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ORIGINAL PAPER
Solomon Tesfamariam ÆRehan Sadiq
Risk-based environmental decision-making using fuzzy analytic hierarchy
process (F-AHP)
Published online: 18 March 2006
Springer-Verlag 2006
Abstract Environmental risk management is an integral
part of risk analyses. The selection of different mitigat-
ing or preventive alternatives often involve competing
and conflicting criteria, which requires sophisticated
multi-criteria decision-making (MCDM) methods.
Analytic hierarchy process (AHP) is one of the most
commonly used MCDM methods, which integrates
subjective and personal preferences in performing anal-
yses. AHP works on a premise that decision-making of
complex problems can be handled by structuring the
complex problem into a simple and comprehensible
hierarchical structure. However, AHP involves human
subjectivity, which introduces vagueness type uncer-
tainty and necessitates the use of decision-making under
uncertainty. In this paper, vagueness type uncertainty is
considered using fuzzy-based techniques. The traditional
AHP is modified to fuzzy AHP using fuzzy arithmetic
operations. The concept of risk attitude and associated
confidence of a decision maker on the estimates of
pairwise comparisons are also discussed. The method-
ology of the proposed technique is built on a hypo-
thetical example and its efficacy is demonstrated through
an application dealing with the selection of drilling fluid/
mud for offshore oil and gas operations.
Keywords Analytic hierarchy process (AHP) ÆFuzzy
arithmetic ÆMulti-criteria decision-making (MCDM) Æ
Risk attitude ÆDrilling fluids
List of notation
(a,b,c) Triangular fuzzy number
CI Consistency index
CR Consistency ratio
~
FAi Final fuzzy AHP score
~
GkFuzzy global preference weights
~
JFuzzy judgment matrix
~
jij Pairwise comparison index in fuzzy
judgment matrix
RI Random index
RkRI
aRisk index value
U
T
(A
i
) Total utility or ordering value (Chen’s
method)
~
wiFuzzy weight (where i=1 to n)
WEigenvector value
X
i,j
k
Risk item, where iis the order of the child in
the level/layer kof hierarchical structure,
and jis the parent of the child
x
O
(A
i
) Geometric center of an alternative (Yager
centroid index)
aAlpha cut of fuzzy number
kEigenvalue
k
max
Maximum eigenvalue
k
RI
Risk attitude
l
x
Membership function of x
1 Introduction
Risk management is an integral part of overall envi-
ronmental risk analyses. It is a process of weighting
alternatives (options), selecting the most appropriate
action, and integrating the results of risk assessment
with engineering data, social, economic, and political
concerns to make an acceptable decision. Generally, risk
assessment process involves objectivity, whereas risk
management involves preferences and attitudes, which
have objective and subjective elements (Asante-Duah
1993). Risk management poses a challenge for decision
maker to select an alternative based on multiple and
often-conflicting criteria that necessitate the use of
decision support aid. Multiple criteria decision-making
(MCDM) is used for this purpose. MCDM techniques
S. Tesfamariam (&)ÆR. Sadiq
Institute for Research in Construction,
National Research Council Canada, Ottawa,
ON K1A 0R6, Canada
E-mail: Solomon.Tesfamariam@nrc-cnrc.gc.ca
E-mail: Rehan.Sadiq@nrc-cnrc.gc.ca
Tel.: +1-613-9932448
Fax: +1-613-9545984
Stoch Environ Res Risk Assess (2006) 21: 35–50
DOI 10.1007/s00477-006-0042-9
deal with the problems whose alternatives are predefined
and the decision-maker ranks available alternatives.
MCDM is proved to be a promising and growing field of
study since early 1970s and many applications in the
fields of engineering, business, and social sciences have
been reported. Carlsson and Fulle
´r(1996) classified
MCDM methods into four distinct types including (i)
outranking, (ii) utility theory, (iii) multiple objective
programming, and (iv) group decision and negotiation
theory. Hwang and Yoon (1981) have also critically re-
viewed these methods in the crisp environment and their
applications for a single decision-maker.
One of the methods classified under utility theory is
the analytic hierarchy process (AHP) that is developed
by (Saaty 1977,1980). AHP has proved to be one of the
most widely applied MCDM methods (Vaidya and
Kumar 2006). There is a growing list of publications on
the application of AHP method in civil and environ-
mental engineering; e.g., Holguin-Veras (1993), Khas-
nabis et al. (2002), Sadiq (2001), Uddameri (2003), Dey
(2002), McIntyre et al. (1999), Ock et al. (2005), Ziara
et al. (2002), etc.
The AHP provides an ideal platform for complex
decision-making problems. The AHP uses objective
mathematics to process the subjective and personal
preferences of an individual or a group in decision-
making (Saaty 2001). The AHP works on a premise that
decision-making of complex problems can be handled by
structuring it into a simple and comprehensible hierar-
chical structure. Once the hierarchical structure is
developed, a pairwise comparison is carried out between
any two criteria. The levels of the pairwise comparisons
range from 1 to 9, where ‘‘1’’ represents that two criteria
are equally important, while the other extreme ‘‘9’’ rep-
resents that one criterion is absolutely more important
than the other. Solution of the AHP hierarchical struc-
ture is obtained by synthesizing local and global pref-
erence weight to obtain the overall priority (Saaty 1980).
The AHP can be summarized into three main steps:
1. structuring of a problem into a hierarchy consisting
of a goal and subordinate features,
2. establishing pairwise comparisons between elements
(criterion) at each level, and
3. synthesis and establishing the overall priority to rank
the alternatives.
The concept of complex system and sub-system
modeling is initially addressed by Kolmogorov’s theo-
rem (Nguyen and Kreinovich 1997). Complex systems
are either sub- or super-additive and therefore they are
difficult to model and describe at one level. To avoid the
overall complexity, the system needs to be divided into
sub-components at various hierarchical levels (based on
their individual complexities) to understand the system
clearly and describe the relationships with lesser ambi-
guity.
Generally, environmental decision-making process is
subjected to scarcity of data and lack of knowledge,
which necessitates making decisions under ignorance
(Harwell et al. 1986). However, even if the data are
available, criteria quite often contain linguistic defini-
tions involving human judgment and subjectivity, which
introduce uncertainties in the decision-making process.
Hence, risk management is a process of informed deci-
sion-making under uncertainty and partial ignorance.
The primary objective of this paper is to present a
methodology to guide decision-making under vagueness
type uncertainty. This task is achieved by proposing a
risk-based fuzzy analytic hierarchy process (F-AHP). In
the next section, vagueness type uncertainty is discussed
and basic information on fuzzy arithmetic operations is
presented. This will be followed by a step-by-step pro-
cedure on the development of risk-based F-AHP in
aiding decision-making process. Finally, the efficacy of
integrating fuzzy-based technique to AHP is illustrated
using an application of selecting/evaluating three generic
types of drilling fluids for offshore oil operations.
