1-s2.0-s0360835207000721-main

Computers & Industrial Engineering 53 (2007) 137–148

www.elsevier.com/locate/dsw

A fuzzy group decision making approach for bridgerisk assessment q

Ying-Ming Wang a,*, Taha M.S. Elhag b

a Institute of Soft Science, Fuzhou University, Fuzhou 350002, PR Chinab School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, P.O. Box 88, Manchester M60 1QD, UK

Received 20 June 2006; received in revised form 19 April 2007; accepted 26 April 2007Available online 3 May 2007

Abstract

This paper proposes a fuzzy group decision making (FGDM) approach for bridge risk assessment. The FGDMapproach allows decision makers (DMs) to evaluate bridge risk factors using linguistic terms such as Certain, Very High,High, Slightly High, Medium, Slightly Low, Low, Very Low or None rather than precise numerical values, allows them toexpress their opinions independently, and also provides two alternative algorithms to aggregate the assessments of multiplebridge risk factors, one of which offers a rapid assessment and the other one leads to an exact assessment. A case study isinvestigated using the FGDM approach to illustrate its applications in bridge risk assessment. It is shown that the FGDMapproach offers a flexible, practical and effective way of modelling bridge risks.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Bridge risk assessment; Fuzzy logic; Group decision making; Fuzzy weighted average

1. Introduction

Bridge risk assessment is often conducted to determine the priority or the optimal scheme of bridge main-tenance. For instance, Adey, Hajdin, and Bruhwiler (2003) presented a risk-based approach to determiningthe optimal intervention for a bridge subject to multiple hazards. Johnson and Niezgoda (2004) presented arisk-based method for ranking, comparing and choosing the most appropriate bridge scour countermeasuresusing failure mode and effects analysis (FMEA) and risk priority numbers (PRNs). Stein, Young, Trent, andPearson (1999) developed a risk-based method for assessing the risk associated with scour threat to bridgefoundations. The risk of scour failure was defined as the product of the probability of scour failure or heavydamage and the cost associated with the failure, adjusted by a risk adjustment factor based on foundationand span types. Shetty, Chubb, Knowles, and Halden (1996) proposed a risk-based framework for assess-

0360-8352/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.cie.2007.04.009

q This research was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) under Grant No. GR/S66770/01.

* Corresponding author. Tel.: +86 591 87893307; fax: +86 591 87892545.E-mail address: [email protected] (Y.-M. Wang).

mailto:[email protected]

138 Y.-M. Wang, T.M.S. Elhag / Computers & Industrial Engineering 53 (2007) 137–148

ment and prioritization of bridges in need of remedial work, which involves risk evaluation, rankings ofbridges in terms of risk, design of remedial action for each bridge, and optimal allocation of resourcesfor remedial work on different bridges. Risk is quantified as the product of probability of failure and con-sequences of failure. Lounis (2004) presented a risk-based approach for bridge maintenance optimizationthat takes into account several and possibly conflicting criteria, with emphasis on the risk of failure as a gov-erning criterion. The optimal maintenance strategy was defined as the solution that achieved the best com-promise among the three selected relevant and conflicting criteria: minimization of risk of failure,minimization of maintenance costs, and minimization of traffic disruption. Compromise programmingwas used to determine the optimal ranking of maintenance strategies in terms of their effectiveness in riskreduction, cost minimization, and traffic control. A multi-criteria optimality index was proposed as a mea-sure of the effectiveness of the optimal maintenance strategy in achieving a satisfactory trade-off between therelevant and competing maintenance criteria.

Bridge risk is usually assessed against multiple criteria (or called risk factors) and by a group of decisionmakers (DMs) and is therefore a typical group decision making problem. Due to the fact that bridge risk can-not be precisely measured and can only be assessed using DMs’ knowledge and subjective judgments, it can bewell modelled using linguistic variables and fuzzy set theory. In fact, fuzzy logic has been widely used to assessvarious risks such as construction project risk (Carr & Tah, 2001; Cho, Choi, & Kim, 2002; Kuchta, 2001),software development risk (Chen, 2001; Lee, 1996a; Lee, 1996b; Lee, 1999; Lee, Lee, Lee, & Chen, 2003), soft-ware operational risk (Xu, Khoshgoftaar, & Allen, 2003), forest fire risk (Iliadis, 2005), e-commerce develop-ment risk (Ngai & Wat, 2005), environmental risk (Sadiq & Husain, 2005), investment risk (Serguieva &Hunter, 2004), tourist risk (Tsaur, Tzeng, & Wang, 1997), and son on. It seems to us, however, no attempthas been made so far to model bridge risk using fuzzy set theory. This paper aims at developing a practicaland effective fuzzy group decision making (FGDM) approach and providing two alternative algorithms forbridge risk assessment.

