judged by experts under different emergency alternatives, we calculated evolved fuzzy probabilities of different event states under different emergency alternatives. Then determined the weights of attributes by FAHP and obtained comprehensive evaluation value of each emergency alternative. Finally, we can obtain a ranking of feasible emergency alternatives by comparing the triangular fuzzy numbers. At the end of the article, a case study on the accident of crane hitting the high bent is given to illustrate feasibility and validity of proposed method.

The remainder of the paper is organized as follows: Section 2 briefly introduces FTA and Fuzzy theory which are used to establish fault tree model. The FAHP is given to solve two key issues in this section. At the end of this section, the procedure of the hybrid approach is proposed. In Section 3, a case study on the accident of crane hitting the high bent is given. In Section 4, sensitivity analysis is conducted to demonstrate the proposed method’s effectiveness. The paper ends with some concluding remarks in Section 5.

2 Methodology

2.1 Problem description

According to distinct characteristics of risk decision making problem in work system mentioned, we briefly depict the risk decision-making problem as a decision tree in Fig. 1. The mathematical symbols in Fig. 1 are illustrated in Table 2 and the explanation for the decision tree is given as follows.

As shown in Fig. 1, if ith emergency alternative is implemented, either the emergency will evolve into scenario S ₁ from S ₀ with probability ${\tilde{P}}_{i 1}$ or end with probability $1 - {\tilde{P}}_{i 1}$ . Furthermore, if the emergency evolves into scenario S ₁, there are also two possible results, either the emergency will evolve into scenario S ₂ from S ₁ with probability ${\tilde{P}}_{i 2}$ or end with probability ${\tilde{P}}_{i 1} - {\tilde{P}}_{i 2}$ . Similarly, if the emergency evolves into scenario S _n-1, there are also two possible results, either the emergency will evolve into scenario S _n from S _n-1 with probability ${\tilde{P}}_{in}$ or end with probability ${\tilde{P}}_{i (n - 1)} - {\tilde{P}}_{in}$ . If the emergency ends with scenario S _j, then damage result vector is $({\tilde{l}}_{j 1}, {\tilde{l}}_{j 2}, \dots, {\tilde{l}}_{jt})$ , the expected evaluation under ith emergency alternative is $({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{it})$ . Then the expected result including the cost and loss with respect to each emergency alternative can be obtained as $({\tilde{r}}_{1}, {\tilde{r}}_{2}, \dots, {\tilde{r}}_{m})$ . To solve the risk decision-making problem described above, FAHP is applied to calculate probability ${\tilde{P}}_{ij}$ .

2.2 Conventional fault tree analysis

In FTA, we can draw the cause of the accident and the accident form into a tree structure diagram using the logical volume or logical sum through detailed analysis of the entire system. All accident modes of fault tree (FT) top accident and basic events can be identified through qualitative analysis. Then the FT is constructed to depict the logical relations among conditions and factors resulting in the evolvement of emergency. To construct the FT, Boolean operators are used to connect the top, intermediate and basic events together. Commonly used Boolean operators are AND and OR are defined as follows: $P_{AND} = AND (P_{1}, P_{2}, \dots, P_{n}) = \prod_{i = 1}^{n} P_{i}$ (1) $P_{OR} = OR (P_{1}, P_{2}, \dots, P_{n}) = 1 - \prod_{i = 1}^{n} (1 - P_{i})$ (2)

In the constructed FT, the top event which is the most undesirable event state is denoted as □. The intermediate events which are the cause of the top event are denoted as same as the top event. The basic events which are the primary emergency events and the cause of the top event or intermediate events are denoted as ∘. Logic gate ‘OR’ is denoted as .Logic gate ‘AND’ is denoted as .

According to the FT, the probabilities of different states under different emergency alternatives can be obtained through quantitative analysis. Furthermore, the overall ranking value of each alternative is calculated according to the probabilities of different states and the loss of each state. Finally, a ranking of the feasible emergency alternatives can be determined.