1.1 Vagueness and fuzzy arithmetic
Uncertainty is an unavoidable and inevitable component
of decision-making process. The typology and definition
of uncertainty within artificial intelligence and engi-
neering community is vast, and often, conflicting taxo-
nomies are provided (e.g., Klir and Yuan 1995;
Jousselme et al. 2003). Klir and Yuan (1995) taxonomy
identifies uncertainties as fuzziness (lack of definite or
sharp distinction, vagueness), non-specificity (two or
more alternatives are left unspecified), and discord
(disagreement in choosing among several alternatives,
conflict). AHP technique involves subjectivity in pair-
wise comparisons therefore vagueness type uncertainty
dominates in this process. The following discussion is
limited to uncertainty due to vagueness.
Maupin and Jousselme (2004) have classified vague-
ness into three main categories: ontological, linguistic,
and epistemic vagueness. Ontological vagueness deals
with physical nature of objects. Linguistic vagueness
arises due to the limitation of the natural languages.
Epistemic vagueness is due to the limitation of sensorial
apparatus, lack of knowledge or computational limita-
tions. Indeed, each definition of the vagueness, where
subjective opinion is used in the AHP knowledge solic-
itation, is exhibited in different stage of the decision-
making process.
Vagueness type uncertainties can be handled using
the fuzzy set theory (Zadeh 1965). Fuzzy-based tech-
niques are a generalized form of interval analysis used to
address uncertain and/or vague information. A fuzzy
number describes the relationship between an uncertain
quantity xand a membership function l
x
, which ranges
between 0 and 1. A fuzzy set is an extension of the
classical set theory (in which xis either a member of set
Aor not) in that an xcan be a member of set Awith a
certain membership function l
x
. Fuzzy sets qualify as
36
fuzzy numbers if they are normal, convex, and bounded
(Klir and Yuan 1995). Different shapes of fuzzy numbers
are possible (e.g., bell, triangular, trapezoidal, Gaussian,
etc.). In order to simplify the implementation, in this
paper, triangular fuzzy numbers (TFNs) are used. TFN
is represented by three points (a,b,c) on the universe of
discourse (scale Xon which criterion is defined), repre-
senting the minimum, most likely, and maximum values,
respectively.
One important feature of fuzzy numbers (sets) is the
concept of a-cut. The a-cut of a fuzzy set Ais a crisp set
A
a
that contains all the elements of the universal set X
whose membership grades in Aare greater than or equal
to the specified value of an a, i.e., Aa¼xjlx>a
fg
(Klir
and Yuan 1995). Operations on the fuzzy number can be
performed on the real number or the membership
function (l
x
). Fuzzy operations are carried out on the
fuzzy numbers using fuzzy arithmetic. Fuzzy arithmetic
is based on two properties of fuzzy numbers (Klir and
Yuan 1995): (1) each fuzzy number can fully and un-
iquely be represented by its a-cut; and (2) a-cuts of each
fuzzy number are closed intervals of real numbers for all
a2(0, 1 ]. Hence, once the interval numbers is ob-
tained, a well-established operation of interval analysis
can be utilized (Ferson and Hajagos 2004). Some com-
monly used fuzzy arithmetic operations are listed in
Table 1.
1.2 Fuzzy analytic hierarchy process
The AHP helps the decision maker to manage compli-
cated problem by converting it into manageable and
comprehensible small problems. However, as discussed
earlier, the subjective pairwise comparison is prone to
vagueness type uncertainty, consequently raise the need
for exploring fuzzy-based techniques.
The first attempt in the integration of fuzzy-based
technique with AHP appeared in van Laarhoven and
Pedrycz (1983), who compared fuzzy ratios described by
TFNs and weight computed using logarithmic least
squares method. Buckley (1985) highlighted the short-
comings in their method and proposed a geometric mean
to derive fuzzy weights and performance scores. Boender
et al. (1989) modified van Laarhoven and Pedrycz’s
normalization method by implementing a regression
equation. Cheng (1999) introduced the entropy concepts
to calculate aggregate weights. Chang (1996) introduced
the use of the extent analysis method for the synthetic
extent values of the pairwise comparisons. Later, Zhu
et al. (1999) improved on the extent analysis. Deng
(1999) presented a fuzzy-based approach for handling
multi-criteria analysis problems by incorporating the
decision maker’s attitude toward risk. Lee et al. (1999)
introduced the concept of comparison interval and
proposed a methodology based on stochastic optimiza-
tion to achieve global consistency. Leung and Cao
(2000) also discussed the consistency issue and proposed
a concept of fuzzy consistency and tolerance deviation.
Yu (2002) proposed goal programming to compute the
fuzzy priority vectors. Arslan and Khisty (2005) pro-
posed a set of if–then rules to select the cognitive com-
parisons made between each alternative. Recently,
Wang et al. (2005) presented a two-stage logarithmic
goal programming method for generating weights from
interval comparison matrices. On the application side,
Kuo et al. (2002) and Mikhailov and Tsvetinov (2004)
presented evaluation of services using F-AHP.
Table 1 Common fuzzy arithmetical operations using two TFNs
Operators
a,b
Formulae Results
Summation A+B (a
1
+b
1
,a
2
+b
2
,a
3
+b
3
)
Subtraction AB(a
1
b
3
,a
2
b
2
,a
3
b
1
)
Multiplication A·B(a
1
·b
1
,a
2
·b
2
,a
3
·b
3
)
Division A/B (a
1
/b
3
,a
2
/b
2
,a
3
/b
1
)
Scalar product QÆB(Q·b
1
,Q·b
2
,Q·b
3
)
a
A=(a
1
,a
2
,a
3
); B=(b
1
,b
2
,b
3
)
b
The values of A and B are positive, if negative numbers are used,
the corresponding min and max values have to be selected
a
1
<a
2
<a
3
;b
1
<b
2
<b
3
;a
i
and b
i
(i= 1 to 3)>0; n>0; Q>0
Step 1: Formulate the
hierarchic tree
Step 2: Create fuzzy pairwise
comparison matrix
()
J
~
Step 3: Check for consistency
(CI) for the most likely value
CI < 0.1?
No
Step 5: Aggregate individual
preferences
Yes
Adjust
values
Step 6: Fuzzy
defuzzification
Step 7: Risk attitude
Step 8: Final ranking and decision making
Step 4: Calculate the fuzzy
weight
Fig. 1 An 8-step proposed methodology for fuzzy AHP
37
This paper builds on the interpretation of F-AHP by
incorporating the decision maker’s attitude in the final
decision-making process as was originally proposed by
Deng (1999). Further, this paper also introduces two
ranking methods to prioritize alternatives. An eight-step
procedure for F-AHP is formulated in this paper and
schematically provided in Fig. 1. These eight steps are
followed through a simple hierarchical structure exam-
ple shown in Fig. 2. A step-by-step description of the
methodology is presented as following.
