Fuzzy AHP in Prioritizing Feeders for Maintenance in Nuclear Power Plants
A. Srividya1, H.N. Suresh2 and A.K. Verma1
Professor, Reliability Engineering Group, Department of Electrical Engineering,
Indian Institute of Technology Bombay, Mumbai – 400076, INDIA
2
Research Scholar, Reliability Engineering Group, Department of Electrical Engineering,
Indian Institute of Technology Bombay, Mumbai – 400076, INDIA
1
Abstract - A fuzzy version of prioritization from
among several alternatives under different decision
criteria of Saaty’s pair wise comparison method is
presented in this paper. Each ratio expressing the relative
significance of a pair of factors is displayed in a matrix
from which suitable weights can be extracted. Since these
ratios are essentially fuzzy, they express the opinion of a
decision-maker on the importance of a pair of factors.
This method is used in such a way that information from
experts, who are asked to express their opinions in fuzzy
numbers with triangular membership functions, is
embedded in it. The method is applied at two levels:
beginning with the finding of fuzzy weights for the
decision criteria, followed by finding the fuzzy weights for
the alternatives under each of the decision criteria. Fuzzy
scores for the alternatives are obtained. Using the fuzzy
scores, experts will be able to prioritize the alternatives for
maintenance activities based on the listed criteria. The
method is illustrated for outlet feeders in a nuclear power
plant with representative values.
failure rates and have different costs of inspection,
depending on the number of welds present. During
testing operations, the technicians are subjected to
radiation exposure with limit-dose prescribed by the
regulatory body. Since the feeders are placed in a
radioactive environment, the radiation exposure to the
workers is of concern and hence, a restriction is
imposed on the exposure time allowed to the workers,
along with the cost constraint.
These feeders are inspected once in five years, with
individual inspection planned during the planned
shutdown [3]. Usually ISI activity is taken up during the
planned shutdown period of the plant, which leaves a
restriction on the choice of feeders to be inspected
every year. In order to select the feeders (alternatives)
that should be inspected with respect to the criteria such
as risk, exposure time, accessibility, and cost, fuzzy
prioritization using AHP has been framed. Feeders in a
typical PHWR [2] are shown in Fig.1, which describes
the complexity involved due to the positioning of
feeders.
Keywords – Prioritization, feeders, fuzzy numbers,
analytic hierarchy process, multi-criteria decision
analysis, triangular fuzzy number.
I. INTRODUCTION
Nuclear power plants are under continuous demand
to meet performance standards and safety by regulatory
authorities. Maintenance plays an important role in
achieving this goal. A comprehensive approach for
solving this type of problem is decision theory, which
provides a systematic and consistent method to evaluate
alternatives and make optimal choices using Saaty’s
pair wise comparison method [1]. This paper presents a
methodology of prioritizing the different outlet feeders
for maintenance activities using fuzzy concepts in
Analytic Hierarchical Process. The methodology is
illustrated with a case study with representative values.
Fig.1. Schematic of feeder layout in PHWR
II. PROBLEM FORMULATION
The fuzzy decision framework for feeder prioritization
is shown in Fig. 2.
This study originated with an aim to prioritize the
feeders in Nuclear power plants with criterion such as
risk, cost, inspection frequency and exposure time [2].
In Pressurized Heavy Water Reactors (PHWRs),
there are 306 inlet and 306 outlet feeders. The failure of
feeder results in Small Loss of Coolant Accident
(SLOCA), which can pose a threat to reactor core. The
feeder failures are prevented by conducting In Service
Inspections (ISI). The pipe segments can have different
1-4244-1529-2/07/$25.00 ©2007 IEEE
III FUZZY NUMBERS
Special class of fuzzy numbers suitable for this
application is used in this paper. Triangular fuzzy
numbers and some operations performed on them, such
as addition, multiplication and inversion are defined in
this section. Some of the definitions, being special cases
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of definitions of fuzzy numbers by Dubois and Prade
[4] are given in the next section.