In comparison with the existing approaches for bridge risk assessment, the proposed FGDM approachtreats bridge risk assessment as a fuzzy multiple criteria group decision making problem. It allows a groupof bridge experts (DMs) to make their judgments independently and express their opinions (judgments) infuzzy linguistic terms rather than in precise numerical values that prove to be difficult in practice. So, the pro-posed FGDM approach will be easier to use and more realistic.

The paper is organized as follows: In Section 2, we briefly introduce some basic concepts on fuzzy sets,including fuzzy numbers, fuzzy arithmetics, defuzzification and fuzzy weighted average, to pave the way forthe FGDM approach. In Section 3, we develop the FGDM approach to bridge risk assessment and providetwo alternative algorithms to aggregate risk ratings. Section 4 investigates a case study using the proposedFGDM approach and algorithms to illustrate their potential applications in bridge risk assessment. Conclu-sions are offered in Section 5.

2. Basic concepts on fuzzy sets

Fuzzy sets are generalizations of crisp sets and were first introduced by Zadeh (1965) as a way of represent-ing imprecise or vagueness in real world. A fuzzy set is a collection of elements in a universe of informationwhere the boundary of the set contained in the universe is ambiguous, vague and otherwise fuzzy. Each fuzzyset is specified by a membership function, which assigns to each element in the universe of discourse a valuewithin the unit interval [0, 1]. The assigned value is called degree (or grade) of membership, which specifies theextent to which a given element belongs to the fuzzy set or is related to a concept. If the assigned value is 0,then the given element does not belong to the set. If the assigned value is 1, then the element totally belongs tothe set. If the value lies within the interval (0, 1), then the element only partially belongs to the set. Therefore,any fuzzy set can be uniquely determined by its membership function.

Let X be the universe of discourse. A fuzzy set ~A of the universe of discourse X is said to be convex if andonly if for all x1 and x2 in X there always exists:

l~Aðkx1 þ ð1� kÞx2Þ P Minðl~Aðx1Þ; l~Aðx2ÞÞ; ð1Þ
where l~AðxÞ is the membership function of the fuzzy set ~A and k 2 [0,1].

Y.-M. Wang, T.M.S. Elhag / Computers & Industrial Engineering 53 (2007) 137–148 139

A fuzzy set ~A of the universe of discourse X is said to be normal if there exists a xi 2 X satisfying l~AðxiÞ ¼ 1.Fuzzy sets can also be represented by intervals, which are called a-level sets or a-cuts. The a-level sets Aa of a

fuzzy set ~A are defined as

Aa ¼ fx 2 X jl~AðxÞP ag ¼ ½minfx 2 X jl~AðxÞP ag;maxfx 2 X jl~AðxÞP ag�: ð2Þ

According to Zadeh’s extension principle (Zadeh, 1965), the fuzzy set ~A can be expressed as

~A ¼ [aaAa; 0 < a 6 1: ð3Þ

Fuzzy numbers are special cases of fuzzy sets that are both convex and normal. A fuzzy number is a convexfuzzy set, characterized by a given interval of real numbers, each with a grade of membership between 0 and 1.Its membership function is piecewise continuous and satisfies the following conditions:

(a) l~AðxÞ ¼ 0 for each x 62 [a,d];(b) l~AðxÞ is non-decreasing (monotonic increasing) on [a, b] and non-increasing (monotonic decreasing) on

[c,d];(c) l~AðxÞ ¼ 1 for each x 2 [b,c],

where a 6 b 6 c 6 d are real numbers in the real line R = (�1, +1).The most commonly used fuzzy numbers are triangular and trapezoidal fuzzy numbers, whose membership

functions are respectively defined as

l~A1ðxÞ ¼

ðx� aÞ=ðb� aÞ; a 6 x 6 b;

ðd� xÞ=ðd� bÞ; b 6 x 6 d;

0; otherwise:

8><>: ð4Þ

l~A2ðxÞ ¼

ðx� aÞ=ðb� aÞ; a 6 x 6 b;

1; b 6 x 6 c;

ðd� xÞ=ðd� cÞ; c 6 x 6 d;

0; otherwise:

8>>><>>>: ð5Þ

For brevity, triangular and trapezoidal fuzzy numbers are often denoted as (a,b,d) and (a,b,c,d). It is obvi-ous that triangular fuzzy numbers are special cases of trapezoidal fuzzy numbers with b = c.