1.2.1 Step 1: develop the hierarchical structures
Developing the hierarchical model includes the decom-
position of complex decision problem into smaller
manageable elements of different hierarchical levels/
layers. A four-level hierarchical tree is illustrated in
Fig. 2. The first layer of the hierarchy corresponds to
objective or goal, and the last layer corresponds to the
evaluation alternatives (options), whereas the interme-
diate levels correspond to criteria and sub-criteria. The
nomenclature adopted for each item in the hierarchical
model is X
i,j
k
, where iis the order of the child at the level/
layer k,andjis the parent of the child (Sadiq et al.
2004). For example, X
1,1
2
represents the item is at level k
= 2, is the first child i= 1 and its parent is j=1 (Fig. 2).
Each child, in the intermediate levels, is criterion and
sub-criterion that affect the corresponding parent and
child, respectively. The apostrophe on any intermediate
item (element, factor, sub-criterion), X
i,j
k¢
, indicates that
the element doesn’t have dependent children. The
ensuing derivation and discussions are limited to the
shaded items located at levels 2 and 3 (i.e., X
3,1
2
,X
1,3
3
,
X
2,3
3
and X
3,3
3
). However, similar procedure is followed to
aggregate throughout the network.
1.2.2 Step 2: develop fuzzy judgment matrix using
pairwise comparisons
The elements of a particular level are compared pairwise
with a specific element of an upper level. A fuzzy judg-
ment matrix ð~
JÞis generated using fuzzy pairwise com-
parison index ð~
jijÞ:A relative importance (strength) of
the pairwise comparison is assigned using a scale of 1–9
(Saaty 1977,1980), which are fuzzified to capture
vagueness in perception and meaning (Table 3). For n
number of comparison items, the fuzzy judgment matrix
~
Jis:
~
J¼
~
j11 ~
j12 ~
j1n
~
j21 ..
.~
j2n
.
.
...
..
.
.
~
jn1~
jn2 ~
jnn
2
6
6
6
6
4
3
7
7
7
7
5ð1Þ
For diagonal entries, i.e., i=j,~
jij ¼1:Upper right-
hand triangle entries ~
jijare comparison items needs to be
defined by decision maker, whereas for lower left-hand
triangle entries are derived by taking reciprocals, i.e.,
~
jji ¼1=~
jij:For illustration purpose, a comparison is
sought between three items shown in Fig. 2,X
1,3
3
,X
2,3
3
and X
3,3
3
, using the relative importance given in Table 2;
let the level of importance (or dominance) of X
1,3
3
to X
2,3
3
is a fuzzy number
3;X
1,3
3
to X
3,3
3
is
5andX
2,3
3
to X
3,3
3
is
4:
X
11,0
Level 1
(Goal)
X
21,1
X
23,1
X
31,1
X
31,3
X
32,1
X
32,3
X
33,3
Level 2
Level 3
Level 4
(Evaluation
alternatives)
A
1
A
2
A
3
Legend
X
ki,j
Level
ParentChild
X
2’2,1
X
k’i,j
The item doesn’t have
dependent children.
Fig. 2 Hierarchical tree example for 4-level structure
38
Hence the judgment matrix is populated as following:
X3
1;3X3
2;3X3
3;3
~
J¼
X3
1;3
X3
2;3
X3
3;3
1
3
5
1=
3
1
4
1=
51=
4
1
2
43
5
The concept of fuzzification factor Dis introduced in
Table 2. For this example, the value of fuzzification
factor Dis assumed ‘‘1’’, i.e.,
3 meaning a TFN (2, 3, 4).
1.2.3 Step 3: check for consistency
Consistency is important in human thinking, which en-
ables us to order the world according to dominance
(Saaty 2005). It is paramount to ensure that there is
consistency in the pairwise comparisons. The pairwise
comparisons of the fuzzy judgment matrix in Eq. 1 are
prone to inconsistency and error in the preference re-
sponses of people (Zahedi 1986), and as a result often
become inconsistent. The AHP introduces a consistency
measure to avoid this problem and estimate the relative
weight in the presence of inconsistency in responses.
Once the judgment matrix is populated (Step 2), the
eigenvalue kand eigenvector value Ware obtained by
solving eigenvalue formulation (JkI)W=0. Accord-
ingly, the maximum eigenvalue is obtained by k
max
=
max(k). Saaty (1977,1980) has shown that in a consis-
tent judgment matrix, k
max
=n, where nis the dimen-
sion of the judgment matrix. Consistency index (CI)
indicates whether a decision maker provides consistent
values (comparisons) in a set of evaluation. The CI is
defined as
CI ¼kmax nðÞ
=n1ðÞ ð2Þ
The final inconsistency in the pairwise comparisons is
solved using consistency ratio CR = CI/RI, where RI is
the random index, which is obtained by averaging the CI
of a randomly generated reciprocal matrix (Saaty 1980).
The values of RI are tabulated in Table 3. The threshold
of the CR is 10%, and in case of exceedance a three-step
procedure is followed (Saaty 2005): (1) identify the most
inconsistent judgment in the decision matrix, (2) deter-
mine a range of values the inconsistent judgment can be
changed to so that would reduce the associated incon-
sistency, and (3) ask the decision maker to reconsider the
judgment to a ‘reasonable value’. In this paper, though
the pairwise comparison indices (relative importance) of
the judgment matrix are TFNs, however, the CI is
evaluated for the most likely value.
Following the example of the judgment matrix illus-
trated in Step 2, the CI is computed. The maximum
eigenvalue evaluated is (k
max
=3.086). Thus, for n=3,
the CI from Eq. 2 is (CI=0.043) and the random index
from Table 3, RI=0.52. Finally, the consistency ratio
CR is computed to be 8%. This value is below the 10%
threshold and hence, the judgment matrix is acceptable.
Same procedure is followed throughout the hierarchical
structure.