In order to eliminate ambiguity and imprecision, the uncertainties are represented by TFNs in the nextSection.

2.3 Fuzzy analytic hierarchy process

Definition 1. [45] We define $\tilde{M}$ on R to be a TFN if its membership function $μ_{\tilde{M}} (x)$ is equal to $μ_{\tilde{M}} (x) = {\begin{matrix} 0, x < a \\ \frac{x - a}{m - a}, a \leq x \leq m \\ \frac{b - x}{b - m}, m \leq x \leq b \\ 0, x > b \end{matrix}$ (3) a and b stand for lower and upper value of the support of $\tilde{M}$ respectively, and m for the barycenter value. Assume ${\tilde{p}}_{1}$ and ${\tilde{p}}_{2}$ are respectively expressed as (a ₁, m ₁, b ₁) and (a ₂, m ₂, b ₂), the algebraic algorithms about the TFN ${\tilde{p}}_{1}$ and ${\tilde{p}}_{2}$ are showed as follows: ${\tilde{p}}_{1} + {\tilde{p}}_{2} = (a_{1} + a_{2}, m_{1} + m_{2}, b_{1} + b_{2})$ (4) ${\tilde{p}}_{1} - {\tilde{p}}_{2} = (a_{1} - a_{2}, m_{1} - m_{2}, b_{1} - b_{2})$ (5) ${\tilde{p}}_{1} \times {\tilde{p}}_{2} = (a_{1} a_{2}, m_{1} m_{2}, b_{1} b_{2})$ (6) ${\tilde{p}}_{1} \div {\tilde{p}}_{2} = (a_{1} / b_{2}, m_{1} / m_{2}, b_{1} / a_{2})$ (7) $1 - {\tilde{p}}_{1} = (1 - b_{1}, 1 - m_{1}, 1 - a_{1})$ (8) ${\tilde{p}}_{1}^{- 1} = (1 / b_{1}, 1 / m_{1}, 1 / a_{1})$ (9) $λ \times {\tilde{p}}_{1} = (λ a_{1}, λ m_{1}, λ b_{1}) λ > 0 and λ \in R$ (10)

Fuzzy Operator of TFNs showed as follows: ${\tilde{P}}_{AND} = \prod_{i = 1}^{n} {\tilde{P}}_{i} = (\prod_{i = 1}^{n} a_{i}, \prod_{i = 1}^{n} m_{i}, \prod_{i = 1}^{n} b_{i})$ (11)

$\begin{matrix} {\tilde{P}}_{OR} & = & 1 - \prod_{i = 1}^{n} (1 - {\tilde{P}}_{i}) = (1 - \prod_{i = 1}^{n} (1 - a_{i}), \\ 1 - \prod_{i = 1}^{n} (1 - m_{i}), 1 - \prod_{i = 1}^{n} (1 - b_{i})) \end{matrix}$ (12)

To get the overall evaluation value of emergency alternatives, the weights of cost and criteria must be determined. The FAHP method is used to solve this problem. The process of applying FAHP can be divided into three steps. First, establish a hierarchical structure. Second, construct the pairwise comparison matrix. The conversion scale used to convert linguistic judgments in TFNs is shown in Table 1. Third, calculate the weights of cost and criteria according to the pairwise comparison matrix by using the Chang’s extent analysis method on fuzzy AHP mentioned in [41]. Chang’s method is described as follows:

Assume m extent analysis values for each object can be obtained, with sign ${\tilde{M}}_{i}^{j}$ . The value of fuzzy synthetic extent with respect to the ${\tilde{M}}_{i}^{j}$ is defined as: ${\tilde{S}}_{i} = \sum_{j = 1}^{m} {\tilde{M}}_{i}^{j} \otimes {[\sum_{i = 1}^{n} \sum_{j = 1}^{m} {\tilde{M}}_{i}^{j}]}^{- 1}$ (13) $\sum_{j = 1}^{m} {\tilde{M}}_{i}^{j} = (\sum_{j = 1}^{m} a_{j}, \sum_{j = 1}^{m} m_{j}, \sum_{j = 1}^{m} b_{j})$ (14) $\sum_{i = 1}^{n} \sum_{j = 1}^{m} {\tilde{M}}_{i}^{j} = (\sum_{i = 1}^{n} a_{i}, \sum_{i = 1}^{n} m_{i}, \sum_{i = 1}^{n} b_{i})$ (15) ${[\sum_{i = 1}^{n} \sum_{j = 1}^{m} {\tilde{M}}_{i}^{j}]}^{- 1} = (\frac{1}{\sum_{i = 1}^{n} b_{i}}, \frac{1}{\sum_{i = 1}^{n} m_{i}}, \frac{1}{\sum_{i = 1}^{n} a_{i}})$ (16)

Assume that ${\tilde{M}}_{1} = (a_{1}, m_{1}, b_{1})$ and ${\tilde{M}}_{2} = (a_{2}, m_{2}, b_{2})$ , the degree of possibility of ${\tilde{M}}_{2} \geq {\tilde{M}}_{1}$ is defined as: $V ({\tilde{M}}_{2} \geq {\tilde{M}}_{1}) = {\begin{matrix} 1, if m_{2} \geq m_{1} \\ 0, if a_{1} \geq b_{2} \\ \frac{a_{1} - b_{2}}{(m_{2} - b_{2}) - (m_{1} - a_{1})}, else \end{matrix}$ (17)

$\begin{matrix} V (\tilde{M} \geq {\tilde{M}}_{1}, \tilde{M_{2}}, \dots, {\tilde{M}}_{k}) = min V (\tilde{M} \geq {\tilde{M}}_{i}), \\ i = 1, 2, \dots, k \end{matrix}$ (18) $d^{'} (A_{i}) = min V (S_{i} \geq S_{k})$ (19)

Then the weight vector is given by $W^{'} = {(d^{'} (A_{1}), d^{'} (A_{2}), \dots, d^{'} (A_{n}))}^{T}$ (20)

Via 0-1 normalization, the normalized weight vectors are $W = {(d (A_{1}), d (A_{2}), \dots, d (A_{n}))}^{T}$ (21)

Because of the obtained expected overall evaluation of each emergency alternative is a triangular fuzzy number, we use the Endpoint Method mentioned in [35] to compare the fuzzy numbers. The Endpoint Method is described as follows:

Assume two TFNs ${\tilde{M}}_{1}$ and ${\tilde{M}}_{2}$ , $P ({\tilde{M}}_{1}, {\tilde{M}}_{2})$ denotes the total possibility degree that ${\tilde{M}}_{1}$ is smaller than ${\tilde{M}}_{2}$ , $R_{1} ({\tilde{M}}_{1}, {\tilde{M}}_{2})$ denotes the part possibility degree that ${\tilde{M}}_{1}$ is smaller than ${\tilde{M}}_{2}$ for the interval points, $R_{2} ({\tilde{M}}_{1}, {\tilde{M}}_{2})$ denotes the part possibility degree that ${\tilde{M}}_{1}$ is smaller than ${\tilde{M}}_{2}$ for the middle points. Then we can define $P ({\tilde{M}}_{1}, {\tilde{M}}_{2})$ as following: $\begin{matrix} P ({\tilde{M}}_{1}, {\tilde{M}}_{2}) \\ = {\begin{matrix} R_{1} ({\tilde{M}}_{1}, {\tilde{M}}_{2}), if {\tilde{M}}_{1} and {\tilde{M}}_{2} are nonoverlapped \\ \int_{0}^{1} [r (R_{1} ({\tilde{M}}_{1}, {\tilde{M}}_{2}) + R_{2} ({\tilde{M}}_{1}, {\tilde{M}}_{2}))] dr, else \end{matrix} \end{matrix}$ (22) $R_{1} ({\tilde{M}}_{1}, {\tilde{M}}_{2}) = min {1, \frac{max {0, b_{2} - a_{1}}}{(b_{1} - a_{1}) + (b_{2} - a_{2})}}$ (23) $R_{2} ({\tilde{M}}_{1}, {\tilde{M}}_{2}) = \frac{m_{2}}{m_{1} + m_{2}}$ (24)