Let ~A ¼ ða1; a2; a3Þ and ~B ¼ ðb1; b2; b3Þ be two positive triangular fuzzy numbers. Then basic fuzzy arithme-tic operations on these fuzzy numbers are defined as (Dubois & Prade, 1980; Kauffman & Gupta, 1991)

Addition: ~Aþ ~B ¼ ða1 þ b1; a2 þ b2; a3 þ b3Þ;Subtraction: ~A� ~B ¼ ða1 � b3; a2 � b2; a3 � b1Þ;Multiplication: ~A� ~B � ða1b1; a2b2; a3b3Þ;Division: ~A� ~B � ða1

b3; a2

b2; a3

b1Þ.

An important concept related to the applications of fuzzy numbers is defuzzification, which converts afuzzy number into a crisp value. Such a transformation is not unique because different methods are possible.The most commonly used defuzzification method is the centroid defuzzification method, which is also knownas center of gravity or center of area defuzzification. The centroid defuzzification method can be expressed asfollows (Yager, 1981):

�x0ð~AÞ ¼R d

a xl~AðxÞdxR da l~AðxÞdx

; ð6Þ

where �x0ð~AÞ is the defuzzified value. For trapezoidal fuzzy numbers (a,b,c,d), the centroid-based defuzzifiedvalue turns out to be

�x0ð~AÞ ¼1

3aþ bþ cþ d� dc� ab

ðdþ cÞ � ðaþ bÞ

� �: ð7Þ

Especially when b = c, the above formula becomes


�x0ð~AÞ ¼aþ bþ d

3; ð8Þ

which is the defuzzification formula of triangular fuzzy numbers (a,b,d) and will be used in this paper.The fuzzy weighted average of fuzzy numbers is referred to as fuzzy weighted average (FWA), which is

defined as

~y ¼ ~w1~x1 þ ~w2~x2 þ � � � þ ~wn~xn

~w1 þ � � � þ ~wn; ð9Þ

where ~x1; . . . ;~xn are n fuzzy numbers to be weighted and ~w1; . . . ; ~wn are fuzzy weights. Fuzzy arithmetic oper-ations are found not suitable for computing ~y because the weight variables appear in both denominator andnumerator simultaneously. Lots of research has been done on how to compute ~y. The most commonly usedapproach is to calculate ~y using the extension principle. Let xia ¼ ½xL

ia; xUia�, wia ¼ ½wL

ia;wUia� and ya ¼ ½yL

a ; yUa � be

the a-level sets of ~xi, ~wi and ~y, respectively. Then ya ¼ ½yLa ; y

Ua � can be derived by the following pair of fractional

programming models:

yLa ¼Min

w1xL1a þ w2xL

2a þ � � � þ wnxLna

w1 þ w2 þ � � � þ wn

s:t: wLia 6 wi 6 wU

ia; i ¼ 1; . . . ; n:ð10Þ

yUa ¼Max

w1xU1a þ w2xU

2a þ � � � þ wnxUna

w1 þ w2 þ � � � þ wn

s:t: wLia 6 wi 6 wU

ia; i ¼ 1; . . . ; n:ð11Þ

Let

z ¼ 1=ðw1 þ w2 þ � � � þ wnÞ; ð12Þvi ¼ zwi; i ¼ 1; . . . ; n: ð13Þ

The above fractional programming models can be simplified as (Kao & Liu, 2001)

yLa ¼Min v1xL

1a þ v2xL2a þ � � � þ vnxL

na

s:t: v1 þ v2 þ � � � þ vn ¼ 1;

z � wLia 6 vi 6 z � wU

ia; i ¼ 1; . . . ; n;

z P 0:

ð14Þ

yUa ¼Max v1xU

1a þ v2xU2a þ � � � þ vnxU

na

s:t: v1 þ v2 þ � � � þ vn ¼ 1;

z � wLia 6 vi 6 z � wU

ia; i ¼ 1; . . . ; n;

z P 0:

ð15Þ

These are linear programming models and are easy to solve using MS Excel Solver or LINDO software package.