µT ( Z ) =
sup
(min( µM ( x ), µ N ( y ))) (3)
x ,y∈\ 2 :z = f ( x,y )
µM
1
l
M
u \
Fig.3. Membership function of a triangular fuzzy number
B. Arithmetic Operations on Triangular Fuzzy Numbers
Some important fuzzy arithmetic operations [6]
used in this paper are given in this section.
a) Addition: Consider two TFNs M1 = (l1, m1, u1)
and M2 = (l2, m2, u2). Then,
M 1 ⊕ M 2 = ( l1 + l2 ,m1 + m2 ,u1 + u2 )
(4)
b) Multiplication:
M 1 : M 2 ( l1l2 ,m1 m2 ,u1u2 )
(5)
c) Inverse:
 1 1 1
(6)
( l,m,u )−1  , , 
u m l 
Fig.2. Fuzzy decision framework for feeder prioritization
A. Triangular Fuzzy Numbers
A fuzzy number, which indicates a fuzzy set,
addresses the definition of fuzzy set theory [5]. A
triangular fuzzy number shown in Fig. 3 must have the
following properties:
µM (x) = 0, for all x∈(-∞,L)
µM (x) is increasing on [L, M]
µM (x) =1 for x = M
µM (x) is strictly decreasing on [M, U]
µM (x) = 0, for all x ∈(U,∞)
A fuzzy number M on \ (= (-∞, +∞)) is said to be a
triangular fuzzy number if its membership function µM:
R→ [0, l] is equal to:
µM
l
 1
m − l x − m − l ,

u
 1
( X ) = 
x −
,
m − u
m − u
0,


x ∈ [ l ,m ]
IV COMBINING EXPERT JUDGMENT
Responding to the feeder condition when in the
form of a triangular fuzzy set is indeed a hard task for
inspectors since precision criteria is of decisive
significance. For this, inspection team should be well
conversant in the fuzzy set theory and its analysis.
Otherwise, judging the degraded states in linguistic
terms is a sensible proposition. After getting the
answers from experts in linguistic terms, they can be
transformed to triangular fuzzy sets as defined in the
earlier section. More details about expert judgment
process can be found in [7, 8]. Three methods, based on
experts’ opinions for symmetric triangular fuzzy sets –
Averaging method, Averaging method with feedback
and Combining method – can be used when opinions
are in the form of triangular fuzzy sets or numbers. The
details of these methods can be obtained in [9,10]. In
this paper, opinions of four knowledge engineers from
the field of nuclear industry were involved.
(1)
x ∈ [ m ,u ]
o t h e r w is e
with l ≤ m ≤ u, l and u stand for the lower and the upper
limit of the support M, respectively, and m for the
modal value. The triangular fuzzy number, as given by
(1), will be denoted by (l, m, u), as shown in Fig. 3. The
support of M is the set of elements {x ∈ \ / 1< x < u}.
For any two fuzzy numbers M and N, defined by their
membership functions µM and µN, the membership
function of the fuzzy number is given as:
V ANALYTIC HIERARCHY PROCESS
Analytic Hierarchy Process (AHP) is a decision
analysis approach developed by Thomas L Saaty in
1971 [1]. It is a systematic method for comparing a list
of objectives for decision-making. AHP allows for the
application of data, experience, insight and intuition in a
logical and thorough way. It also helps the decisionmakers in deriving ratio scale priorities or weightages
T = f ( M ,N )
(2)
and evaluated according to the extension principle [4]
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as opposed to arbitrarily assigning them. Hence, AHP is
used to derive the significance of attributes based on the
organizational knowledge of project execution.
Preference rating used is presented in Table I. Fig. 4
shows the fuzzy set scale defined as TFN [10-12] in this
paper.
Discussions with experts from this field lead to the
preceding criteria being of paramount significance.
TABLE I
INTENSITY OF IMPORTANCE SCALE
Intensity of
Importance
Definition
Explanation
Two activities contribute
equally to the objective
Experience and judgment
Weak importance of one
[1,3,4]
slightly favor one activity
over another [WI]
over another
Experience and judgment
Essential or strong
[3,5,6]
strongly favor one activity
importance [SI]
over another
An activity is strongly
Demonstrated importance
[5,7,8]
favored and its dominance
[DI]
demonstrated in practice
The evidence favoring one
Absolute importance
activity over another is of
[8,9,10]
[AI]
highest possible order of
affirmation
If activity i has one of the
above judgments assigned
Reciprocal
An assumption
to it compared with
of above
activity j,
numbers
then j has reciprocal value
compared to i.
