Safety management performance assessment for Maritime Safety Administration (MSA) by using generalized belief rule base methodology

Safety Science 63 (2014) 157–167

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Safety management performance assessment for Maritime Safety Administration (MSA) by using generalized belief rule base methodology Jinfen Zhang a,b,c,d, Xinping Yan a,b,c, Di Zhang a,b,c,⇑, Stein Haugen d, Xue Yang d a

Intelligent Transport Systems Research Center, Wuhan University of Technology, Wuhan 430063, China Engineering Research Center for Transportation Safety (Ministry of Education), Wuhan University of Technology, Wuhan, China c Research and Development Base on Waterway Transportation Safety and Anti-pollution of CJRDC Ministry of Transport, Wuhan University of Technology, Wuhan, China d Department of Production and Quality Engineering, Norwegian University of Science and Technology, Trondheim, Norway b

a r t i c l e

i n f o

Article history: Received 30 January 2013 Received in revised form 30 September 2013 Accepted 29 October 2013 Available online 2 December 2013 Keywords: Waterway transportation Maritime safety administration Belief rule base Bayes Network

a b s t r a c t Waterway transportation plays an essential role for economic development. Maritime Safety Administration (MSA) shoulders the responsibility to maintain waterway transportation safety and efficiency and to avoid environmental contamination. This paper focuses on the assessment of MSA performance in term of safety with Belief Rule-base (BRB) methodology. A generalization of traditional BRB theories, which is called G-BRB, is introduced in this paper at first. G-BRB focuses on the fact that experts’ subjective standards on one quantitative attribute may be different. The qualitative data of safety and cost attributes from expert questionnaires is transformed into antecedent belief structure (A-BS), which can well reflect the distinctions among experts’ knowledge. The match degree and activation weight are then proposed to deal with A-BSs to make inference. After that, the proposed method is used to assess the performance of one MSA in China during the year 2007–2011. The factors used for MSA performance assessment are divided into two sub-groups and assessed separately. One group mainly focuses on safety situation and the other group pays attention to cost of maintaining safety. Then the outputs of the two sub-groups are used as the inputs for MSA performance assessment to get final results. After the comparison with experts knowledge (which is believed to be expert opinions of MSA performance), the results show that G-BRB can deal with experts’ different criteria on some factors successfully. The results also show that the proposed method performs well in terms of precision and reliability. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Maritime transportation plays a significant role in the integrated transportation system, especially in international trade system. Safety is a crucial factor due to the fact that various accidents like ship collisions and groundings often result in great economic loss, fatalities and the environmental contamination. Consequently, maritime risk is one of the most important focus areas for waterway transportation. Researchers and maritime authorities have been paying special attention to the risky areas where traffic density is high for a long time. The areas include gulfs (Kujala et al., 2009; Jakub et al., 2010; Jason et al., 2003), straits (Özgecan et al., 2009), ports (Yeo et al., 2007; Wen and Liu, 2010), busy waterways (Mou et al., 2010) and so on. These water areas share similar characteristics, such as high traffic density along with limited space, high sensitivity and vulnerability to accidents. The high density

⇑ Corresponding author at: Intelligent Transport Systems Research Center, Wuhan University of Technology, Wuhan 430063, China. Tel.: +86 013986160037. E-mail addresses: [email protected], [email protected] (D. Zhang). 0925-7535/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssci.2013.10.021

and high risk for transportation increase the complexity of navigation safety management. There can be a lot of elements that have implications on safety. A lot of theories and models have been developed for maritime risk assessment. The International Maritime Organization (IMO) proposed a standardized risk assessment procedure called Formal Safety Assessment (FSA) early in 1997 (IMO MSC/ Circ.829) and then the guideline for it was later proposed in 2002 (IMO MSC/ Circ.1023). It is called ‘‘a rational and systematic process for assessing the risk related to maritime safety and the protection of the marine environment and for evaluating the costs and benefits of IMO’s options for reducing these risks’’. FSA is a standardized maritime risk assessment procedure, which gives a systematic guideline for risk assessment and risk control strategy selection. The procedure is divided into five steps: (1) Hazards identification; (2) Assessment of risks associated with possible hazards; (3) Development of programs of managing the risks estimated; (4) Cost-benefit analysis for the risk control options (RCOs); and (5) Making decisions on which options to select to keep risk under acceptable level. Christos and Harilaos (2009) gave a critical review of this method and proposed possible improvements for each step.

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The traffic characteristics in a waterway, such as ships reaching pattern and geographical distribution and so on, could have an influence on safety. Some models were proposed to predict collision or grounding probability for strait or intersection waterways based on traffic flow theory. Macduff (1974) proposed a collision and grounding probability prediction model by using the idea of Buffon Needle. The model takes ship size, course distribution, and channel width into consideration. Pedersen et al. (2002) also tried to find an analytical solution to collision and grounding probability of waterways that are parallel head-on, intersecting or crossing based on the traffic density and geographical distributions. The above analytical models gave a general picture of risk in the waterways, but not detailed information such as spatial and temporal distributions of accidents. Monte Carlo simulations (Jason et al., 2003; Pentti et al., 2009; Jakub et al., 2010) have become increasingly popular in recent years. The idea is to simulate ship traffic at first based on the geographical distribution in each entrance gate and ship’s behavior during navigating. Then risk assessment could be carried out by analyzing the outputs from simulations such as number of encounters, collision angle and so on. Some other models, such as ship maneuverability model (Jakub et al., 2010) and traffic volume fluctuation model (Jason et al., 2003), could also be incorporated in the simulations. It can also predict risk tendency along with changes of traffic volume. One challenge for this method is how to simulate ship’s behavior, especially when it is influenced by other encountered ships. Historical data is usually very useful for risk assessment. Many researchers (Wang et al., 2005; Antão et al., 2008; Mou et al., 2010) tried to find the laws of accident by statistically analyzing data collected over years. The studies were usually carried out by statistics of some indicators that can reflect accident severity, such as number of accidents, accident type, fatalities or economic loss and so on. Although statistics for historical data are very important, some scholars (Jakub et al., 2010; Wang et al., 2012a,b) believed that it is not enough to explain the essence of accident. For major accidents (Haugen and Seljelid, 2011, 2012), there is very little historical data due to the fact that such accidents are rare. However, major accident risk cannot be neglected because of its severe consequences. Under this circumstance, experts’ knowledge on risk indicators is significant as well. Several models have been introduced to combine historical data with expert knowledge. Analytic Hierarchy Process (AHP) (Saaty, 1977), usually combined with fuzzy set theory (Karahalios et al., 2011), is a useful tool to solve problems like Multi Attribute Decision Making (MCDM). AHP and fuzzy theory have also been used comprehensively in risk-related problems in many fields, such as industries (Cebeci, 2009; Abdelgawad and Fayek, 2010; Wang et al., 2012a,b; Zheng et al., 2012; Caputo et al., 2013) and Small and Medium Enterprises (SMEs) (Fera and Macchiaroli, 2010; Bharathi et al., 2012; Hosseinzadeh et al., 2013; Floyde et al., 2013). Industries and SMEs are often confronted with the problem of risk identification, assessment and reduction with both qualitative and quantitative data. To mention some, Fera and Macchiaroli (2010) applied the AHP method to risk assessment for SMEs. The proposed model was then applied to steel industry and logistics provider to test its performance in terms of precision and sensitivity. Abdelgawad and Fayek (2010) used fuzzy logic and fuzzy AHP to address the limitations of traditional Failure Mode and Effect Analysis (FMEA) used in many industry areas. The proposed method can help project management teams take correct actions more effectively. Wang et al. (2012a,b) studied the risk assessment of implementing initiatives in the fashion supply chain by combining fuzzy logic and fuzzy AHP to overcome the qualitative nature and uncertainty in the fashion industry. Caputo et al. (2013) studied the safety devices of industrial machinery selection