Calculate the value of $P ({\tilde{r}}_{i}, {\tilde{r}}_{j})$ , then count number of $P ({\tilde{r}}_{i}, {\tilde{r}}_{j}) < 0.5$ . The larger the number is, the better the emergency alternative.

2.4 The procedure of the hybrid approach

Based on above analysis, the solution procedure of risk decision-making problem in emergency response is shown in Fig. 2. Major steps are shown as follows:

Determine feasible emergency alternatives, potential states of emergency event, decision criteria, and cost of alternative.

Determine the fault tree top event, intermediate events and basic events, plot the logical tree diagram. Estimate probabilities of the basic events.

Combining logic tree diagram, we can obtain the fuzzy probability ${\tilde{P}}_{ij}$ from fuzzy probability of basic events by applying these arithmetic rules of TFN described in Section 2.3.

In order to have same range Standard of different attributes, cost and loss values need to be normalized to the context of 0-1. The calculation formulas of ${\tilde{o}}_{i}^{,}$ and ${\tilde{l}}_{jk}^{,}$ can be respectively expressed by

{\tilde{o}}_{i}^{,} = \frac{{\tilde{o}}^{max} {\tilde{o}}_{i}}{{\tilde{o}}^{max} {\tilde{o}}^{min}}, i = 1, 2, \dots, m

(25)

$\begin{matrix} {\tilde{l}}_{jk}^{,} & = & \frac{{\tilde{l}}_{k}^{max} - {\tilde{l}}_{jk}}{{\tilde{l}}_{k}^{max} - {\tilde{l}}_{k}^{min}}, \\ j & = & 0, 1, 2, \dots, n; k = 1, 2, \dots, t \end{matrix}$ (26)

e) Obtain expected evaluation of each emergency alternative under each criteria by ${\tilde{r}}_{ik} = (1 - {\tilde{P}}_{i 1}) {\tilde{l}}_{0 k}^{,} + \sum_{j = 1}^{n - 1} ({\tilde{P}}_{ij} - {\tilde{P}}_{i (j + 1)}) {\tilde{l}}_{jk}^{,} + {\tilde{P}}_{in} {\tilde{l}}_{nk}^{,}$ (27)

f) Calculate the exact value of the weight vector using the method mentioned in Section 2.3. Then calculate the expected overall evaluation result of each emergency alternative ${\tilde{r}}_{i}$ . ${\tilde{r}}_{i}$ could be expressed by ${\tilde{r}}_{i} = w^{cos t} {\tilde{o}}_{i}^{,} + \sum_{k = 1}^{t} w_{k} {\tilde{r}}_{ik} i = 1, 2, \dots, m$ (28)

g) Sort of ${\tilde{r}}_{i}$ using Endpoint Method mentioned in Section 2.3. Finally, we can obtain a ranking of feasible emergency alternatives.

3 Case study

3.1 Construct the FT for the case

High bent is used in high-rise buildings to facilitate high operations [46]. Collapse and shock events occur frequently. This case study accident of cranehitting high bent in a hydropower construction project, the proposed method will be applied. The following three possible evolved states of emergency event can be summed up according to similar cross-job accidents:

S ₁ : Few safety facilities of high bent were damaged. Such as minor damage of local safety net, one or two fences are broken, dangerous source was not corrected in time.

S ₂ : Safety facilities of high bent were widely damaged, safety net shed, protection wall was collapsed, fence was extensively broken, accident of against objects or fall occurrences as larger sources of danger were not effectively corrected.