3. The FGDM approach for bridge risk assessment

According to the British Highways Agency (2004), bridge risk can be expressed as any event or hazard thatcould hinder the achievement of business goals or the delivery of stakeholder expectations and is defined as theproduct of the likelihood and consequences of an event occurring. That is

Risk ¼ Likelihood� Consequences: ð16Þ
Risks associated with bridge structures maintenance activities include deterioration, failure to meet the Agency’sobligations for freedom of movement on the network and failure of a component, element or structure.
To identify and assess bridge risks, maintenance needs have to be identified, based on which projects can bedeveloped, each project addressing either singular or multiple structures maintenance needs. All maintenanceneeds can be classified as being either:


• Essential – work required to maintain safety standards. Work is required to be carried out on structures orstructural elements because they are considered to be unsafe or structurally inadequate, e.g., major concreterepairs or replacement of structural elements.

• Preventative – maintenance work that is not essential now but is justified on economic grounds as it pro-vides minimum whole life cost maintenance. By timely intervention, preventative work will reduce the riskof essential work arising prematurely in the future, e.g., painting of steelwork.

• Upgrading – work resulting from changes in requirements of faults, e.g., parapet replacement, pierupgrading.

Associated with the identified maintenance needs are the risk events that could occur if nothing were doneabout them in the short term of 3–4 years. The risk events can be defined as protective coating failure, dete-rioration, failure to meet obligation for required operational capacity, concrete spalling, equipment failure,non-structural component failure (waterproofing and non-structural joints), structural component failure(structural joints, bearings and parapets), component failure, element failure, or structural failure (global col-lapse). Each of them can be broken down further into a risk event chain.

For each defined risk event, its likelihood and consequences need to be assessed on the basis of evidence andengineering judgment. This can usually be done by a team of bridge experts or called decision makers (DMs)in the terminology of decision analysis. It turns out to be not easy, if not impossible, to require the DMs toprovide precise numerical judgments about the likelihood and consequences of each risk event. So, fuzzy lin-guistic terms are much easier to be accepted and adopted by the DMs. Table 1 shows the linguistic termsdefined for the likelihood of bridge risk event in this paper, whose membership functions are shown inFig. 1. In the case that a risk event is broken down into an event chain, the likelihood of the risk event isdefined to be the product of the likelihood ratings within the event chain.

Note that how to define the membership functions (particularly the parameters) of the linguistic terms isalso a very important issue. It is most desirable that the DMs achieve a consensus on these definitions. If theydisagree with each other, then average values should be used for the definitions.

The consequences of a risk event depend on the level and type of traffic using a structure or route, the fea-tures surrounding the structure (i.e. what it crosses or supports), and the availability of alternative routes, andmight include human injury, network disruption, disruption to spur water or gas mains or other major utilitysupply lines, disruption to other transport networks adjacent to the structure (e.g. rail), repair/replacement ofa structure/component/element, and environment damage.

Table 1Descriptions of likelihood ratings

Likelihood rating Description Fuzzy number

Certain Certainty (either already happened or certain to happen) C = (1.0,1.0,1.0)Very high Very highly likely VH = (0.85,1.0,1.0)High Highly likely H = (0.7,0.85,1.0)Slightly high Likely SH = (0.5,0.7,0.85)Medium Possible, and likely M = (0.3,0.5,0.7)Slightly low Possible, but slightly unlikely SL = (0.15,0.3,0.5)Low Possible, but unlikely L = (0,0.15,0.3)Very low Possible, but very unlikely VL = (0,0,0.15)

VL L SL M SH H VH

0 0.15 0.3 0.5 0.7 0.85 1

Fig. 1. Membership functions of likelihood ratings.

……

Bridge Risk Assessment

Safety Functionality Sustainability Environment

Bridge Structure 1

Bridge Structure 2

Bridge Structure n

Fig. 2. Hierarchical structure of bridge risk assessment.


All the consequences are assessed against the following four criteria (see Fig. 2):

• Safety – safety of the public.• Functionality – effects on the level of service/availability of the network for use.• Sustainability – sustainability of both expenditure and workload, where the aim is to reach a state of steady

expenditure and workload, avoiding the built-up of a backlog of unavoidable, essential work by doing effec-tive, targeted preventative maintenance, i.e. painting steelwork, silane of new concrete and preventativemaintenance on all bridges in new/as new condition. Sustainability will also require the timely replacementof structures.

• Environment – effects on the environment, including the (aesthetic) appearance of the structures.

Consequences must be credible and reasonable and should not be based on an extreme event. Table 2 showsthe linguistic terms defined for the consequences of bridge risk event, whose membership functions are pre-sented in Fig. 3. In the case that a risk event has multiple consequences, the most likely one should be used.