problem by using AHP method, which allows a rapid ranking of alternatives and selecting the most suitable one by combining experts’ knowledge. As mentioned above, fuzzy theory and AHP are popular tools and have great application potential in solving risk-related decision making problems in most aspects of industries. Bayes Network (BN) (Jensen and Nielsen, 2007; Trucco et al., 2008; Li et al., 2012) is one of the most popular methods been used to risk assessment in maritime field. BN builds the relationships among factors by conditional probability tables (CPTs). The assessment is then carried out by combining CPT with prior values of all variables on the basis of Bayes theory. One limitation for BN is that the size of the CPT grows exponentially with the number of variables, which makes it very difficult to obtain (Li et al., 2012). One possible solution is that the CPT can be decomposed and calculated separately if the variables are independent of each other (Wang et al., 2012a,b). Belief rule base (BRB) (Wang et al., 2006; Yang et al., 2009; Yang et al., 2012a,b; Yang et al., 2008) is another method that has a different theoretical foundation compared to BN. BRB makes inferences by combining all the activated BRBs based on evidence theory, which is nonlinear in essence (Chin et al., 2009). The BRBs are quite similar with CPTs in BN. Although BRB theory is faced with similar challenges, the evidence theory makes it possible to obtain BRBs by combining various experts’ knowledge, so that the BRBs can be more reliable. Furthermore, the factors which have impact on risks can be divided into groups and assessed separately in an iterative way, which can also reduce the dimensions of BRBs substantially. BRB theory has been successfully applied to the maritime domain such as accident analysis (Wang et al., 2012a,b), technique selection for ship emission reduction (Yang et al., 2012a,b), maritime security assessment (Yang et al., 2009) and so on. Several researchers (Yang et al., 2008; Wang et al., 2012a,b) have also tried to integrate BN and evidence reasoning to carry out quantitative risk assessment. With respect to some variables, experts’ opinions on its degree of severity are different in many cases. However, most of the present research treats them equally. This paper takes such distinctions among experts’ opinions into consideration and tries to extend the ‘‘IF’’ part of the BRBs into belief structures. Then a novel match degree and activation weight computing method under the generalized version is proposed for rule activation in BRB. Maritime Safety Administration (MSA) shoulders the responsibility to maintain waterway transportation safety. Consequently MSA’s performance could be evaluated by the safety situation of the area it is in charge of to a large extent. What’s more, safety situation and cost can further be divided into several specific indexes. Take the safety situation as an example. It can be assessed according to accident number, fatality, economic loss, environment pollution and so on. According to the introduction to its structure and basic idea, BRB methodology is quite suitable to solve such a problem. In this paper, a possible extension for traditional BRB method is analyzed in detail at first. Then a MSA performance evaluation has been carried out as a case study. The results from the G-BRB are also compared with other traditional used methods to evaluate its validity and reliability. The rest of the paper is organized as follows: Section 2 gives theoretical fundamentals of evidence theory and evidential reasoning used. Section 3 introduces the possible extension of the traditional BRB theories and how to extend it to a generalized version. One MSA in China is selected as a case study and the results of safety situation, cost and MSA’s performance from different methods are further analyzed and the sensitivity analysis is also carried out in Section 4. Conclusions and possible extensions of the work are discussed in Section 5.

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been transformed into the above forms. After combination, the probability assigned to each subset can be obtained by using the following equations (Wang et al., 2006):

2. Theory preliminaries 2.1. Evidence theory Evidence theory, which is also called D-S Theory (DST), was first proposed by Dempster (1967) and was later modified by Shafer (1976). DST is a theory which has been used to combine evidences from different sources under uncertainties to make rigorous estimations. Suppose there are N possibilities for a problem that need to be predicted, which are represented by H = {H1,. . .,HN}. H is usually called frame of discernment. A basic probability assignment (BPA) assigns a number between 0 and 1 to each element of the power set 2H. The BPA should satisfy:

X

mðAÞ ¼ 1 and mð/Þ ¼ 0;

ð1Þ

A#H

m0i ðAÞ

þ

m0i ðHÞ





L Y

i¼1

MðHÞ ¼ k

" L Y

MðHÞ ¼ k

;

A 2 2H ; A–H;

ð8Þ

i¼1

m0i ðHÞ



i¼1

" L Y

# m0i ðHÞ

L Y

#  0i ðHÞ ; m

ð9Þ

i¼1

#  0i ðHÞ m

ð10Þ

;

i¼1

where

2

where / means null set and A is any subset of H. It should be noted that the parameter m(H) reflects the degree of ignorance of a evidence. If m(H) = 0, then the evidence is called complete. Otherwise it is called incomplete. The belief function Bel and plausible measure Pl, which can reflect some attributes of the evidence, are defined as follows:

BelðAÞ ¼

MðAÞ ¼ k

" L Y 

X

X

mðBÞ and PlðAÞ ¼

B#A

mðBÞ

ð2Þ

A22H i¼1

ð11Þ

i¼1

After that, the above probability assignments should be normalized to get the final prediction results as follows:

M0 ðAÞ ¼

MðAÞ ; 1  MðHÞ

M0 ðHÞ ¼

MðHÞ : 1  MðHÞ

A 2 2H ; A–H;

ð12Þ

A\B¼/

It can be seen that Bel(A) and Pl(A) are the lower and upper bound of the probability assigned to A in one evidence respectively. Suppose there are two belief structures m1 and m2 transformed from independent sources. According to DST, the combination can be defined as follows:

8 < 0;P

ðm1  m2 ÞðAÞ ¼

A ¼ /;

: 1PB\C¼A

m1 ðBÞm2 ðCÞ

B\C¼/

m1 ðBÞm2 ðCÞ

; A–/:

ð3Þ

where ‘’ represents evidence combination operator. It has been proved that DST cherishes the properties of both commutative and associative (Shafer, 1976). Therefore, multiple pieces of evidence can be combined in an iterative way in any order. 2.2. Evidential reasoning theory The results from DST may be obviously unreasonable when two pieces of evidence conflict seriously. Wang et al., (2006) gave an example to illustrate such a problem. Evidential reasoning (ER) algorithm solves the problem by giving each belief structure a weight before combination. At first, the BPA in DST is transformed by considering the weight given to each piece of evidence as follows:

m0 ðAÞ ¼ wi mðAÞ; m0 ðHÞ ¼ 1 

31 L L XY Y  0  0 0 k¼4 mi ðAÞ þ mi ðHÞ  ðN  1Þ mi ðHÞ5 :

X

A 2 2H ; A–H

m0 ðAÞ

ð4Þ ð5Þ

ð13Þ

It can be conveyed that if all evidences are complete, then m0i ðHÞ ¼ 0; ði ¼ 1; . . . ; LÞ, which means that  0i ðHÞ; ði ¼ 1; . . . LÞ. Consequently, MðHÞ ¼ 0 and furtherm0i ðHÞ ¼ m more M0 ðHÞ ¼ 0. This result shows that the combination of completed evidences is also a complete version. 3. Generalization of belief rule base (BRB) theories BRB theory is an inference tool based on a tree-structured framework. It can combine quantitative data with qualitative data (such as expert knowledge) to make the optimum judgment or assessment for the target attribute. Several steps, including attribute data transformation, BRBs construction, rule activation and inference, are needed to make the judgment. During the process, each attribute is divided into several grade scales. Under this circumstance, only qualitative data can be used directly. Consequently fuzzy theory should be used at first to deal with quantitative data. With respect to the rule activation, the basic idea is that the more similar the BRB is with the input belief structure, the higher the probability that it is activated. At last, evidential reasoning (ER) approach is used to combine all the activated BRBs to make rigorous inference. 3.1. Basic framework of BRB

h

A22

0

m ðHÞ ¼ 1  w 0 m ðHÞ ¼ w@1  0

ð6Þ X

1 mðAÞA

ð7Þ

A22h

where w is the weight assigned to BPA. 2H = {/, {H1}, ...{HN}, {H1, H2}..., H} is the power set of H. The sum of all BPAs’ weights  0 ðHÞ þ m0 ðHÞ is comshould be 1. It should be noted that m0 ðHÞ ¼ m posed of two parts. The first part denotes the relative weight given to the evidence and the second part denotes the incompleteness in the evidence itself. Now suppose there are L BPAs which all have

Suppose one assessment attribute can be inferred by a group of sub attributes as shown in Fig. 1. Each sub attribute data should at first be divided into several discrete subjective grades such as High, Average and Low. By doing this subjective data can be directly used. After that, the belief rule base can be constructed based on historical data or expert knowledge following the approach introduced by Liu et al. (2005) and Yang et al. (2012a,b). A belief rule base is defined as follows:

IF A1 ^ . . . ^ Am THEN A : fðC 1 ; b1 Þ; . . . ; ðC N ; bN Þg

ð14Þ

where Ai 2 fC i1 ; . . . ; C iT i g (i = 1,. . ., m) and Cij (j = 1,. . ., Ti) are all possible grades of the ith attribute. Ti is the ith attribute’s grade number. For example, if the grade scale set is {High, Average, Low}, then

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J. Zhang et al. / Safety Science 63 (2014) 157–167

year. A similar situation can also be relevant with distress number and death toll. If we want to combine their knowledge to form a new belief rule base with the same standard, then each antecedent attribute should be transformed into uncertain version, which is called Antecedent Belief Structure (A-BS) in this paper. Under this situation, the definitions of match degree, rule activation and so on proposed in Liu et al. (2005) and Yang et al. (2012a,b) is no longer applicable. A novel approach must be developed to deal with A-BS. In order to illustrate the problem clearly, the generalization of traditional belief rule base is shown as follow:

IF fðC 11 ; a11 Þ; . . . ; ðC 1T 1 ; a1T 1 Þg ^ . . . ^ fðC m1 ; am1 Þ; . . . ; ðC mT m ; amT m Þg THEN fðC 1 ; b1 Þ; . . . ; ðC N ; bN Þg ð15Þ In the above definition, the meaning of C-BS is the same with traditional belief rule base method. The sign ‘‘ ^ ’’ means ‘‘AND’’. Cij and aij (i = 1,. . .,m; j = 1,. . .,Ti) are the possible grade for the ith sub attribute and the possibility that the ith sub attribute belongs to grade Cij respectively.

Fig. 1. Prototype structure of BRB.

Ti is 3. Ci (i = 1,. . .,N) is the possible consequence of the assessment attribute and bi is the possibility that the consequence is Ci. It is easy P P to know that 0 6 bi 6 1 and Ni¼1 bi 6 1. If Ni¼1 bi ¼ 1 then the belief rule base is called complete. Otherwise it is incomplete, which means that it is possible that the consequences are totally unknown. According to the concept of belief rule base introduced above, we can divide a belief rule base into two parts. The first part is A1 ^ . . . ^ Am . The second part is fðC 1 ; b1 Þ; :::; ðC N ; bN Þg, which is called Consequence Belief Structure (C-BS). By doing this we can see that belief rule base is a generalized version of the traditional IF-THEN rule by extending the certain consequence to C-BS. That’s to say, only the C-BS of belief rule base is extended to uncertain version. The ‘IF’ part of belief rule base is still certain version. However, in some applications, the antecedent attribute of ‘IF’ part can be uncertain version as well. The reason is that uncertainty can happen due to many factors such as the uncertainty of historical data, expert knowledge conflict and so on. For example, suppose there are two belief structures from two experts: Expert 1: IF accident number is high AND distress number is average AND death toll is low, THEN safety situation is {(very poor: 0.2), (poor: 0.4), (average: 0.4), (good: 0), (very good: 0)} Expert 2: IF accident number is high AND distress number is average AND death toll is low, THEN safety situation is {(very poor: 0), (poor: 0.4), (average: 0.6), (good: 0), (very good: 0)}

The ‘IF’ part of the above belief rule bases are taken into consideration to illustrate the uncertainty. Although they are the same, the two experts’ opinions on ‘‘accident number is high’’ are not quite the same. The first expert assumes that accident number is high if it is no less than 200 per year. However, the second expert believes that accident number is high if it is no less than 150 per