S ₃ : Safety facilities and infrastructure of high bent fundamentally damaged, major hazard was not timely and effectively corrected, resulting in high trestle overall collapsed and heavy casualties.

We summarized following emergency measures according to above forecast states and similar cross-job accidents. To estimate cost of corresponding alternative, we collect 34 cases of emergency drills report and consider the cost as calculated mathematical expectation value. The unit of cost is ten thousand RMB.

A ₁ : Do not take any emergency measures. The cost is estimated as ${\tilde{o}}_{1} = (0, 0, 0)$ .

A ₂ : Repair local damaged safety facilities of high bent. Such as damaged safety net, safety railing. The cost is estimated as ${\tilde{o}}_{2} = (1, 2, 4)$ .

A ₃ : Repair damaged safety facilities of high bent in a large scale. Rescue fall staffs, strengthen management of the team. The cost of alternative is estimated as ${\tilde{o}}_{3} = (8, 12, 15)$ .

A ₄ : Stop the construction of high bent. Repair safety facilities and infrastructure of high bent. Rescue fall staffs, strengthen management. The cost is ${\tilde{o}}_{4} = (20, 25, 30)$ .

A ₅ : Stop the construction of high bent and lifting operations. Repair safety facilities and infrastructure of high bent in a large scale. Rescue fall staffs, strengthen management and training. The cost is ${\tilde{o}}_{5} = (40, 45, 50)$ .

A ₆ : Stop the construction of high bent and lifting operations. Rescue fall staffs. Comprehensively repair safety facilities and infrastructure of high bent. Comprehensively repair lifting. Strengthen management and training. Alert person near the accident, evacuate all workers nearby. The cost is ${\tilde{o}}_{6} = (90, 100, 110)$ .

The thought of “people-oriented” must be followed when considering measures of different emergency alternatives. So casualties should be considered firstly. Secondly, the damage of construction equipment, the stagnation of the construction progress and the impact of social pressure should be considered in project management. Thus the following criteria need to be considered in emergency decision-making process:

C ₁ : Casualties in construction Accident;

C ₂ : Construction progress stalled;

C ₃ : Construction equipment and facilitiesdamaged;

C ₄ : The pressure of negative social impact.

According to the description of different states and the severity of the evolution, the FT of this case can be constructed as shown in Fig. 3. The meanings of corresponding events are shown in Table 2.

3.2 Calculate probability of state transition ${\tilde{P}}_{ij}$

The following equations can be obtained through Fig. 3, if we select emergency alternative A _i: $S_{i 1} = X_{i 1} + X_{i 2}$ (29) $S_{i 2} = S_{i 1} M_{i 2} X_{i 6} = (X_{i 1} + X_{i 2}) (X_{i 3} + X_{i 4}) X_{i 5} X_{i 6}$ (30)

$\begin{matrix} S_{i 3} = (X_{i 1} + X_{i 2}) (X_{i 3} + X_{i 4}) \\ X_{i 5} X_{i 6} (X_{i 7} + X_{i 8}) X_{i 9} X_{i 10} \end{matrix}$ (31)

By analyzing 186 cases of related accident, probabilities of basic events determined by indirect elicitation technique and Delphi method from expert’s knowledge and experience are shown in Table 3. We have analyzed the 186 cases in our previous work [47].

Then we can calculate the probability of state transition according to the Equations. (11), (12), (29), (30) and (31). Where the probability are denoted by ${\tilde{P}}_{ij}$ .

3.3 Normalized the vector of cost and loss

According to Equation (25), normalized cost of emergency alternative was obtained. Combined withrelevant case studies, the loss extent of accident states are estimated by experts. They are shown in Table 4. The values in column C ₁ represents the number of falling person in state S _j, values in column C ₂, C ₃ and C ₄ represents severity of loss in the scale of 0–100. S ₀ represents initial state. Through Equation (26), we obtained normalized severity of loss in different states. Then the expected result of criteria C _k was obtained byEquation (27). The weights of costs and criteria should be considered to get overall rating of the evaluation value.