Let ~RðkÞij be the risk ratings of a risk event of the ith bridge structure under evaluation with respect to thefour criteria, provided by DMk (the kth decision maker), k = 1, . . .,m, where m is the number of DMs. ByEq. (16), we have

~RðkÞij ¼gLRðkÞi �gCR

ðkÞij ; i ¼ 1; . . . ; n; j ¼ 1; . . . ; 4; k ¼ 1; . . . ;m; ð17Þ

where gLRðkÞi is the likelihood rating of the risk event provided by DMk and is the same for the four criteria andgCRðkÞij (j = 1, . . ., 4) are the consequences ratings of the risk event. Due to the fact that both gLRðkÞi and gCRðkÞij

are fuzzy numbers, the risk ratings ~RðkÞij are also fuzzy numbers.To determine the overall risk of the risk event, the relative importance weights of the four criteria need to be

specified. This can be done by directly assigning a crisp or fuzzy weight or linguistic term to each criterion.Fig. 4 shows the linguistic terms defined for the relative importance weights of the four criteria.

Let ~wðkÞj (j = 1,. . .,4) be the fuzzy weights assigned by DMk to the four criteria. Note that crisp weights canbe seen as a special case of fuzzy weights. The overall risk (or called aggregative risk) of each risk event can bedetermined in different ways. In what follows, we provide two alternative algorithms.

• Algorithm-1

(1) Average likelihood and consequences ratings as well as the weights of criteria according to fuzzy addi-tion operation:

gLR i ¼1

m

Xm

k¼1

gLRðkÞi ¼1

m

Xm

k¼1

LRðkÞiL ;1

m

Xm

k¼1

LRðkÞiM ;1

m

Xm

k¼1

LRðkÞiU

!; i ¼ 1; . . . ; n; ð18Þ

gCRij ¼1

m

Xm

k¼1

gCRðkÞij ¼1

m

Xm

k¼1

CRðkÞijL ;1

m

Xm

k¼1

CRðkÞijM ;1

m

Xm

k¼1

CRðkÞijU

!; i ¼ 1; . . . ; n; j ¼ 1; . . . ; 4; ð19Þ

~wj ¼1

m

Xm

k¼1

~wðkÞj ¼1

m

Xm

k¼1

wðkÞjL ;1

m

Xm

k¼1

wðkÞjM ;1

m

Xm

k¼1

wðkÞjU

!; j ¼ 1; . . . ; 4; ð20Þ

Table 2Descriptions of consequences of bridge risk events

Consequence type Consequence rating Description Fuzzy number

Safety Very high Potential for great number of fatalities VH = (85, 85,100)High Potential for high number of fatalities H = (50,85,100)Medium Potential for a number of fatalities or high number of serious injuries M = (15,50,85)Low Potential for a number of serious injuries L = (0,15,50)Very low Potential for small number of serious injuries VL = (0,0,15)None No fatalities and injuries N = (0,0,0)

Functionality Very high Closure of a strategic route VH = (85,85,100)High Disruption of a strategic route or closure of a regional route H = (50,85,100)Medium Restriction of a strategic route or disruption of a regional route M = (15,50,85)Low Restriction of a regional route or closure of a local route L = (0,15,50)Very low Restriction of a local route VL = (0,0,15)None No closure, disruption and restriction to a strategic, regional or local route N = (0,0,0)

Sustainability Very high Cost and/or work implications are unacceptable if delay VH = (85,85,100)High Cost and/or work implications are excessive if delay H = (50,85,100)Medium Cost and/or work implications are significant if delay M = (15,50,85)Low Cost and/or work implications increase if delay L = (0,15,50)Very low Cost and/or work implications slightly increase if delay VL = (0,0,15)None No extra cost and/or works if delay N = (0,0,0)

Environment Very high Unacceptable environmental damage VH = (85,85,100)High Significant environmental damage H = (50,85,100)Medium Environmental damage M = (15,50,85)Low Slightly environmental damage L = (0,15,50)Very low Very slightly environmental damage VL = (0,0,15)None No environmental damage N = (0,0,0)

VL L M H VH

0 15 50 85 100

Fig. 3. Membership functions of consequence ratings.

VL L M H VH

0 0.25 0.5 0.75 1.0

Fig. 4. Membership functions of relative importance weights.


where gLRðkÞi ¼ ðLR

ðkÞiL ;LR

ðkÞiM ;LR

ðkÞiU Þ, gCR

ðkÞij ¼ ðLR

ðkÞijL ;LR

ðkÞijM ;LR

ðkÞijU Þ and ~wðkÞj ¼ ðw

ðkÞjL ;w

ðkÞjM ;w

ðkÞjU Þ are respec-

tively the triangular fuzzy numbers on likelihood and consequences ratings as well as the relative impor-tance weights of criteria provided by DMk.