[1,2,3]
Equal Importance [EI]
Fig.5. Schematic of prioritization hierarchy
B
AHP is normally used to capture experts’
knowledge. Regular AHP cannot reflect experts’
thinking. In decision-making environment of AHP, the
input and the relation between criteria and alternatives
are uncertain and inaccurate. To overcome this
disadvantage of AHP, Fuzzy Logic is used, which has
the ability to deal with inaccurate, uncertain and
subjective problems in pair-wise comparison.
Therefore, fuzzy AHP is proposed to solve the
hierarchical and the multi criteria decision making
(MCDM) problems in this prioritization of feeders [6].
VI USING FUZZY SETS IN MULTI-CRITERION
DECISION MAKING
Membership grade
1
Let rij denote the numerical value assigned to the
relative significance/ importance (i.e. ratios) of criteria
Ci and Cj according to the “intensity of importance”
fuzzy scale given in Table I. If Ci and Cj are equally
important, then rij =1; if Ci is more important than Cj,
then rij > 1; and if Ci is less important than Cj, then rij <
1; Table II has positive entries everywhere and it
satisfies the reciprocal property i.e. rji = 1/rij. Here rij are
in the form of TFNs and therefore, inverse operation on
TFNs is used to get their reciprocals. Normalized
average weights (priorities) are computed from the
matrix, as shown in Table III. Lootsama [13] showed
that normalized column and row weights are as good
enough as normalized eigen vectors. In this, the average
of the two (row and column) normalized weights is
used as the final weight. Alternatives (feeders) are also
compared in pair-wise manner under each criterion.
These matrices are given in Tables IV-VI. Priority of
criteria and priority of maintenance strategies are then
multiplied (fuzzy multiplication) as shown in Table VII
and added for each of the alternative feeders to obtain
the final scores. The defuzzified values are depicted in
Fig. 4.
0.8
EI
WI
0.6
SI
0.4
DI
AI
0.2
0
1
2
3
4
5
6
7
8
9
10
Rating
Fig.4. Fuzzy set scale
A.
Fuzzy Analytic Hierarchy Process
Hierarchy of Prioritization
The hierarchy of prioritization of the example case
is shown in Fig. 5. The top goal is the ranking of
feeders for maintenance actions based on the following
criteria: Remaining Life (C1), Cost of Inspection (C2),
Inspection frequency (C3) and Exposure Time (C4),
which is as depicted in the middle layer of Fig. 5, and
the alternatives (feeders, F1-F4) form the bottom layer.
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TABLE II
MATRIX OF RELATIVE SIGNIFICANCE OF DECISION CRITERIA
Criteria
C1
C2
C3
C4
RS
C1
C2
C3
C4
CS
ICS
NICS
[1,1,1]
[1,3,4]
[3,5,6]
[5,7,8]
[10,16,19]
[0.05,0.06,0.10]
[0.08,0.13,0.25]
[0.25,0.33,1]
[1,1,1]
[0.10,0.11,0.14]
[3,5,6]
[4.35,6.44,8.14]