3.2. Generalization of BRB Before consequence distribution inference by rule activation and combination, traditional version of BRBs is generalized. They are based on historical data and expert knowledge. Several approaches such as fuzzy membership functions, rule or utility based transformation (Yang et al., 2006) can be used to treat various types of data. In this paper, all antecedent attributes are quantitative. So utility based method is used. Suppose there are T grades for an attribute, the utility value for each grade is u(Ci) (i = 1,. . .,T). Let x be a value which lies between u(Ci) and u(Ci+1) (i = 1,. . ., T1). The probability that x belongs to the ith and (i + 1)th grade can be computed by the following linear piece-wise function:

ai ¼

uðC iþ1 Þ  x uðC iþ1 Þ  uðC i Þ

and aiþ1 ¼

x  uðC i Þ uðC iþ1 Þ  uðC i Þ

ð16Þ

Consequently, the attribute value x can then be transformed into a A-BS version: {(C1, 0),. . ., (Ci, ai), (Ci+1, ai+1),. . ., (CT, 0)}. If x is larger than C1, then the A-BS will be {(C1, 1),. . ., (CT, 0)}. If x is smaller than CT, then the A-BS will be {(C1, 0),. . ., (CT, 1)}. It is easy to see that ai + ai+1 = 1, which means the A-BS is complete. In order to better illustrate the transformation, the example used in Section 3.1 is used again here. Table 1 shows the two experts’ opinions on the values of different grades for three attributes. A unified standard for each attribute should be created. Suppose the average of the two experts’ grade values (the last row of Table 1) is treated as the unified standard, the conditions of the two BRBs should be transformed into an uncertain version. With regard to expert 1, by ‘‘accident number is high’’ he means no less than 200, which is larger than the average high grade (175). So the A-BS for accident number is transformed into {(high: 1), (average: 0), (low: 0)}. By ‘‘distress number is average’’ he means that distress number is 850, which lies between standard value of high (1100) and average (775). So according to the generalized approach introduced above, it can be transformed into {(high:

Table 1 Experts’ grade standard (an example). Attribute

Accident number

Grade standard

High

Average

Low

High

Distress number Average

Low

High

Death toll Average

Low

Expert 1 Expert 2 Average

200 150 175

125 95 110

50 40 45

1200 1000 1100

850 700 775

500 400 450

14 10 12

9 6 7.5

4 2 3

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J. Zhang et al. / Safety Science 63 (2014) 157–167

(850–775)/(1100–775)), (average: (1100–850)/(1100–775)), (low: 0)}, that is {(high: 0.2308), (average: 0.7692), (low: 0)}. All the other attribute values can be dealt with in the same way. After disposal, the two BRBs are transformed into the following versions: Expert 1: IF accident number is {(high: 1), (average: 0), (low: 0)} AND distress number is {(high: 0.2308), (average: 0.7692), (low: 0)} AND death toll is {(high: 0), (average: 0.2222), (low: 0.7778)}, THEN safety situation is {(very poor: 0.2), (poor: 0.4), (average: 0.4), (good: 0), (very good: 0)} Expert 2: IF accident number is {(high: 0.6154), (average: 0.3846), (low: 0)} AND distress number is {(high: 0), (average: 0.7692), (low: 0.2308)} AND death toll is {(high: 0), (average: 0), (low: 1)}, THEN safety situation is {(very poor: 0), (poor: 0.4), (average: 0.6), (good: 0), (very good: 0)}

Ti X

ckj ¼

ð17Þ

l¼1

It is worthy of noting that 0 6 ckj 6 1. If and only if the two distributions are exactly the same, then ckj ¼   P PT i PT i j Ti j j ¼ min a ; a a ¼ a ¼ 1. If either a or a kl kl kl kl is l¼1 l¼1 kl l¼1 kl zero for all l = 1,. . .,Tk, then cij = 0, which means that they are totally different. This is intuitively reasonable because there is no overlap between the two distributions. It also can be seen that the match degree value derived from the above equation will be the same as the traditional one discussed in Yang et al. (2012a,b) when the state of antecedent attribute is certain. So the match degree defined here is a generalization of the traditional version. The match degree for other attributes can be derived in the same way. What’s more, when defining the overall match degree, another parameter di (i = 1,. . .,m) must also be taken into consideration because it reflects the importance of the role that the attribute plays on A’s consequence. Consequently, the overall match degree between the new A-BS and the kth belief rule base is defined as:

ck ¼

Ti m Y X i¼1

It is worth noting that only A-BS is changed for the two BRBs. The C-BS is not changed. That’s to say, this type of operation only has impact on the rule activation phase and has no influence on the methodology of consequence inference. The reason is that consequence inference is only related with the C-BS of BRBs. Until now, the generalization of belief rule bases (G-BRB) can be built by handling all experts’ knowledge in the same way. If we want to give different weights to different evidence, an extension is required. Table 2 shows a series of G-BRB for attribute A’s consequence inference. There are M rule bases in total with weight hi (i = 1,. . .,M). di (i = 1,. . .,m) is antecedent attribute weight. A larger di means that the attribute has greater influence on the consequence. There are Ti possible grades for attribute Ai. C kij and akij (i = 1,. . .,m; j = 1,. . ., Ti; k = 1,. . .,M) are the jth grade of the ith attribute in rule base k and its corresponding possibility respectively. bki (i = 1,. . .,N; k = 1,. . .,M) is the possibility that the consequence belongs to grade Ci in the kth rule base. Suppose now there is a new A-BS ffðC 11 ; a11 Þ; . . . ; ðC 1T 1 ; a1T 1 Þg; . . . ; fðC m1 ; am1 Þ; . . . ; ðC mT m ; a1T m Þgg, which has been transformed from a set of antecedent attribute values (x1,. . ., xm) with the generalized approach discussed above. What we want to do is to predict possible consequence distribution of the new A-BS. The first thing that needs to be done is determining activation weight for each rule base according to the match degree. The idea is that the more similarity between the input A-BS and rule base, the higher the active weight will be given. In order to make the problem simpler, only one attribute is taken into consideration at first. For the kth antecedent attribute Ak, the matchndegree between fðC k1; o ak1 Þ; . . . ; ðC kT k ; akT k Þg and the jth  is defined as following: A-BS C jk1 ; ajk1 ; . . . ; C jkT k ; ajkT k

  min akl ; ajkl :



k ij ;

min a aij

!  di

ð18Þ

j¼1

di ¼ di = maxðd1 ; . . . ; dm Þ, which reflects the relative weight for where  the ith attribute. It is easy to see that 0 6 ck 6 1 and ck = 1 if and only if the new A-BS and the kth A-BS are exactly the same for all antecedent attributes. If the match degree is zero for at least one antecedent attribute, then ck = 0, which means that the belief rule base will not be activated and it will make no contribution to consequence inference. All the belief rule bases with nonzero match degree will be activated. The activation weight is not only related to the match degree ck, but also to the rule weight hk (k = 1,. . .,M). So according to Yang et al. (2012a,b) the activation weight for the kth belief rule base can be computed as follow:

2 ! 3 Ti m   di Y X k 4 5 hk min aij ; aij

wk ¼

j¼1 i¼1 hk ck ¼ 2 ! 3 ; M X Ti M m   di X Y X hl c l l 5 hl 4 min aij ; aij l¼1

l¼1

i¼1

k ¼ 1; . . . ; M

j¼1

ð19Þ where the denominator is used to normalize all G-BRBs, so that the sum of all the activated weights will be equal to 1. After this process, the consequence inference is carried out by combining all the activated rule bases based on evidential reasoning, which will be introduced in the following section. 3.3. Consequence inference by aggregating the activated belief rules Due to the fact that process mode for C-BS of BRB is not changed in this paper, consequence inference is carried out based on evidential reasoning (ER) theory. According to the ER theory described

Table 2 Generalized belief rule bases for attribute A. Rule

1

Rule weight

Antecedent attributes(weight) A2(d2) n   o C 121 ; a121 ; :::; C 12T 2 ; a12T 2 n   o C 221 ; a221 ; :::; C 22T 2 ; a22T 2

...

h1

A1(d1) n   o C 111 ; a111 ; :::; C 11T 1 ; a11T 1 n   o C 211 ; a211 ; :::; C 21T 1 ; a21T 1 ... n   o M M M CM 11 ; a11 ; :::; C 1T 1 ; a1T 1

... n   o M M M CM 21 ; a21 ; :::; C 2T 2 ; a2T 2

... ...

2

h2

... M

... hM

Attribute A’s Consequences

... ...

Am(dm) n   o C 1m1 ; a1m1 ; :::; C 1mT m ; a1mT m n   o C 2m1 ; a2m1 ; :::; C 2mT m ; a2mT m

C1

C2

...

CN

b11

b12

...

b1N

b21

b22

...

b2N

... n   o M M M CM m1 ; am1 ; :::; C mT m ; amT m

...

...

bM 1

bM 1

... ...

bM N

...

162

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in Section 2.2, the probability that the consequence is Cj can be computed according to the following equations:

l aj ¼

" M  Y k¼1

# M  Y wk bkj þ 1  wk  ð1  wk Þ k¼1 " # M Y ð1  wk Þ 1l

;

j ¼ 1; . . . ; N

ð20Þ

k¼1

" #1 N Y M  M  X Y k l¼ wk bl þ 1  wk  ðN  1Þ ð1  wk Þ l¼1 k¼1

ð21Þ

k¼1

where wk is the activation weight of the kth belief rule base. After all the activated G-BRBs are combined, the consequence distribution for antecedent attribute A can be inferred. It should be mentioned that the G-BRBs derived from different experts can also be combined with the above approach. Take the above two experts’ G-BRBs as an example, if they are treated as two pieces of evidence, they can be combined by the above ER approach as well. Before combination, the relative weight for the two experts should be confirmed. Suppose the weights given to the two experts are 0.2 and 0.8 respectively, the two BRBs can be combined into one by the above rule combination. The combined BRB is as follow: IF

accident number is {(high: 0.6838), (average: 0.3162), (low: 0)} AND distress number is {(high: 0.0119), (average: 0.7974), (low: 0.1907)} AND death toll is {(high: 0), (average: 0.011), (low: 0.989)}, THEN safety situation is {(very poor: 0.0108), (poor: 0.4), (average: 0.5892), (good: 0), (very good: 0)}

In many real applications, problems can be much more complex. If there are too many factors which have impact on one attribute, they can be divided into several groups. The factors in the same group can be assessed separately at first. Then the outputs from each group could be treated as the inputs to assess the final attribute. By doing this the complex problem can be divided into several simpler ones. What’s more, the number of BRBs grows exponentially with the number of sub attributes. So it also can make it much easier for experts to give their opinions to build BRBs.

volved ships’ assistance. The framework for MSA performance assessment is shown in Fig. 2. The assessment framework can be divided into three parts, which can be carried out step by step. Firstly, safety situation and cost are assessed by the methodology proposed in Section 3 separately. Both of them are treated as an independent structure. Then their outputs are treated as a input belief structure for MSA performance assessment at the top of the figure. 4.1. Safety situation assessment Under the above safety situation assessment framework, three experts were inquired by using a questionnaire. At first, they were required to give their own standard on the three factors on ‘‘High’’ and ‘‘Low’’ for each factor. The grade value of ‘‘Average’’ is the mean value of ‘‘High’’ and ‘‘Low’’. The data on experts’ grade value is shown in Table 3. The weights given to the three experts are based on their experience of maritime management. Expert 2 is the most experienced one and expert 3 has been working with the maritime management work for only two years. Consequently, the biggest weight (0.5) is given to the second expert while the smallest weight (0.2) is given to the third expert. Due to the fact that the three factors influence safety situation in different degrees, a weight is given to each attribute. Preventing maritime accidents and saving people’s lives are the top concerns for MSA. So the attribute accident number and death are given the largest weight (0.4). In term of distress number, according to the historical data, the probability that a person in distress is saved is around 99% during these years. That is to say, almost all people in distress were saved. So a smaller weight (0.2) is given to this factor. It can be seen from Table 3 that expert 2’s opinion on accident number is the most conservative while expert 3’s is the least conservative. Expert 3’s standard on death toll is the highest and expert 1’s standard is the lowest. The data given by experts describes their standards on safety situation. Expert 2’s standard is the highest while the first expert is the lowest. The weighted average of the three experts’ opinions is treated as the unified standard. Table 4 shows the experts’ grade standards transformation result under the unified standards. In the above table, ‘‘H’’ means ‘‘High’’, ‘‘A’’ means ‘‘Average’’ and ‘‘L’’ means ‘‘Low’’. It can be seen from Table 4 that most crisp grade values have been transformed into uncertain versions under the unified standards. Some grade values have changed to a large degree. The bold distributions are typical ones. Take the expert 1’s grade value on ‘‘Average’’ as an illustration. After transformation, the probability for ‘‘High’’ becomes much higher (0.8936) while

4. MSA performance assessment based on G-BRB According to MSA’s primary responsibility, its performance is assessed by two indicators: safety situation of the waterway that it is in charge of and the cost put into maintaining transportation safety. In general, maritime safety situation is determined by several factors related to humans, ships and the environment. MSA pays special attention to avoiding accidents and to reduce economic loss. It also pays attention to avoid fatalities or injuries caused by accidents such as collision and grounding. In this paper, three factors are used to assess the safety situation, which are accident number, distress number and death/missing toll. In term of cost, it mainly includes conventional maritime patrol during normal times and search and rescue when accidents happen. Due to the fact that conventional patrol is almost the same during the years, it is not taken into consideration. Salvage is the main cost for MSA. Furthermore, when situation is too difficult for salvage ships to deal with, other ships’ assistance is needed. Therefore, total cost is evaluated by the cost of salvage and the cost of other in-

Fig. 2. Framework for MSA performance assessment.