(2) Calculate risk ratings by Eq. (17) and fuzzy multiplication operation:

~Rij ¼gLR i �gCRij ¼ ðLRiL � CRijL;LRiM � CRijM ;LRiU � CRijU Þ; i ¼ 1; . . . ; n; j ¼ 1; . . . ; 4: ð21Þ


(3) Defuzzify the risk ratings and the weights of criteria by Eq. (8):

�Rij ¼1

3ðRijL þ RijM þ RijU Þ; i ¼ 1; . . . ; n; j ¼ 1; . . . ; 4; ð22Þ

�wj ¼1

3ðwjL þ wjM þ wjU Þ; j ¼ 1; . . . ; 4; ð23Þ

where ~Rij ¼ ðRijL;RijM ;RijU Þ and ~wj ¼ ðwjL;wjM ;wjU Þ are the triangular fuzzy numbers on risk ratings andthe relative importance weights of criteria, respectively.

(4) Generate overall risk score by weighting and averaging the risk ratings:

RSi ¼X4

j¼1

�wj�Rij=

X4

j¼1

�wj; i ¼ 1; . . . ; n: ð24Þ

(5) Rank and prioritize bridge structures according to their overall risk scores: big risk score means high riskand high priority.

• Algorithm-2

(1) Average likelihood and consequences ratings and criteria weights by Eqs. (18)–(20).(2) Compute the a-level sets of the above averaged likelihood and consequences ratings and the weights of

criteria by Eq. (2):

ðLRiÞa ¼ ½ðLRiÞLa ; ðLRiÞUa � ¼ ½LRiL þ aðLRiM � LRiLÞ;LRiU þ aðLRiU � LRiMÞ�; ð25ÞðCRijÞa ¼ ½ðCRijÞLa ; ðCRijÞUa � ¼ ½CRijL þ aðCRijM � CRijLÞ;CRijU þ aðCRijU � CRijMÞ�; ð26ÞðwjÞa ¼ ½ðwjÞLa ; ðwjÞUa � ¼ ½wjL þ aðwjM � wjLÞ;wjU þ aðwjU � wjMÞ�: ð27Þ

(3) Compute the a-level set, ðRSiÞ ¼ ½ðRSiÞLa ; ðRSiÞUa �, of the overall risk score by FWA:

ðRSiÞLa ¼Min

P4j¼1wjððLRiÞLa � ðCRijÞLaÞP4

j¼1wj

;

s:t: ðwjÞLa 6 wj 6 ðwjÞUa ; j ¼ 1; . . . ; 4;

ð28Þ

ðRSiÞUa ¼Max

P4j¼1wjððLRiÞUa � ðCRijÞUa ÞP4

j¼1wj

;

s:t: ðwjÞLa 6 wj 6 ðwjÞUa ; j ¼ 1; . . . ; 4;

ð29Þ

which can be transformed into the following pair of linear programming (LP) models:

ðRSiÞLa ¼MinX4

j¼1

vjððLRiÞLa � ðCRijÞLaÞ;

s:t:X4

j¼1

vj ¼ 1;

ðwjÞLa � z 6 vj 6 ðwjÞUa � z; j ¼ 1; . . . ; 4;

z P 0:

ð30Þ

ðRSiÞUa ¼MaxX4

j¼1

vjððLRiÞUa � ðCRijÞUa Þ;

s:t:X4

j¼1

vj ¼ 1;

ðwjÞLa � z 6 vj 6 ðwjÞUa � z; j ¼ 1; . . . ; 4;

z P 0:

ð31Þ


(4) Compute ranking index by the following equation (Chen & Klein, 1997):

RIi ¼Pn1

l¼0½ðRSiÞUal� c�Pn1

l¼0½ðRSiÞUal� c� �

Pn1

l¼0½ðRSiÞLal� d�

¼Pn1

l¼0ðRSiÞUal� ðnþ 1ÞcPn1

l¼0ðRSiÞUai�Pn1

l¼0ðRSiÞLalþ ðn� 1Þðd� cÞ

; i ¼ 1; . . . ; n; ð32Þ

where c ¼ mini;lfðRSiÞLalg ¼ minifðRSiÞLa0

g, d ¼ maxi;lfðRSiÞUalg ¼ maxifðRSiÞUa0

g and n1 is the number ofa levels minus one, satisfying 0 ¼ a0 < a1 < � � � < an1

¼ 1.
(5) Rank and prioritize bridge structures according to their ranking indices: the bigger the ranking index, the
higher the risk and the priority.