[0.12,0.16,0.23]
[0.18,0.33,0.58]
[0.17,0.20,0.33]
[7,9,10]
[1,1,1]
[0.17,0.20,0.33]
[8.34,10.40,11.66]
[0.09,0.10,0.12]
[0.13,0.20,0.30]
[0.13,0.14,0.20]
[0.17,0.20,0.33]
[3,5,6]
[1,1,1]
[4.30,6.34,7.53]
[0.13,0.16,0.23]
[0.19,0.33,0.59]
[1.55,1.67,2.53]
[9.17,13.20,15.33]
[7.10,11.11,13.14]
[9.17,13.20,15.33]
[26.99,39.18,46.33]*
[0.39,0.47,0.68] **
Adjusted
Average
score
[0.03,0.04,0.09]
[0.06,0.09,0.17]
[0.20,0.34,0.57]
[0.19,0.33,0.58]
[0.15,0.28,0.49]
[0.14,0.24,0.40]
[0.20,0.34,0.57]
[0.20,0.34,0.58]
* Row Sum
** Inverted Column Sum
NRS
TABLE III
MATRIX OF RELATIVE SIGNIFICANCE OF FEEDERS WITH RESPECT TO C1
Criteria
F1
F2
F3
F4
RS
F1
F2
F3
F4
CS
ICS
NICS
[1,1,1]
[1,3,4]
[3,5,6]
[0.10,0.11,0.14]
[5.1,9.11,11.10]
[0.09,0.11,0.20]
[0.16,0.28,0.59]
[0.25,0.33,1]
[1,1,1]
[1,3,4]
[5,7,8]
[7.25,11.30,14]
[0.07,0.09,0.14]
[0.12,0.23,0.42]
[0.17,0.20,0.33]
[0.25,0.33,1]
[1,1,1]
[7,9,10]
[8.42,10.53,12.33]
[0.08,0.10,0.12]
[0.14,0.24,0.36]
[7,9,10]
[0.13,0.14,0.20]
[0.10,0.11,0.14]
[1,1,1]
[0.09,0.10,0.12]
[0.13,0.16,0.23]
[0.15,0.25,0.37]
[8.42,10.53,12.33]
[2.38,4.47,6.2]
[5.1,9.11,11.14]
[13.1,17.11,19.14]
[29,41.22,48.81] *
[0.33,0.39,0.57]**
NRS
Adjusted Average
score
[0.17,0.26,0.43]
[0.16,0.27,0.51]
[0.05,0.11,0.21]
[0.09,0.17,0.32]
[0.10,0.22,0.38]
[0.12,0.23,0.37]
[0.27,0.42,0.66]
[0.21,0.33,0.51]
* Row Sum
** Inverted Column Sum
TABLE IV
MATRIX OF RELATIVE SIGNIFICANCE OF FEEDERS WITH RESPECT TO C2
Criteria
F1
F2
F3
F4
RS
F1
F2
F3
F4
CS
ICS
NICS
[1,1,1]
[3,5,6]
[5,7,8]
[0.25,0.33,1.00]
[9.25,13.3,16.0]
[0.06,0.08,0.11]
[0.08,0.15,0.27]
[0.17,0.20,0.33]
[1,1,1]
[1,3,4]
[0.13,0.14,0.20]
[2.3,4.34,5.53]
[0.18,0.23,0.43]
[0.23,0.47,1.08]
[0.13,0.14,0.20]
[0.25,0.33,1.00]
[1,1,1]
[7,9,10]
[8.38,10.47,12.20]
[0.08,0.10,0.12]
[0.10,0.19,0.30]
[1,3,4]
[5,7,8]
[0.10,0.11,0.14]
[1,1,1]
[7.1,11.11,13.14]
[0.08,0.09,0.14]
[0.09,0.18,0.35]
[2.3,4.34,5.53]
[9.25,13.33,16.00]
[7.1,11.11,13.14]
[8.38,10.47,12.20]
[27.03,39.25,46.87]*
[0.40,0.49,0.80] **
Adjusted
Average
score
[0.05,0.11,0.20] [0.06,0.13,0.24]
[0.20,0.34,0.59] [0.21,0.40,0.84]
[0.15,0.28,0.49] [0.13,0.24,0.39]
[0.18,0.27,0.45] [0.14,0.23,0.40]
* Row Sum
** Inverted Column Sum
NRS
TABLE V
MATRIX OF RELATIVE SIGNIFICANCE OF FEEDERS WITH RESPECT TO C3
Criteria
F1
F2
F3
F4
RS
F1
F2
F3
F4
CS
ICS
NICS
[1,1,1]
[3,5,6]
[3,5,6]
[0.13,0.14,0.23]
[7.13,11.1,13.2]
[0.08,0.09,0.14]
[0.10,0.18,0.34]
[0.17,0.20,0.33]
[1,1,1]
[1,1,1]
[7,9,10]
[9.17,11.2,12.33]
[0.08,0.09,0.11]
[0.11,0.18,0.26]
[0.17,0.20,0.33]
[1,1,1]
[1,1,1]
[1,3,4]
[3.17,5.2,6.33]
[0.16,0.19,0.32]
[0.22,0.39,0.76]
[5,7,8]
[0.10,0.11,0.14]
[0.25,0.33,1.00]
[1,1,1]
[6.35,8.44,10.14]
[0.10,0.12,0.16]
[0.14,0.24,0.38]
[6.34,8.40,9.66]
[5.1,7.11,8.14]
[5.25,7.33,9.00]
[9.13,13.14,15.23]
[25.82,35.98,42.03]*