1.2.4 Step 4: calculate the fuzzy weights
Various techniques are used to compute the final fuzzy
weights, such as, computation of the eigenvector (as
described in Step 3), arithmetic mean, geometric mean,
etc. Preliminary investigation carried out using these
techniques showed no significant difference. Conse-
quently, for the ease of implementation, the geometric
mean is adopted to estimate the weights. Fuzzy arith-
Table 3 Random index used to compute consistency ratio (CR)
N12345678910
Random
index (RI)
0 0 0.52 0.89 1.11 1.25 1.35 1.40 1.45 1.49
Table 2 Fuzzy numbers used for making pairwise comparisons
Relative
importance
a
Fuzzy scale
b
Definition Explanation
1 (1, 1, 1) Equal importance Two activities contribute equally to objective
3(3-D
c
,3,3+D) Weak importance Experience and judgment slightly favour one
activity over another
5(5-D,5,5+D) Essential or strong importance Experience and judgment strongly favor one
activity over another
7(7-D,7,7+D) Demonstrated importance One activity is strongly favoured and
demonstrated in practice
9 (8, 9, 9) Extreme importance The evidence favouring one activity over another
is of highest possible order of affirmation
2;
4;
6;
8(x-D,x,x+D) Intermediate values between two
adjacent judgments
When compromise is needed
1=
x(1/(x+D), 1/ x, 1/(x-D))
1=
9 (1/9, 1/9, 1/8)
a
The intensity of importance definition is in accordance with the description proposed by Saaty (1977,1980)
b
Minimum, most likely, and maximum values
c
Dis a fuzzification factor
39
metic operations (described in Table 1) are utilized over
matrix ~
Jto compute the fuzzy weights. Following the
previous example for ~
J;the geometric mean is computed
for each row ~
Ji:Given ~
Jfrom Eq. 1, the corresponding
fuzzy weights are computed as:
~
Ji¼ð
~
ji1~
jinÞ1=n
~
wi¼~
Jið~
J1~
JnÞ1ð3Þ
where ~
wiis the fuzzy weight (where i=1 to n). There-
fore, for X
1,3
3
,X
2,3
3
and X
3,3
3
, the fuzzy weight are com-
puted as:
~
J1¼ð
1
3
5Þ1=3¼ð1;1;1Þð2;3;4Þð4;5;6Þ½
1=3
¼2:0;2:5;2:9ðÞ)
~
w1¼~
J1ð~
J1~
J2~
J3Þ1
¼ð0:43;0:63;0:89Þ;
similarly
~
J2¼ð1=
3
1
4Þ1=3
¼ð1=4;1=3;1=2Þð1;1;1Þð6;4;5Þ½
1=3
¼0:32;0:37;0:44ðÞ)
~
w2¼ð0:19;0:28;0:42Þ
and
~
J3¼ð1=
51=
4
1Þ1=3
¼ð1=6;1=5;1=4Þð1=5;1=4;1=3Þð1;1;1Þ½
1=3
1:58;1:73;1:87ðÞ
)~
w3¼ð0:07;0:09;0:14Þ:
Sum of the most likely values of weights ~
wi;i¼1;2;3;is
equals to 1 (=0.63+0.28+0.09), which is the basic
axiom of AHP. Therefore, crisp AHP is a special case
of F-AHP, when fuzzification factor reduces to zero.
The difference between sum of minimum values 0.69
(=0.43+0.19+0.07) and maximum value 1.45
(=0.89+0.42+0.14) represent a range of uncertainty or
fuzziness in the computed weight, and can be viewed as
belief and plausibility, respectively (Alim 1988).
1.2.5 Step 5: establish hierarchical layer sequencing
The local priorities at each level are aggregated to obtain
final preferences of the alternative. This computation is
carried out from the evaluation alternatives (bottom
level) to the top level (goal or objective). As depicted in
Fig. 2, each of the three alternatives (level 4), A
i
,,i=1, 2,
3 are aggregated through level 3, level 2 and finally to
level 1 (goal). Therefore, following Fig. 2at each level of
k, the fuzzy global preference weights ð~
GkÞare computed
as:
~
Gk¼~
wk~
Gk1;ð4Þ
~
G1¼~
w1;~
G2¼~
w2~
G1;~
G3¼~
w3~
G2;~
G4¼~
w4~
G3ð5Þ
The final fuzzy AHP score ð~
FAiÞ;for each alternative A
i
is obtained by carrying out fuzzy arithmetic sum over
each global preference weights:
~
FAi ¼X
n
k¼1
~
Gk
for each alternative A
i
.
1.2.6 Steps 6 and 8: order the alternatives using fuzzy
ranking methods
Fuzzy defuzzification methods can be used for ranking
fuzzy numbers. The defuzzification entails converting the
final fuzzy AHP score ~
FAi into a crisp value. Once the
final fuzzy AHP score ð~
FAiÞof each alternative is de-
fuzzified, the crisp numbers are compared and ranked
accordingly. Various techniques are used for defuzzifi-
cation; however, each technique extracts different levels
of information from the fuzzy numbers, consequently
may give different ranking orders (Prodanovic and Si-
monovic 2002). Therefore, an alternative ranked the best
may be ranked differently upon changing the defuzzifi-
cation technique. This is commonly called rank reversal,
which is a common concern in AHP analysis (Saaty
2001). This problem is further aggravated with the fuzzy
0
0.5
1
06
A1A2A3
xo(A1)=1.83
a1c1
b1a2a3c3c2
b2b3
xo(A2)=3. 42 xo(A3)= 3.32
centroidal method (Yager 1980)
Chen’s rankin
g
method (Chen 1985)
0
0.5
1
b
a
06
A
Lmin Lma
x
1A2A3
UT(A1)=0.21
a1a3
a2
b1c1c3c2
b2b3
UT(A2)=0.44 UT(A2)=0. 48
Fig. 3 Fuzzy ranking methods applied to TFNs
40
outputs and using different defuzzification techniques.
This induces a dilemma on the decision maker’s part for
the selection of an alternative. For illustration purpose,
two commonly used defuzzification techniques are con-
sidered and compared in this paper, namely, Yager’s
centroid index method (Yager 1978,1980) and Chen’s
ranking method (1985).
The Yager (1978,1980) centroid index is a geometric
center x
O
(A
i
)of the fuzzy number of alternativeA
i
, where
geometric center corresponds to an xvalue on the hor-
izontal axis X(universe of discourse), as depicted in
Fig. 3a. For a given TFN (a
1
,b
1
,c
1
), Yager’s (1978,
1980) proposed centroid index as following:
xOðAiÞ¼R1
0AilAiðxÞdx
R1
0lAiðxÞdx
¼b1a1
ðÞa1þ2
3b1a1
ðÞ
þc1b1
ðÞb1þ1
3c1b1
ðÞ
b1a1
ðÞþc1b1
ðÞ
ð6Þ
where A
i
is treated as a moment arm (weight function)
measuring the importance of the value x. The denomi-
nator serves as a normalizing factor whose value is equal
to the area under the membership function lAifor an
alternativeA
i
. The value of x
O
(A
i
) may be seen as the
weighted mean value of the fuzzy number A
i
. Hence, the
bigger the x
O
(A
i
) values are, better will be ranking of an
alternative.
To illustrate the concept of centroid method, three
triangular fuzzy alternatives are compared: A
1
=(1,1.5,
3); A
2
= (2, 2.5, 5.75); A
3
= (2.25, 3, 4.5). The maximum
value of A
2
is intentionally skewed to the right (very high
uncertainty) in order to show sensitivity of the ranking
methods. Using Eq. 6, the values of x
O
(A
1
), x
O
(A
2
) and
x
O
(A
3
) are estimated as 1.83, 3.42 and 3.25. The corre-
sponding final rankings are 3, 1, and 2, respectively
(Fig. 3a).