3.4 Determine the weights

To determine the weights of each criterion, three experts of emergency management were invited to participate in the decision-making process. Pairwise compare the importance of costs and each criterion by Delphi methods, Table 5 was obtained.

According to Equations (13) to (21), normalized vector of weight was obtained: W = (0.0801, 0.4647, 0.1027, 0.2606, 0.0919) ^T. For the problem under study, ${\tilde{λ}}_{max} = (3.14, 5.12, 7.86)$ , convert it into crisp value using centroid defuzzification method called center of gravity [48], then λ _max = 5.37, C . I . =0.0925, R . I . =1.12, C . R . =0.083 < 0.1. Hence, the consistency of matrix is acceptable. Then according to Equation (28), the expected overall evaluation of emergency alternatives was obtained: $\begin{matrix} {\tilde{r}}_{1} & = & (0 . 8106, 0 . 9545, 1 . 1343), \\ {\tilde{r}}_{2} & = & (0 . 8334, 0 . 9615, 1 . 1175), \\ {\tilde{r}}_{3} & = & (0 . 8452, 0 . 9658, 1 . 1047), \end{matrix}$ $\begin{matrix} {\tilde{r}}_{4} & = & (0 . 8454, 0 . 9620, 1 . 0913), \\ {\tilde{r}}_{5} & = & (0 . 8210, 0 . 9449, 1 . 0784), \\ {\tilde{r}}_{6} & = & (0 . 8103, 0 . 9067, 1 . 0091) \end{matrix}$

3.5 Comprehensive ranking of emergency alternatives

In this Section, we solved the problem of comparing triangular fuzzy numbers.

The total extent of the possibility of ${\tilde{r}}_{i} < {\tilde{r}}_{j}$ can be calculated according to the Equation (22) to (24). The results are shown in Table 6.

According to what was mentioned in 2.3, $P ({\tilde{r}}_{i}, {\tilde{r}}_{j})$ denotes the total possibility degree that ${\tilde{r}}_{i}$ is smaller than ${\tilde{r}}_{j}$ . If $P ({\tilde{r}}_{i}, {\tilde{r}}_{j}) > 0.5$ , then the ordering of ${\tilde{r}}_{i}$ and ${\tilde{r}}_{j}$ is defined as ${\tilde{r}}_{i} < {\tilde{r}}_{j}$ . From this table, we can get the following information: there are three numbers bigger than ${\tilde{r}}_{1}$ ; ${\tilde{r}}_{2}$ is only smaller than ${\tilde{r}}_{3}$ and bigger than the other four numbers; ${\tilde{r}}_{3}$ is bigger than the other five numbers; only two numbers smaller than ${\tilde{r}}_{4}$ ; ${\tilde{r}}_{5}$ is only bigger than ${\tilde{r}}_{6}$ and smaller than the other four numbers; ${\tilde{r}}_{6}$ is the smallest one. The count number of ${\tilde{r}}_{i} > {\tilde{r}}_{j} (where j \neq i)$ is shown in Table 7.

The larger the number is, the better the emergency alternative. So the sort of emergency alternatives is expressed as ${\tilde{r}}_{3} ≻ {\tilde{r}}_{2} ≻ {\tilde{r}}_{1} ≻ {\tilde{r}}_{4} ≻ {\tilde{r}}_{5} ≻ {\tilde{r}}_{6}$ , namely A ₃ ≻ A ₂ ≻ A ₁ ≻ A ₄ ≻ A ₅ ≻ A ₆.