163

J. Zhang et al. / Safety Science 63 (2014) 157–167 Table 3 Experts’ grade standard for safety-related factors. Expert (weight)

Expert 1 (0.3) Expert 2 (0.5) Expert 3 (0.2) Weighted average

Accident (weight: 0.4)

Distress (weight: 0.2)

Death toll (weight: 0.4)

High

Average

Low

High

Average

Low

High

Average

Low

300 250 340 283

200 175 235 194.5

100 100 130 106

2500 2000 1500 2050

2000 1500 1150 1580

1500 1000 800 1110

25 25 15 23

17.5 9.5 10 12

10 8 5 8

the probability for ‘‘Average’’ is rather low. Although this result is contradicted with our intuition, it is reasonable because a high value in one standard may be low under another standard. After the transformation, G-BRBs for safety situation assessment can be built by combining all experts’ knowledge with ER approach proposed in Section 3.3. In the following case studies, the traditional method, which treats all experts’ knowledge equally, is also studied. In the traditional methodology, the A-BS of BRBs are all crisp ones and the unified grade standard for the three factors is used to deal with historical data as well. Table 5 shows the historical data on accident, distress and death toll from 2007 to 2010. The accident data includes all the reported accidents that have happened in the area where the MSA is in charge of. It should be mentioned that the majority of accidents have small consequences. The data shows that both maritime accidents and death toll in the waterways that the MSA is in charge of were decreasing while the number of people in distress fluctuated during the years. Before the safety situation assessment, all the historical data should be transformed into uncertain versions under the unified standards. After that, the uncertain versions of historical data would be compared with all G-BRBs with the methodology proposed in Section 3.2. By doing so, the match degree between historical data and each BRB can be obtained and furthermore the activation weight for each BRB can also be determined. In this paper, all the G-BRBs are treated equally. The safety situation assessment results of the four years with traditional method and the G-BRB proposed in this paper are shown in Fig. 3. What is more, Bayes Networks (BN) is also used to make safety situation assessment. The prior probability distributions used here are obtained by transforming historical data into uncertain versions. BN theory makes assessment by combining prior probability with condition probability. The same structure is used for BN. In the simulations, all the BRBs are treated as the CPT in BN. The ‘‘Traditional method’’ in the figures means that all experts’ grade standards are treated equally. That is to say, the A-BSs used in traditional method are all crisp versions. The results are also obtained by rule combination based on evidential reasoning. It can be seen from Fig. 3 that the results from all methods show that safety situation was becoming better during the years 2007 and 2010. The figures also show that safety situation results from traditional method are better than that from G-BRB for all years except 2008. This result implies that the traditional method tends to be more optimistic than G-BRB. Take the results of 2010 as an illustration,

Table 5 Historical data for the five factors. Year

Accident

Distress

Death toll

Salvage

Other ships’ assistance

2007 2008 2009 2010

261 203 136 179

1304 822 1123 1653

17 20 14 5

255 181 114 157

8 191 33 37

the traditional method shows the safety situation is ‘‘Good’’ almost certain (97.6%), while the probability of ‘‘Good’’ is only 64.6% for the G-BRB. 4.2. Cost assessment The questionnaires asking for experts’ opinions on safety are also asked for opinions on cost. The values that the three experts have given to different standards on the two factors used for cost assessment are shown in Table 6. It also can be seen that the second expert’s standard is the highest while the first expert is the lowest. In order to maintain the waterway’s transportation safety situation, MSA need to put a lot of resource into salvage. The resource includes humans, emergency facilities such as search and rescue ships, pollution prevention equipment and so on, which is a high expenditure for MSA. However, when other ships’ assistance is needed in an accident rescue, the extra cost for MSA is not too high in most cases, even though sometimes they need to pay for ships involved in search and rescue. The weights given to Salvage and Other ships’ assistance are 0.7 and 0.3 respectively. Data transformation is carried out in the same way as the safety situation assessment. The generalized versions for experts’ grade value are shown in Table 7. For the factor Salvage, the first expert’s grade value on ‘‘Average’’ is the same as the second expert’s grade value on ‘‘High’’. For the factor Other ships’ assistance, the second expert’s grade value on ‘‘Average’’ is the same as the first and third experts’ grade value on ‘‘Low’’. These are good examples of the distinctions when people are making subjective judgments, showing the need for the method proposed in this paper. In this subsection, the three methods used in the safety situation assessment are used in a similar way. Historical data of salvage and other ships’ assistance in Table 5 is used here. The results are shown in Fig. 4. The figures show that in most situations the results from three methods are quite similar. The results show

Table 4 Generalization results for experts’ grade values on safety-related factors. Expert

1 1 1 2 2 2 3 3 3

(High) (Average) (Low) (High) (Average) (Low) (High) (Average) (Low)

Antecedent attributes Accident

Distress

Death toll

{(H: {(H: {(H: {(H: {(H: {(H: {(H: {(H: {(H:

{(H: {(H: {(H: {(H: {(H: {(H: {(H: {(H: {(H:

{(H: {(H: {(H: {(H: {(H: {(H: {(H: {(H: {(H:

1), (A: 0), (L: 0)} 0.0621), (A: 0.9379), (L: 0)} 0), (A: 0), (L: 1)} 0.6271), (A: 0.3729), (L: 0)} 0), (A: 0.7797), (L: 0.2203)} 0 (A: 0), (L: 1)} 1), (A: 0), (L: 0)} 0.4576), (A: 0.5424), (L: 0)} 0), (A: 0.2712), (L: 0.7288)}

1), (A: 0), (L: 0)} 0.8936), (A: 0.1064), (L: 0)} 0), (A: 0.8298), (L: 0.1702)} 0.8936), (A: 0.1064), (L: 0.0204)} 0), (A: 0.8298), (L: 0.1702)} 0), (A: 0), (L: 1)} 0), (A: 0.8298), (L: 0.1702)} 0), (A: 0.0851), (L: 0.9149)} 0), (A: 0), (L: 1)}

1), (A: 0), (L: 0)} 0.5), (A: 0.5), (L: 0)} 0), (A: 0.5), (L: 0.5)} 1), (A: 0), (L: 0)} 0), (A: 0.375), (L: 0.625)} 0), (A: 0), (L: 1)} 0.2727), (A: 0.7273), (L: 0)} 0), (A: 0.5), (L: 0.5)} 0), (A: 0), (L: 1)}

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Fig. 3. Safety situation assessment results.