The Chen’s ranking method (1985) is carried out by
computing total utility or ordering value U
T
(x) to rank
the alternatives. The concept is based on maximising and
minimising set. If there are ‘‘ m’’ alternatives, there will
0
2
4
6
0 2 4 6
A1
A3
a) b)
Alternatives A1 and A2 Alternatives A2 and A3
c)
Alternatives A1 and A3
0
2
4
6
0 2 4 6
A1
A2
0
2
4
6
0 2 4 6
A2
A3
(1.0, 2.0) (3.0, 2.0)
(1.0, 5.75)
(1.5, 2.5)
(2.0, 2.25)
(2.0, 4.5)
(1.0, 5.75)
(2.5, 3.0)
(1.0, 2.25) (3.0, 2.25)
(1.0, 4.5)
(1.5, 3.0)
Fig. 4 One-to-one comparison
of alternatives using 2D fuzzy
numbers
41
be mfuzzy numbers A
i
,i=1, 2,..., m. For a TFN, the
ranking of these ‘‘m’’ alternatives is carried out by
maximising (c
max
= max (c
1
,c
2
,..., c
m
)) and minimising
(a
min
= min (a
1
,a
2
,..., a
m
)) a set or alternatives. The
alternative, which has the highest utility or ordering
value, U
T
(x), is selected as the best alternative. The total
utility or ordering value, U
T
(x), given for a TFN (a
1
,b
1
,
c
1
), as depicted in Fig. 3b, is given in Eq. (7):
UTðxÞ¼1
2ðc1aminÞ
ðcmax aminÞðb1c1Þþ1
ðcmax a1Þ
ðcmax aminÞþðb1a1Þfor x¼1;...;m:ð7Þ
The concept of Chen’s ranking method (1985)is
illustrated as follows. Using the three triangular fuzzy
alternatives, A
1
=(1, 1.5, 3); A
2
=(2, 2.5, 5.75);
A
3
=(2.25, 3, 4.5), a
min
and c
max
are computed as: a
min
= min (1, 2, 2.25) = 1, and c
max
= max (3, 5.75, 4.5) =
5.75. Following this, using Eq. 7, the corresponding
utilities are computed to be U
T
(A
1
) = 0.21, U
T
(A
2
)=
0.44 and U
T
(A
3
) = 0.46. The final corresponding final
ranking is 3, 2, 1, respectively. Both methods rank A
1
as
the least desired option, however, there is a rank reversal
between A
2
and A
3
.
To compare fuzziness in the estimate of ~
FAi;fuzzy
outputs can be illustrated through a 2D plot of the fuzzy
numbers (Fig. 4). This figure is a 2D representation of a
pyramid. For the three alternatives, A
1
,A
2
and A
3
, three
comparisons A
1
–A
2
,A
2
–A
3
, and A
1
–A
3
, are shown in
Fig. 4a–c, respectively. Both horizontal and vertical axes
represent the universe of discourse of the evaluation ~
FAi:
The point within the rectangle is the tip of the pyramid,
which represents most likely value of two alternatives,
whereas horizontal and vertical sides of the rectangle
represent fuzziness associated with each of the two
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
α-cut
Risk Index for A3
2.4
2.55
2.7
2.7
2.85
2.85
2.85
3
3
3
3.15
3.15
3.15
3.3
3.3
3.3
3.45
3.45
3.6
3.6
3.75
3.75
3.9
3.9
4.05
4.2
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Risk attitude ( )
α-cut
Risk Index for A2
2.24
2.48
2.48
2.48
2.72
2.72
2.72
2.96
2.96
2.96
3.2
3.2
3.2
3.44
3.44
3.44
3.68
3.68
3.92
3.92
4.16
4.16
4.4
4.4
4.64
4.88
5.36
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
α-cut
Risk Index for A1
1.2
1.3
1.3
1.4
1.4
1.4
1.5
1.5
1.5
1.6
1.6
1.6
1.7
1.7
1.7
1.8
1.8
1.8
1.9
1.9
2
2
2
2.1
2.1
2.2
2.2
2.3
2.3
2.4
2.5
2.6
2.8
a) b)
c)
Alternative A1 Alternative A2
Alternative A3
(λRI)
Risk attitude ( )
(λRI)
Risk attitude ( )
(λRI)
Fig. 5 Contours for risk index
integrating both decision
maker’s risk attitude and
confidence level
42
alternatives. Bigger sides of the rectangle refer higher
uncertainties and vice versa. If the shape is square-like, it
refers that influence of both alternatives are similar. If
the tip of the pyramid is in the middle of the side of a
rectangle, it refers that fuzziness is equally distributed
for that alternative.
1.2.7 Steps 7 and 8: incorporate risk attitude
in decision-making
The final decision-making based on fuzzy output of an
alternative A
i
, induces undue burden on the decision
maker since there is an infinite solution space. One
simple way of ranking alternatives is described in Step 6.
Alternatively, the final ranking can be undertaken by
incorporating the decision maker’s risk tolerance (atti-
tude) and confidence over the evaluation, which is more
subjective. The a-cut concept described earlier represents
the decision maker’s degree of confidence in the fuzzy
assessment (i.e., a=0 entails lack of confidence over the
fuzzy assessment and hence utilize the full range of
uncertainty, whereas the higher value of arepresents a
more confident decision maker, and reaches maximum
when the value approaches to the most likely value). For
any given a-cut on a TFN, the notation used is (a
a
,b,c
a
).
Further, given the desired confidence over the data, the
risk attitude has a significant effect on the defuzzified
value. This is carried out using the decision maker’s risk
attitude index, k
RI
, (Deng 1999),
RkRI
a¼kRIaaþ1kRI
ðÞcað8Þ
where RkRI
ais final risk index value (a defuzzified value) for
a given a-cut and associated risk attitude k
RI
. The values
of k
RI
are (0 < k
RI
< 1) and its interpretation is context
dependent based on the risk scale definition. In this paper,
higher risk index value represents better (i.e., lower) risk.
Hence, following this definition, k
RI
=1 represents a
pessimistic decision-making attitude; k
RI
=0.5 represent
a compromising decision-making attitude; and k
RI
=0
represents an optimistic decision-making attitude.