The difference between Algorithm-1 and Algorithm-2 is that the former changes a fuzzy multiple criteriadecision making (MCDM) problem into a crisp (non-fuzzy) one through defuzzification, which is the mostcommon and simplest way of dealing with fuzzy MCDM problems and has been widely used in fuzzy multiplecriteria decision analysis (Carr & Tah, 2001; Chen, 2001; Cho et al., 2002; Lee, 1996a, 1996b, 1999; Lee et al.,2003; Ngai & Wat, 2005; Sadiq & Husain, 2005), while the latter solves the fuzzy MCDM problem using fuzzyextension principle and only defuzzifies the final results for comparison and/or ranking purpose. So, the for-mer produces a crisp overall risk score for each bridge structure, while the latter yields a fuzzy overall riskscore, which is thought to be the exact solution to the fuzzy MCDM problem under discussion.

Algorithm-1 proves to be simple, effective and easy to use and offers a rapid assessment to fuzzy MCDMproblems. Algorithm-2 is relatively complicated, but provides an exact assessment to fuzzy MCDM problemsand more information about assessment results. Users can choose either of the two algorithms or both of themto solve their problems.

In Algorithms-1 and 2, we average each DM’s opinions (ratings and criteria weights) equally, which impliesthat the DMs are equally important. In practice, if the DMs are of different importance, then their opinionsshould be weighted. The weights can be either crisp or fuzzy numbers like those defined by Fig. 4.

4. A case study

The British Highways Agency has identified thousands of risk events associated with bridge structures inthe past. From their database, we randomly select five risk events associated with five bridge structures asour case study and consider three bridge experts as a group/team of DMs for simplicity. The three expertsare asked to assess the five risk events against safety (X1), functionality (X2), sustainability (X3) and environ-ment (X4) criteria independently using the fuzzy linguistic terms defined in Tables 1 and 2. The ratings given bythe three experts are shown in Table 3. These ratings are hypothetical for illustrative purpose because the Brit-ish Highways Agency utilizes simple rating technique rather than fuzzy technique and group decision makingapproach to evaluate bridge risks. The relative importance weights of the four criteria are shown in Table 4,where the linguistic terms are defined in Fig. 4.

The two algorithms both average the likelihood and consequences ratings given by the three experts so thata group decision making problem can be simplified as a decision making problem with only one DM. Theaveraged ratings are also shown in Table 3.

Algorithm-1 converts the fuzzy MCDM problem into a crisp MCDM, which is shown in Table 5, where theweights are normalized. From the overall risk scores of the five bridge structures, it is clear that bridge struc-ture 1 has the highest risk and should be given top priority for maintenance, followed by bridge structures 2and 3. The priority ranking of the five bridge structures is obtained as BS1 > BS2 > BS3 > BS4 > BS5.

To generate an exact fuzzy assessment for each bridge structure, eleven a levels are set for computation, i.e.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. Accordingly, eleven a-level sets are derived for the overall riskscore of each bridge structure, which are shown in Table 6 and depicted in Fig. 5. The overall risk scores of thefive bridge structures are all fuzzy numbers. This makes sense because the original ratings are all fuzzy num-bers. Obviously, Algorithm-2 provides more information on the overall risk score of each bridge structure

Table 4The relative importance weights of four criteria assessed by the three DMs

Criterion DM1 DM2 DM3 Average

Safety (X1) VH VH VH (0.75,1.0,1.0)Functionality (X2) M H H (0.42,0.67,0.92)Sustainability (X3) L M M (0.17,0.42,0.67)Environment (X4) VL L L (0,0.17,0.42)

Table 5Bridge risk assessments of the five bridge structures by Algorithm-1

Bridge structure Safety Functionality Sustainability Environment Overall risk score Rank0.4177 0.3038 0.1899 0.0886

BS1 77.17 63.25 74.25 47.08 69.72 1BS2 62.43 62.43 53.67 39.81 58.76 2BS3 52.05 72.45 36.36 44.50 54.60 3BS4 1.67 80.17 80.17 58.56 45.46 4BS5 43.99 7.28 24.38 31.18 27.98 5

Table 3Likelihood and consequences ratings for five risk events (bridge structures) assessed by three DMs

Bridge structure Risk factor Likelihood Consequences

DM1 DM2 DM3 Average DM1 DM2 DM3 Average

BS1 X1 H H VH (0.75,0.90,1.00) H VH VH (73,85,100)X2 H H M (38,73,95)X3 VH H H (62,85,100)X4 M M M (15,50,85)