Table 6 Experts’ grade standard for cost-related factors. Expert (weight)

Expert 1 (0.3) Expert 2 (0.5) Expert 3 (0.2) Weighted average

Salvage (weight: 0.7)

Other ships’ assistance (weight: 0.3)

High

Average

Low

High

Average

Low

280 200 250 234

200 150 185 172

120 100 120 110

80 50 75 64

55 30 52.5 42

30 10 30 20

Table 7 Generalization results for experts’ grade values on cost-related factors. Expert

1 1 1 2 2 2 3 3 3

(High) (Average) (Low) (High) (Average) (Low) (High) (Average) (Low)

Antecedent attributes Salvage

Other ships’ assistance

{(H: {(H: {(H: {(H: {(H: {(H: {(H: {(H: {(H:

{(H: {(H: {(H: {(H: {(H: {(H: {(H: {(H: {(H:

1), (A: 0), (L: 0)} 0.4516), (A: 0.5484), (L: 0)} 0), (A: 0.1613), (L: 0.8387)} 0.4516), (A: 0.5484), (L: 0)} 0), (A: 0.6452), (L: 0.3548)} 0 (A: 0), (L: 1)} 1), (A: 0), (L: 0)} 0.2097), (A: 0.7903), (L: 0)} 0), (A: 0.1613), (L: 0.8387)}

that the cost that MSA put into emergencies has been decreasing over the period. The cost in 2009 was much lower than the other three years. 4.3. MSA performance assessment In this sub section, MSA performance is assessed based on the results of safety situation and cost in Sections 4.1 and 4.2. Since both safety situation and cost outputs are quantitative and generalized versions, it is not necessary to transform the input data. In MSA performance assessment, five grades are used, which are ‘‘Very Poor’’, ‘‘Poor’’, ‘‘Average’’, ‘‘Good’’, and ‘‘Very Good’’. The three experts are still required to give their opinions on C-BSs for BRBs used to assess MSA performance. Due to the fact that both safety situation and cost are qualitative, it is not necessary to use generalized versions for MSA performance assessment. The primary responsibility for MSA is to guarantee the safety of waterway transportation. A much larger weight (0.7) is therefore given to the ‘‘Safety’’ and a smaller weight (0.3) is given to ‘‘Cost’’. Both traditional and G-BRBs activate BRBs in the same way. The

1), (A: 0), (L: 0)} 0.5909), (A: 0.4091), (L: 0)} 0), (A: 0.4545), (L: 0.5455)} 0.3636), (A: 0.6364), (L: 0)} 0), (A: 0.4545), (L: 0.5455)} 0), (A: 0), (L: 1)} 1), (A: 0), (L: 0)} 0.4773), (A: 0.5227), (L: 0)} 0), (A: 0.4545), (L: 0.5455)}

difference is that their input for safety situation and cost are from the corresponding results in Sections 4.1 and 4.2. So their results for MSA performance are not the same. In term of BN methodology, the C-BSs are used as the CPT and the results of safety situation and cost are treated as prior probability distribution. The results are shown in Fig. 5. According to the figures, it is almost sure that the MSA performance was not better than ‘‘Average’’ in 2007 and 2008 and its performance was no worse than ‘‘Average’’ in 2010. The distribution in 2009 is rather scattered, which implies that the uncertainty is high. According to the results, it can be seen that the MSA performance improved to a large extent. The historical data shows that factors of accident number, the death toll and salvage share similar trend with MSA performance. This phenomenon implies that these factors have greater influence on MSA’s performance than others. Consequently, special attention should be paid to them to improve safety management quality. In order to evaluate the results from the above three methodologies, 20 extra experts in maritime transportation were required to give their opinions on MSA performance during the years 2007 and

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J. Zhang et al. / Safety Science 63 (2014) 157–167

Fig. 4. Cost assessment results for MSA.

2010. Before making such a decision, all the historical data are given to them. After that, the MSA’s performance can be obtained in the following way. If there are three experts believing that the MSA performance is ‘‘Poor’’, then the probability given to ‘‘Poor’’ is 3/ 20 = 0.15. All the data from experts is treated in the same way. The results are also shown in Fig. 5, which are called expert opinions. In order to compare the results from different methods with expert opinions, a utility function is introduced. Suppose the consequence distribution is {(C1, a1), . . ., (CN, aN), the utility function is defined as follow:

f ðfðC 1 ; a1 Þ; :::; ðC N ; aN ÞgÞ ¼

N X

ai uðC i Þ

ð22Þ

i¼1

where u(Ci) is value given to the corresponding grade. The utility value given to the worst grade ‘‘Very Poor’’ is 1. The following utility values are given with an increment of 1 at each step and utility value given to ‘‘Very Good’’ is 5. The relative discrepancy to the 20 experts’ knowledge for each method can be computed by using the following equation:

e¼

 1 Þ; . . . ; ðC N ; a  N ÞgÞ f ðfðC 1 ; a1 Þ; . . . ; ðC N ; aN ÞgÞ  f ðfðC 1 ; a  1 Þ; . . . ; ðC N ; a  N ÞgÞ f ðfðC 1 ; a  100%

ð23Þ

100%

Traditional BRB Bayes Networks G-BRB Expert opinions

80% 60%

where {(C1, a1), . . ., (CN, aN) is the result from any method (Bayes Networks, traditional and G-BRB). It is generally believed that results from more experts tend to be more precise. So results from different methods are compared with expert opinions separately in term of precision. Table 8 shows the discrepancies of the three methods for the four years. It can be seen from Table 8 that the results of the G-BRB are closest to expert opinions. The G-BRB is more reliable than the traditional method, especially in the year 2009 and 2010. In general, both traditional and G-BRBs are better than BN. The results show that the G-BRB is more precise than traditional BRB method as a whole, which implies that it is more reasonable to treat experts’ standards separately and transform them into uncertain versions. The average discrepancy is also the lowest compared with the other two methods. The results also show that both traditional and G-BRB methods are better than Bayes Networks in solving the problem. It is interesting to see that the results obtained by dealing with fewer experts’ detailed knowledge are quite similar with results from more experts’ questionnaires on MSA performance directly. The proposed G-BRB is quite useful when there are only few experts who are experienced in the domain that need to be assessed. It should be mentioned that in the proposed method, the largest weight is given to the second expert (0.5), who had the highest standard on both safety and cost. Most of the 20 extra experts

100%

60%

40%

40%

20%

20%

0%

Traditional BRB Bayes Networks G-BRB Expert opinions

80%

0% Very Poor

Poor

Average

Good

Very Good

Very Poor

Poor

(2007) 100% 80% 60%

Average

Good

Very Good

Good

Very Good

(2008) 100%

Traditional BRB Bayes Networks G-BRB Expert opinions

80% 60%

40%

40%

20%

20%

0%

Traditional BRB Bayes Networks G-BRB Expert opinions

0% Very Poor

Poor

Average

Good

Very Good

Very Poor

Poor

(2009)

Average

(2010) Fig. 5. MSA performance assessment results.