Using Eq. 8 and simulating through intervals k
RI
[0,
1] and a-cut [0, 1], a risk index contour map can be
generated for a given scenario. Continuing with the three
comparison discussed in Step 5,A
1
=(1, 1.5, 3); A
2
=(2,
2.5, 5.75); A
3
= (2.25, 3, 4.5), the risk index R
a
k
is
computed and plotted in Fig. 5a–c, respectively. The
risk index contour plot is illustrated as follows; e.g., the
decision maker identifies his/her risk attitude as
Table 4 Comparative risk assessment results for drilling fluids (modified after Meinhold 1998)
Impacts Activity Risk type Risk factors OBFs WBFs SBFs
Operational impacts Drilling Occupational Accidents 2 3 2
Chemical exposure 2 1 1
Public Air emissions 1 2 1
Environmental Air emissions 1 2 1
Spills 2 1 1
Energy use 1 2 1
Offshore discharge/ solids control Occupational Accidents 2 1 1
Chemical exposure 2 1 1
Public Bioaccumulation and ingestion 0 1 0
Environmental Water column effects 0 1 0
Bioaccumulation and effect 0 1 0
Benthic effects 1 1 1
Energy use 1 1 1
Loading and transportation Occupational Accidents 3 0 1
Chemical exposure 2 0 1
Public Air emissions 1 0 0
Accidents 1 0 1
Environmental Spills 3 0 1
Water emissions 1 0 0
Air emissions 2 0 0
Energy use 2 0 0
Onshore disposal Occupational Accidents 3 0 0
Chemical exposure 2 0 0
Public Air emissions 2 0 0
Groundwater contamination 1 0 0
Environmental Air emissions 2 0 0
Groundwater contamination 1 0 0
Energy use 2 0 0
Resource impacts (landfill space/injection capacity) 2 1 0
Economic impacts (and Cost) 31 2
Liabilities 31 1
0No risk because this activity is not involved or negligible value of risk is expected
1Low risk
2Medium risk
3High risk
43
Fig. 6 Hierarchical structure for comparison of drilling fluids
44
k
RI
=0.25 (a risk averted attitude between pessimistic
and compromising; and the confidence over the data is
at a level of 0.6 a-cut. Hence, the corresponding risk
index values of A
1
,A
2
, and A
3
are 1.73, 3.25, and 3.13,
respectively (Fig. 5). This can now be interpreted as for
a=0.6 and k
RI
=0.25, the alternative A
2
is the best
alternative, while A
1
is still the least desired. This anal-
ysis helps identifying sensitivities in decision-making
with respect to confidence on the estimates and the
personal attitude of a decision-maker. It also provides
an insight in the decision-making process to understand
risk management alternatives.
1.3 Selection of drilling fluids using F-AHP:
an application and discussion
Previously, Sadiq et al. (2003a) formulated a method-
ology to guide decision-making in the selection and
evaluation of three generic types of drilling fluids using
the risk-based AHP. This example is now extended to
risk-based fuzzy AHP methodology as described in the
previous section.
The composition of drilling muds is based on a
mixture of clays and additives in a base fluid. There are
three generic types of base fluids—water- (WBFs), oil-
(OBFs), and synthetic-based fluids (SBFs). The compo-
sition of mud used in a particular application depends
on the well conditions and requirements of the envi-
ronmental regulatory compliance. Water-based muds or
fluids are relatively environmentally benign, but drilling
performance is better with oil-based fluids. Implemen-
tation of advanced drilling techniques sometimes de-
mands a fluid with better lubricating characteristics than
WBFs can provide. Hence, this unfolds a typical
MCDM problem where the decision maker has to select
among the three alternatives that may have conflicting
criteria, levels of environmental friendliness and per-
formance enhancement.
Drilling wastes associated with SBFs are less dis-
persible in marine water than WBFs and tend to sink to
the seafloor where they constitute a potential environ-
mental concern to the benthic community (settling and
dispersion characteristics depend in part on the relative
amount of adhering fluids). It is believed that environ-
mental impacts include smothering by the drill cuttings,
changes in grain size and composition, and anoxia caused
by the decomposition of organic matter (US EPA 1999;
Sadiq et al. 2003b). The environmental impacts associ-
ated with the zero discharge of OBFs can be more
harmful than the discharge of SBFs due to non-water
quality environmental impacts, like air pollution and
groundwater pollution in the case of incineration and
land-based disposal, respectively (US EPA 1999). A
qualitative comparison of OBFs, WBFs and SBFs for
various offshore/onshore activities is provided in Ta-
ble 4. The OBFs, WBFs and SBFs are compared for four
major impacts (and their associated risks): operations,
resources, economics, and liabilities. The operational
impacts are further categorized into four major activities:
drilling, discharging offshore, disposing onshore, and
loading and transporting. Each activity involves risks
related to occupational injuries (safety), general public
health, environmental impacts and energy use. Further,
each risk type is divided into elementary risk factors. The
risk factors are ranked on a scale from 0 to 3, where a
higher value represents a greater risk potential of that
activity (Meinhold 1998). These risk factors are used for
pairwise comparison using the methodology provided in
Sadiq et al. (2003a).
Following the formulation of Table 4, a hierarchical
tree of the risks involved in the selection of three dif-
ferent types of drilling fluids is illustrated in Fig. 6. The
hierarchical tree comprised of a six-level structure; the
first level (goal) is selecting the best drilling fluid type;
and at the second level comprised of factors including
operational, resource, economics and liabilities impacts.
In the next level, only operational factors are divided
into four major impacts (criteria) including drilling,
discharge offshore and onshore, and loading and trans-
portation. At the fourth and fifth levels, each of these
major criteria is further divided into children risk fac-
tors. The final (sixth) level is where the three drilling
fluid alternatives are placed.
After the hierarchical structure is formulated (Step 1)
to conduct F-AHP methodology, Steps 2–8are applied
as described in previous section. For brevity, only the
final results are presented. Step 2entails carrying out the
fuzzy pairwise comparisons. The pairwise comparisons
are established based on linguistic measures of impor-
tance (Saaty 1980) as shown in Table 2. A pairwise
comparison of the first level, selecting the best fluid type
(X
1,0
1
), is summarized in Table 5. The pairwise compar-
isons are carried out among the criteria at level 2, lia-
bility (X
1,1
2¢
), economic impact (X
2,1
2¢
), resource impact (X
3,1
2
¢
) and operational impact (X
4,1
2
). Given the pairwise
values, the corresponding fuzzy weights are computed
and also shown in Table 5. The rest of the judgment
matrix comparisons are not included in this paper. The
corresponding fuzzy weights, ~
wii=1, 2...5 are summa-
rized in Table 6.