4 Discussion

In the following, a sensitivity analysis is conducted to demonstrate the proposed method’s effectiveness. In the FFTA, the top-event probability provides only an idea about system conditions. There is a need to evaluate the percentage contribution of each basic event that leads to the top event and unavailability and failure frequencies of the basic events by sensitivity analysis [27]. In the sensitivity analysis, three important decisions for the system are identified which include the weakest link of the system, a better design alternative, and how to evaluate the effect of the adopted solution on system safety [49]. In this study, the sensitivity analysis for the work system alternatives is carried out on the basis of investigating the fuzzy weighted index (FWI) [49, 50] which is used to evaluate the contribution of the each basic event.

The FWI is a form of improvement index when the failure probability of a basic event is defined through fuzzy functions. It is estimated by evaluating the impact on the tree when eliminating each basic event from the tree. According to the description of the risk decision-making problem in emergency response mentioned above, the nth state of emergency event which is denoted as S _n is the top event of this system. Suppose ${\tilde{P}}_{T}$ is the failure possibility of the top event, ${\tilde{P}}_{Tk}$ denotes the failure possibility of the top event when kth basic event is eliminated from the tree. Then the FWI of a basic event $D ({\tilde{P}}_{T}, {\tilde{P}}_{Tk})$ is determined by the ranking value of ${\tilde{P}}_{T} Θ {\tilde{P}}_{Tk}$ : $D ({\tilde{P}}_{T}, {\tilde{P}}_{Tk}) = R ({\tilde{P}}_{T} Θ {\tilde{P}}_{Tk})$ (32)

In these calculations, the largest $D ({\tilde{P}}_{T}, {\tilde{P}}_{Tk})$ corresponds to the most important basic event. In this study, the triangular fuzzy number is adopted as similar as [50]. Thus, the method mentioned by them can be used to calculate $D ({\tilde{P}}_{T}, {\tilde{P}}_{Tk})$ .

Consider two triangular fuzzy number ${\tilde{P}}_{T} (a_{1}, b_{1}, c_{1})$ and ${\tilde{P}}_{Tk} (a_{2}, b_{2}, c_{2})$ , $δ ({\tilde{P}}_{T}, {\tilde{P}}_{Tk})$ denotes the distance between ${\tilde{P}}_{T}$ and ${\tilde{P}}_{Tk}$ which can be represented by the FWI. $δ ({\tilde{P}}_{T}, {\tilde{P}}_{Tk})$ is defined as follows:

$\begin{matrix} D ({\tilde{P}}_{T}, {\tilde{P}}_{Tk}) = δ ({\tilde{P}}_{T}, {\tilde{P}}_{Tk}) \\ = 0.5 {max (| a_{1} - a_{2} |, | b_{1} - b_{2} |) + | c_{1} - c_{2} |} \end{matrix}$ (33)

Thus the ranking of the FWI will determine the fuzzy importance of basic events with respect to each pre-selected feasible emergency alternative. The results can help to make suitable design modifications for the system and demonstrate effectiveness of the proposed method.

A sensitivity analysis technique has been utilized to sort the importance of basic events by evaluating the percentage contribution of them that lead to the top event. According to the Equation (33), the FWI can be seen in Table 8.

According to Table 8, it can be concluded that there are four most critical basic events such as X ₅, X ₆, X ₉ and X ₁₀. The sort of all the basic events in terms of importance is expressed as (X ₅, X ₆, X ₉, X ₁₀) ≻ X ₈ ≻ X ₃ ≻ X ₁ ≻ X ₂ ≻ X ₇ ≻ X ₄. According to the practical operation for emergency alternatives and the sort of the basic events by importance, we demonstrate effectiveness of the proposed method as follows:

If implemented emergency alternative A ₃, repair damaged safety facilities of high bent in a large scale, rescue fall staffs, strengthen management of the team. According to the barycenter value of ${\tilde{P}}_{ij}$ , the probability of evolving into S ₁ is 4 to 14 times bigger than that of evolving into S ₂ but 64 to 580 times bigger than that of evolving into S ₃ if emergency alternative A _i is taken. In other words, according to past experience of evolution accident, in most cases, safety facilities of high bent were damaged during the accident of crane hitting the high bent, while rarely evolving into a more serious accident. From this point of view, both A ₃ and A ₂ are effective emergency measures. As safety management has a significant impact on the top event, A ₃ is more effective than A ₂. Both the expected overall evaluation of emergency alternative A ₅ and A ₆ are small, as this emergency measures are taken for severe accident states such as S ₃, Much more costs and resources of human and material are needed than other four emergency alternatives. Both of them take measures such as stopping the construction of high bent and lifting operations, rescuing fall staffs, comprehensively repairing safety facilities and infrastructure of high bent, strengthening management and training of the team. In addition to this, the measure of comprehensively repairing lifting will be taken in A ₆. This measure can protect personal safety of construction workers in principle. Apart from this, it not only conducive to exclude safety risks in a certain extent but also shorten dead time of construction progress. The measure of alerting person near the accident ensures the safety of personnel life by keeping construction workers away from the high trestle with safety risks temporarily. It complies with the “People-oriented” principle of construction. But consider both weights and the severity of the evolution of events, A ₅ and A ₆ are not very effective, even less effective than do nothing namely A ₁.

As seen from above analysis, we can assess emergency measures of accident case effectively by applying the method proposed in this paper.

5 Conclusion

This paper presents a hybrid approach for sorting of emergency alternatives in risk emergency response decision-making. First draw the logical relationships between basic events and evolution states by constructing fault tree in construction accident. Then experts summed up triangular fuzzy probability of basic events and evolution states under different emergency alternative. The occurrence probability of each evolution state under different emergency alternative can be obtained through the logical relationships obtained above. Then calculate the cost of each emergency alternative and loss of each state under different criteria, get and normalize the weights of costs and criteria by the method mentioned in 2.3. Then the expected overall evaluation of emergency alternatives can be calculated. Finally, we can sort of the emergency alternatives by Endpoint Method. A case study and sensitivity analysis are conducted to demonstrate the proposed method’s effectiveness. The major contributions of this paper are discussed as follows.

First, this paper proposes a new risk decision-making problem in emergency response with dynamic and risky characteristics. It is a new idea for describing the decision-making problem in emergency response and lays a good foundation for further studies on risk decision analysis in emergency response.

Second, in order to estimate probabilities of scenarios given each feasible response action, the principle of FTA is introduced. The FTA method makes it possible to capture the dynamic evolvement process of emergency and estimate the probabilities of emergency scenarios before the response actions are implemented.

Third, compared to the general fault tree analysis, we applied operations of the triangular fuzzy number to represent the probability more objectively. At the same time, fuzzy AHP was used to determine the weight vector more accurately and objectively, Endpoint Method was used to solve the problem of triangular fuzzy number comparison.

In this paper, we treat probabilities of basic events as fuzzy probabilities which incorporate the theory of fuzzy logic as a complement to probability theory [51]. However, many researchers consider probability theory and fuzzy logic as completely different techniques, which solve uncertain problems in a different way. We make an attempt to close the bridge between both approaches that deal with uncertainty and wish lead to some further discussions. And according to [52], the approach of fuzzification operations with probabilities has limitations, it may be a good idea to move from imprecise probabilities to the perception-based probability theory— a theory in which perceptions and their descriptions in a natural language play a pivotal role.

Footnotes

Acknowledgments

This paper is supported by the national natural science fund project (50909045, 51079078), the Fundamental Research Funds for the Central Universities (HUST: 2013QN154).

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A hybrid fuzzy FTA-AHP method for risk decision-making in accident emergency response of work system

Abstract

Abstract

Keywords

1 Introduction

2 Methodology

2.1 Problem description

2.2 Conventional fault tree analysis

3.1 Construct the FT for the case

3.2 Calculate probability of state transition P ˜ ij

3.4 Determine the weights

3.5 Comprehensive ranking of emergency alternatives

4 Discussion

Footnotes

Acknowledgments

References

3.2 Calculate probability of state transition ${\tilde{P}}_{ij}$