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Table 8 Assessment results discrepancy comparison for three methods. Method (%)

Traditional BRB Bayes Networks G-BRB

Year 2007

2008

2009

2010

Average error

13.49 44.83 13.34

7.70 27.74 3.50

9.82 1.16 3.13

4.19 13.72 1.22

8.80 21.86 5.30

Table 9 MSA performance assessment results discrepancies variation by giving different weights to three experts (the year of 2007). Method (%)

Traditional method Bayes Networks G-BRB

Weights given to three experts w1 = 0.3 w2 = 0.5 w3 = 0.2

w1 = 0.35 w2 = 0.45 w3 = 0.2

w1 = 0.4 w2 = 0.4 w3 = 0.2

w1 = 1/3 w2 = 1/3 w3 = 1/3

w1 = 0.3 w2 = 0.6 w3 = 0.1

13.49 44.83 13.34

20.91 50.70 17.63

7.62 38.19 14.63

30.37 33.65 23.30

25.70 33.45 14.06

are experienced maritime safety administrators, who also are likely to have a high standard of safety and cost. Consequently, the results for MSA performance tend to be conservative as well. In order to test the robustness of the proposed method, more simulations are carried out by giving different weights to the three experts. It can be seen from Table 8 that the results of the year 2007 has the lowest precision. So the year of 2007 is selected to study what happens as the weights given to the three experts are changed. Four more scenarios are simulated and the results are shown in Table 9. It can be seen from Table 9 that the discrepancies of the G-BRB stay at low levels with changes of weights given to three experts, while the error fluctuates a lot for the other two methods. In practice, a larger weight tends to be given to experienced experts. But according to Table 9, even when the same weight is given to the three experts, the error is still not too large (23.3%) for the GBRB, which reflects that the reliability of the proposed method is high. 5. Conclusions In this paper, a possible generalization of belief rule base theory is studied. The traditional BRB is first divided into two parts and they are treated separately. This paper focuses on the generalization of antecedent attributes. Due to the fact that people’s standards on a typical factor are usually not identical, experts’ opinions on grade values for a typical attribute are different under many situations. Each antecedent attribute from the ‘IF’ part is extended into a belief structure version (which is called A-BS) instead of certain one to better express experts’ knowledge. The calculations of match degree and activation weight are also modified to deal with A-BSs. The proposed G-BRB is then used to assess the MSA performance by dividing all factors into two groups. The safety situation and cost are assessed separately at first. Then the MSA performance is assessed for the years 2007–2010 with three methods (traditional BRB, BN and the G-BRB). All the results are compared with results from 20 extra experts’ opinions to evaluate the precision. The results show that the proposed method could get satisfactory results and be reliable to weights given to experts’ opinions. The differences between BRB and BN are as follows: (1) The theoretical basis for BRB is D-S theory and evidential reasoning, which are nonlinear in essence, while BN combines the prior knowledge and CPT in a linear way. In general, the nonlinear combinations tend to make the method more reliable. (2) BRB can con-

trol the degree of influence of one factor to its upper level attribute by giving a weight (d) to it, while the BN used in this paper has no such parameter and can only treat factors’ importance to upper level attribute by CPT. All the reported accidents are treated equally in this paper. However, a large percentage of them result in small or negligible consequences, while some of them have brought serious economic loss, fatalities or pollution. So consequence evaluation for historic accidents should be carried out in the future to treat them separately. What is more, factors related to pollution should also be considered to make MSA performance evaluation more comprehensive. These factors are not taken into consideration because of lack of relevant historical data. It must be mentioned that experts’ knowledge on both safety situation and cost will change as time goes by. For instance, the safety situation in one waterway can be improved to a large degree after the MSA’s efforts for many years. Their requirements on safety will also be raised. Under this circumstance, experts’ standards on safety will become much stricter as well. That is to say, experts’ standards on safety and cost will change during the four years. The questionnaires used in this paper are done in 2012. So all the results are based on the same safety and cost standards. If the safety situation or MSA’ cost on maintaining safety change in the future, the results will also be different. Consequently, more study is still needed to further evaluate the model. Acknowledgments The authors would like to thank the anonymous reviewers, whose comments and suggestions are great helpful in improving the quality of this paper. The authors would like to thank all the experts who give support by data collection and questionnaires. Without their help, the work could not have finished. This work is financially supported by the China Scholarship Council (CSC) and Marie Curie Actions (FP7-PEOPLE-2012-IRSES), National Science Foundation of China (NSFC) under Grant Nos. 51209165 and 51179146, and Science and Technology project of transportation from Ministry of Transport (MOT) of China under Grand No. 2011 328 201 90. References Abdelgawad, M., Fayek, A.R., 2010. Risk management in the construction industry using combined fuzzy FMEA and fuzzy AHP. Journal of Construction Engineering and Management 136 (9), 1028–1036. Antão, P., Almeida, T., Jacinto, C., Soares, C.G., 2008. Causes of occupational accidents in the fishing sector in Portugal. Safety Science 46 (6), 885–899. Bharathi, V., Vaidya, O., Parikh, S., 2012. Prioritizing and ranking critical success factors for ERP adoption in SMEs. AIMS International Journal of Management 6 (1), 23–40. Caputo, A.C., Pelagagge, P.M., Salini, P., 2013. AHP-based methodology for selecting safety devices of industrial machinery. Safety Science 53, 202–218. Cebeci, U., 2009. Fuzzy AHP-based decision support system for selecting ERP systems in textile industry by using balanced scorecard. Expert Systems with Applications 36 (5), 8900–8909. Chin, K.S., Yang, J.B., Guo, M., Lam, J.P.K., 2009. An evidential-reasoning intervalbased method for new product design assessment. IEEE Transactions on Engineering Management 56 (1), 142–156. Christos, A.K., Harilaos, N.P., 2009. Formal safety assessment: a critical review. Marine Technology 46, 45–59. Dempster, A., 1967. Upper and lower probabilities induced by a multi-valued mapping. The Annals of Statistics 28, 325–339. Fera, M., Macchiaroli, R., 2010. Appraisal of a new risk assessment model for SME. Safety Science 48 (10), 1361–1368. Floyde, A., Lawson, G., Shalloe, S., Eastgate, R., D’Cruz, M., 2013. The design and implementation of knowledge management systems and e-learning for improved occupational health and safety in small to medium sized enterprises. Safety Science 60, 69–76. IMO MSC/ Circ. 829. 1997. Interim guidelines for the application of formal safety assessment (FSA) to the IMO rule-making process. International Maritime, Organization. IMO MSC/ Circ. 1023. 2002. Guidelines for Formal Safety Assessment (FSA) for use in the IMO Rule-Making process. International Maritime Organization.

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