For a fuzzification factor D=1, the evaluation of the
final global preference weights, ~
G5¼~
w5~
G4;for the three
drilling fluids are summarized in Table 7. The final fuzzy
AHP score ~
FAi (Table 7) of each alternative, OBFs, WBFs
and SBFs, is evaluated to be (0.044, 0.204, 1.572), (0.110,
0.370, 2.257) and (0.127, 0.411, 2.310), respectively. The
sum of the most likely values is equal to one
(0.204+0.37+0.411), whereas the sum of minimum
Table 5 A weighting scheme for major impacts ð~
w1Þ
X
1,1
2¢
X
2,1
2¢
X
3,1
2¢
X
4,1
2
~
w1
X
1,1
2¢
1
3
2
2 (0.21, 0.43, 0.76)
X
2,1
2¢
1=
3
1
1
1 (0.13, 0.18, 0.26)
X
3,1
2¢
1=
2
1
11=
2 (0.10, 0.16, 0.31)
X
4,1
2
1=
2
1
2
1 (0.14, 0.23, 0.41)
45
Table 6 Fuzzy local weights for impacts, activity, risk type, ~
wi(i=1, 2...5)
Level 2 W1 Level 3 W2 Level 4 W3 Level 5 W4 W5 (OBFs) W5 (WBFs) W5 (SBFs) CR
a
X
4,1
2
(CR=3) 0.21 0.43 0.76 X
4,4
3
0.20 0.42 0.80 X
16,4
4
0.13 0.22 0.39 X
24,16
5
0.37 0.67 1.10 0.257 0.400 0.581 0.124 0.200 0.403 0.257 0.400 0.581 0
(CR=1.6) (CR=0.7) X
23,16
5
0.21 0.33 0.63 0.124 0.200 0.403 0.257 0.400 0.581 0.257 0.400 0.581 0
X
15,4
4
0.26 0.48 0.81 X
22,15
5
1.00 1.00 1.00 0.257 0.400 0.581 0.124 0.200 0.403 0.257 0.400 0.581 0
X
14,4
4
0.12 0.20 0.32 X
21,14
5
0.52 0.75 1.04 0.257 0.400 0.581 0.124 0.200 0.403 0.257 0.400 0.581 0
X
20,14
5
0.18 0.25 0.37 0.124 0.200 0.403 0.257 0.400 0.581 0.257 0.400 0.581 0
X
13,4
4¢
0.06 0.11 0.22 0.257 0.400 0.581 0.124 0.200 0.403 0.257 0.400 0.581 0
X
3,4
3
0.13 0.27 0.57 X
12,3
4
0.13 0.22 0.39 X
19,12
5
0.37 0.67 1.10 0.124 0.200 0.403 0.257 0.400 0.581 0.257 0.400 0.581 0
X
18,12
5
0.21 0.33 0.63 0.124 0.200 0.403 0.257 0.400 0.581 0.257 0.400 0.581 0
X
11,3
4
0.26 0.48 0.81 X
17,11
5
1.00 1.00 1.00 0.331 0.429 0.544 0.104 0.143 0.216 0.331 0.429 0.544 0
X
10,3
4
0.12 0.20 0.32 X
16,10
5
0.26 0.40 0.58 0.331 0.429 0.544 0.104 0.143 0.216 0.331 0.429 0.544 0
X
15,10
5
0.12 0.20 0.40 0.331 0.429 0.544 0.104 0.143 0.216 0.331 0.429 0.544 0
X
14,10
5
0.26 0.40 0.58 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0
X
9,3
4¢
0.06 0.11 0.22 (CR=0) 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0
X
2,4
3
0.10 0.16 0.33 X
8,2
4
0.13 0.22 0.39 X
13,8
5
0.37 0.67 1.10 0.061 0.077 0.110 0.508 0.692 0.913 0.160 0.231 0.348 0
X
12,8
5
0.21 0.33 0.63 0.076 0.111 0.186 0.452 0.667 0.965 0.132 0.222 0.364 0
X
7,2
4
0.26 0.48 0.81 X
11,7
5
0.21 0.33 0.63 0.104 0.143 0.216 0.331 0.429 0.544 0.331 0.429 0.544 0
X
10,7
5
0.37 0.67 1.10 0.153 0.200 0.279 0.386 0.600 0.885 0.153 0.200 0.279 0
X
6,2
4
0.12 0.20 0.32 X
9,6
5
0.26 0.40 0.58 0.061 0.077 0.110 0.508 0.692 0.913 0.160 0.231 0.348 0
X
8,6
5
0.12 0.20 0.40 0.104 0.143 0.216 0.331 0.429 0.544 0.331 0.429 0.544 0
X
7,6
5
0.26 0.40 0.58 0.066 0.077 0.093 0.410 0.462 0.518 0.410 0.462 0.518 0
X
5,2
4¢
0.06 0.11 0.22 (CR=0) 0.066 0.077 0.093 0.410 0.462 0.518 0.410 0.462 0.518 0
X
1,4
3
0.09 0.15 0.27 X
4,1
4
0.13 0.22 0.39 X
6,4
5
0.37 0.67 1.10 0.052 0.053 0.059 0.453 0.474 0.492 0.453 0.474 0.492 0
X
5,4
5
0.21 0.33 0.63 0.066 0.077 0.093 0.410 0.462 0.518 0.410 0.462 0.518 0
X
3,1
4
0.26 0.48 0.81 X
4,3
5
0.50 0.50 0.50 0.066 0.077 0.093 0.410 0.462 0.518 0.410 0.462 0.518 0
X
3,3
5
0.50 0.50 0.50 0.104 0.143 0.216 0.331 0.429 0.544 0.331 0.429 0.544 0
X
2,1
4
0.12 0.20 0.32 X
2,2
5
0.50 0.50 0.50 0.066 0.077 0.093 0.410 0.462 0.518 0.410 0.462 0.518 0
X
1,2
5
0.50 0.50 0.50 0.104 0.143 0.216 0.331 0.429 0.544 0.331 0.429 0.544 5.2
X
1,1
4¢
0.06 0.11 0.22 0.066 0.077 0.093 0.410 0.462 0.518 0.410 0.462 0.518 0
X
3,1
2¢
0.13 0.18 0.26
X
2,1
2¢
0.10 0.16 0.31
X
1,1
2¢
0.14 0.23 0.41
a
CR Consistency ratio is in percent (%)
46
Table 7 Evaluation of final global preference weights for the drilling fluids, ~
G5¼~
w:
5~
G4
Impacts Activity Risk type Factors OBFs WBFs SBFs
Operational impacts Drilling Occupational Accidents 0.001 0.011 0.149 0.000 0.005 0.103 0.001 0.011 0.149
Chemical exposure 0.000 0.003 0.060 0.000 0.005 0.086 0.000 0.005 0.086
Public Air emissions 0.003 0.035 0.287 0.001 0.017 0.199 0.003 0.035 0.287
Environmental Air emissions 0.001 0.011 0.119 0.000 0.005 0.083 0.001 0.011 0.119
Spills 0.000 0.002 0.029 0.000 0.004 0.042 0.000 0.004 0.042
Energy use 0.001 0.008 0.078 0.000 0.004 0.054 0.001 0.008 0.078
Offshore discharge/solids
control
Occupational Accidents 0.000 0.003 0.073 0.000 0.007 0.105 0.000 0.007 0.105
Chemical exposure 0.000 0.002 0.042 0.000 0.003 0.061 0.000 0.003 0.061
Public Bioaccumulation and ingestion 0.002 0.024 0.190 0.001 0.008 0.075 0.002 0.024 0.190
Environmental Water column effects 0.000 0.004 0.044 0.000 0.001 0.017 0.000 0.004 0.044
Bioaccumulation and effect 0.000 0.002 0.030 0.000 0.001 0.012 0.000 0.002 0.030
Benthic effects 0.000 0.003 0.027 0.000 0.003 0.027 0.000 0.003 0.027
Energy use 0.001 0.004 0.032 0.001 0.004 0.032 0.001 0.004 0.032
Loading and transportation Occupational Accidents 0.000 0.001 0.011 0.000 0.007 0.095 0.000 0.002 0.036
Chemical exposure 0.000 0.001 0.011 0.000 0.003 0.058 0.000 0.001 0.022
Public Air emissions 0.000 0.002 0.028 0.000 0.005 0.069 0.000 0.005 0.069
Environmental Accidents 0.000 0.004 0.062 0.001 0.013 0.196 0.000 0.004 0.062
Spills 0.000 0.000 0.005 0.000 0.004 0.042 0.000 0.001 0.016
Water emissions 0.000 0.000 0.007 0.000 0.001 0.018 0.000 0.001 0.018
Air emissions 0.000 0.000 0.004 0.000 0.002 0.024 0.000 0.002 0.024
Energy use 0.000 0.001 0.005 0.001 0.003 0.028 0.001 0.003 0.028
Onshore disposal Occupational Accidents 0.000 0.000 0.005 0.000 0.004 0.043 0.000 0.004 0.043
Chemical exposure 0.000 0.000 0.005 0.000 0.002 0.026 0.000 0.002 0.026
Public Air emissions 0.000 0.001 0.008 0.001 0.007 0.044 0.001 0.007 0.044
Groundwater contamination 0.000 0.002 0.018 0.001 0.006 0.046 0.001 0.006 0.046