BS2 X1 M H H (0.57,0.73,0.90) VH H H (62,85,100)X2 H VH H (62,85,100)X3 H M H (38,73,95)X4 M L H (22,50,78)

BS3 X1 VH H SH (0.68,0.85,0.95) M H M (27,62,90)X2 H VH VH (73,85,100)X3 M L M (10,38,73)X4 M M M (15,50,85)

BS4 X1 C VH VH (0.90,1.00,1.00) N VL N (0,0,5)X2 H VH H (62,85,100)X3 VH H H (62,85,100)X4 M M H (27,62,90)

BS5 X1 H M SH (0.50,0.68,0.85) M M H (27,62,90)X2 N VL L (0,5,22)X3 L L M (5,27,62)X4 M M L (10,38,73)


than Algorithm-1 despite the fact that it is more complicated in computation than the latter. The final priorityranking produced by Algorithm-2 is BS1 > BS2 > BS3 > BS4 > BS5, which is the same as the priority rankingproduced by Algorithm-1.

5. Conclusions

In this paper, we have developed a fuzzy group decision making approach called FGDM for bridge riskassessment. Bridge risk is defined as the product of the likelihood and consequences of a risk event that couldoccur if nothing were done about the bridge structure needing maintenance in the short term of 3–4 years.The consequences are assessed against safety, functional, sustainability and environment criteria, respectively.The FGDM approach allows a group of DMs to participate in the risk assessment, allows them to make

Table 6a-Level sets of the overall risk scores of the five bridge structures produced by Algorithm-2

a Bridge structures

1 2 3 4 5

0.0 [35.56,99.00] [26.83,89.64] [19.17,88.81] [20.45,72.58] [5.00,59.44]0.1 [38.94,96.20] [29.63,86.39] [22.22,85.29] [22.63,69.88] [6.38,55.35]0.2 [42.38,93.40] [32.53,83.17] [25.36,81.76] [24.90,67.17] [7.89,51.41]0.3 [45.86,90.59] [35.52,79.99] [28.61,78.23] [27.23,64.47] [9.53,47.62]0.4 [49.38,87.79] [38.60,76.84] [31.95,74.71] [29.64,61.76] [11.31,43.98]0.5 [52.92,84.98] [41.77,73.74] [35.39,71.19] [32.12,59.06] [13.22,40.49]0.6 [56.50,82.18] [45.02,70.67] [38.91,67.68] [34.66,56.35] [15.28,37.13]0.7 [60.11,79.39] [48.36,67.64] [42.53,64.20] [37.27,53.64] [17.49,33.91]0.8 [63.74,76.60] [51.77,64.66] [46.23,60.73] [39.95,50.93] [19.85,30.83]0.9 [67.39,73.82] [55.27,61.73] [50.02,57.30] [42.69,48.21] [22.37,27.88]1.0 [71.06,71.06] [58.85,58.85] [53.89,53.89] [45.50,45.50] [25.06,25.06]

RI 0.6443 0.5612 0.5262 0.4665 0.3237Rank 1 2 3 4 5

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60 70 80 90 100Overall risk score

Alp

ha

BS1BS2BS3BS4BS5

Fig. 5. The overall risk scores of the five bridge structures.


judgments independently, and also allows them to express their judgments on likelihood, consequences and theweights of criteria in linguistic terms rather than in precise numerical values that prove to be not easy. Two alter-native algorithms have been developed to aggregate the assessments of the four criteria to generate an overall riskassessment. Algorithm-1 changes a fuzzy multiple criteria decision making problem into a crisp one and turnsout to be simple enough and effective. Algorithm-2 handles fuzzy MCDM problems using the extension principleand provides exact solutions to the problems. Both algorithms are justified through a case study. It has beenshown that the FGDM approach offers a flexible, practical and effective way of modelling bridge risks.

In some circumstances a bridge structure may have more than one risk event. In these situations, each riskevent needs to be assessed separately and the maximum overall risk score should be assigned to the bridgestructure. So, no matter how many risk events are involved in a bridge structure, the FGDM approach isalways applicable.

Finally, we point out that one important aspect of group decision making is seeking to achieve a maximumconsensus among the DMs. To this end, the proposed FGDM approach can be combined with the Delphitechnique, which should be applied before implementing Algorithm-1 or 2. This will help to achieve a betterand more credible risk priority ranking for bridge structures.

Acknowledgement

The authors thank two anonymous reviewers for their helpful comments and suggestions, which helped toimprove the paper.

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