Environmental Air emissions 0.000 0.000 0.003 0.000 0.003 0.017 0.000 0.003 0.017
Groundwater contamination 0.000 0.001 0.007 0.000 0.003 0.018 0.000 0.003 0.018
Energy use 0.000 0.001 0.004 0.001 0.003 0.024 0.001 0.003 0.024
Resource impacts (landfill space/injection capacity) 0.010 0.020 0.048 0.017 0.039 0.095 0.057 0.117 0.251
Economic impacts (and cost) 0.010 0.026 0.092 0.037 0.097 0.294 0.010 0.026 0.092
Liabilities 0.014 0.033 0.088 0.045 0.099 0.222 0.045 0.099 0.222
~
FAiR~
G50.044 0.204 1.572 0.110 0.370 2.257 0.127 0.411 2.310
47
values (0.044+0.11+0.127)<1 and sum of maximum
values (1.57+2.26+2.31)>1. The difference between
sum of minimums and sum of maximums represents the
overall uncertainty (vagueness) in the decision-making
process.
In the final ranking of fuzzy AHP score ~
FAi;the op-
tion with the highest score is ranked the best. As dis-
cussed earlier, both Yager’s centroid index (1978,1980)
and Chen’s ranking method (1985) are used for defuzz-
ification to rank the alternatives. The final defuzzified
values for OBFs, WBFs and SBFs are summarized in
Table 8. Both ranking methods demonstrate consistent
ranking order of 3, 2 and 1, i.e., SBFs are the most
desirable option, followed by WBFs and OBFs. The
results for each alternative are summarized schemati-
cally in Fig. 7. The risk index contour plots (Fig. 4) can
be generated for each drilling fluid alternative using
decision-maker’s confidence and risk attitude as de-
scribed in Step 7of last section.
To see the impact of fuzzification factor on defuzzified
values, Dis varied from 0 to 2 for both defuzzification
methods. The sensitivity analysis results are shown for
the selection problem of three drilling fluids, OBFs,
WBFs and SBFs. Result of the centroid index and Chen’s
utility value are depicted in Fig. 8a, b, respectively. It can
be discerned that throughout the D, values of SBFs are
the best choice while OBFs is the least desired option. It
is interested to note that for the centroid index, the sen-
sitivity of WBFs and SBFs is not significant. However, at
a lower Dvalue, e.g., for D=0 (crisp number) the utility
value method is drastic difference in the estimates for
three alternatives. This is due to the fact that Chen’s
method (1985) provides a relative estimate. However,
with an increase in D, the difference is minimized and the
values of the utility value converge. Conversely, the
ranking in the centroid index is carried out based on
absolute scale. Hence, it may be more useful where the
defuzzified value has a physical meaning, e.g., acceptable
design factor of safety, threshold of contaminant expo-
sure, etc., centroid index method is preferred.
In general, the final decision-making is a deterministic
process, therefore requires a suitable defuzzification
technique to order the alternatives. At times, where the
final ranking is crucial or has detrimental consequence,
using multiple defuzzification techniques may be useful
to select the parameter that is consistently ranked high.
The risk index values using decision maker’s attitude and
confidence provide a complete snap shot of the decision
action. Risk index can also be used for ranking alterna-
tives where an acceptable risk tolerance and the corre-
sponding risk attitude of a decision-maker are provided.
2 Conclusion
Decision-making process is subject to scarcity of data
and lack of knowledge, and subjectivity. Risk manage-
Table 8 Ranking of drilling fluids using Chen (1985) and Yager
(1980) defuzzification methods
Alternative U
T
(A
i
) Rank Centroid, x
O
(A
i
) Rank
OBFs 0.241 3 0.609 3
WBFs 0.331 2 0.930 2
SBFs 0.343 1 0.961 1
0
0.5
1
01 2 3
OBFs
WBFs
SBFs
Fig. 7 Risk index for three drilling fluids. aOBFs, bWBFs, cSBFs
0
1
2
3
0.0 0.5 1.0 1.5 2.0 2.5
Fuzzification,
∆
Yager's Centroid Index
OBFs
WBFs
SBFs
0
1
0.0 0.5 1.0 1.5 2.0 2.
5
Fuzzification,
∆
Chen's Ranking Method
OBFs
WBFs
SBFs
Ya
g
er’s Centroid index (1980) Chen’s ranking method (1985)
a) b)
Fig. 8 Effect of fuzzification
factor (D) on the deffuzified
values of three alternatives
(drilling fluids) for both ranking
methods
48
ment is an integral part of overall environmental risk
analyses and requires MCDM approaches to select best
alternatives under multiple and conflicting criteria.
As uncertainty is an avoidable and inevitable com-
ponent of decision-making process, the F-AHP is
proposed to enable the decision maker to account for
the impact of uncertainty (vagueness) in the final
decision-making. It is designed to select the best alter-
native among a number of options available on a ra-
tional and intuitive basis. The F-AHP used fuzzy
arithmetic operations to aggregate the fuzzy global
preference weights with respect to each alternative. The
F-AHP methodology integrated decision maker’s atti-
tude as well as confidence in the final decision-making
process. However, for more than one decision-maker,
the problem becomes complex and the best solution is
the one that will be accepted by all decision-makers.
The F-AHP methodology can be extended to multiex-
pert MCDM.
To demonstrate the utility of the F-AHP methodol-
ogy, risk management problem of the selection/evalua-
tion of three generic types of drilling fluids was
presented. A six-level hierarchical structure was devel-
oped to perform MCDM. The alternatives were ranked
based on the score estimated by the summation of final
global preference weights. The final defuzzified scores
for OBFs, WBFs, and SBFs were 0.211, 0.385, and
0.408, respectively. Under the given assumptions the
SBFs were found to be the most desirable option, fol-
lowed by WBFs and OBFs.
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