[0.41,0.49,0.72] **
Adjusted
Average
score
[0.15,0.23,0.37] [0.13,0.21,0.36]
[0.12,0.20,0.32] [0.12,0.19,0.29]
[0.12,0.20,0.35] [0.17,0.30,0.56]
[0.22,0.37,0.59] [0.18,0.30,0.49]
* Row Sum
** Inverted Column Sum
NRS
TABLE VI
MATRIX OF RELATIVE SIGNIFICANCE OF FEEDERS WITH RESPECT TO C4
Criteria
F1
F2
F3
F4
F1
F2
F3
F4
CS
ICS
NICS
RS
Adjusted
Average
score
[0.04,0.05,0.12]
[0.05,0.07,0.14]
[0.07,0.14,0.24]
[0.09,0.15,0.23]
[0.17,0.31,0.52]
[0.16,0.27,0.45]
[0.29,0.44,0.70]
[0.37,0.52,0.71]
* Row Sum
** Inverted Column Sum
NRS
[1,1,1]
[0.25,0.33,1]
[0.25,0.33,1.0]
[0.10,0.11,0.14]
[1.6,1.77,3.14]
[1,3,4]
[1,1,1]
[0.13,0.14,0.20]
[1,1,14]
[3.13,5.14,6.2]
[1,3,4]
[5,7,8]
[1,1,1]
[0.17,0.20,0.33]
[7.17,11.2,13.33]
[7,9,10]
[1,1,1]
[3,5,6]
[1,1,1]
[12.0,16.0,18.0]
[10,16,19]
[7.25,9.33,11.0] [4.38,6.47,8.20] [2.27,2.31,2.47] [23.9,34.11,40.67]*
[0.05,0.06,0.10] [0.09,0.11,0.14] [0.12,0.15,0.23] [0.40,0.43,0.44]
[0.67,0.76,0.91] **
[0.06,0.08,0.15] [0.10,0.14,0.21] [0.13,0.20,0.34] [0.45,0.57,0.66]
Note: F1, F2, F3 and F4: Outlet feeders, C1, C2, C3 and C4: Criteria, C1 = Remaining life, C2 = Cost of inspection,
C3 = Inspection Frequency, C4 = Exposure Time, RS = Row sum, CS = Column sum, NRS = Normalized row sum, ICS = Inverted column sum,
NICS = Normalized inverted column sum.
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TABLE VII
COMPUTATION OF FINAL SCORES FOR PRIORITIZING FEEDERS
Criteria
C1
C2
C3
C4
Average Adjusted
Scores of Criteria
[0.06,0.09,0.17]
[0.19,0.33,0.58]
[0.14,0.24,0.40]
[0.20,0.34,0.58]
F1
[0.16,0.27,0.51]
[0.06,0.13,0.24]
[0.13,0.21,0.36]
[0.05,0.07,0.14]
F2
F3
F4
F1
[0.09,0.17,0.32]
[0.12,0.23,0.37]
[0.21,0.33,0.51]
[0.01,0.02,0.09]
[0.21,0.40,0.84]
[0.13,0.24,0.39]
[0.14,0.23,0.40]
[0.01,0.04,0.14]
[0.12,0.19,0.29]
0.17,0.30,0.56]
[0.18,0.30,0.49]
[0.02,0.05,0.14]
F2
F3
F4
[0.01,0.02,0.05]
[0.01,0.02,0.06]
[0.01,0.03,0.09]
[0.04,0.14,0.48]
[0.02,0.08,0.23]
[0.03,0.07,0.23]
[0.02,0.05,0.12]
[0.02,0.07,0.22]
[0.02,0.07,0.19]
Feeder
Feeder
Total Scores
Crisp Score
[0.09,0.15,0.23]
[0.16,0.27,0.45]
[0.37,0.52,0.71]
[0.01,0.02,0.08]
[0.05,0.14,0.44]
0.2100
[0.02,0.05,0.13]
[0.03,0.09,0.26]
[0.07,0.17,0.41]
[0.08,0.25,0.78]
[0.09,0.26,0.77]
[0.13,0.35,0.92]
0.3709
0.3766
0.4668
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0.5
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Crisp Score
0.4
0.3
0.2
0.1
0
F1
F2
F3
F4
Feeders
Fig.6. Priority of feeders after defuzzification
VII CONCLUSIONS
In this paper, feeder selection for maintenance
strategy is illustrated by incorporating fuzzy sets in
Saaty’s priority theory. As “importance” is normally
expressed in subjective/ linguistic terms, “intensity of
importance” scale is fuzzified and expressed in the form
of TFN. Four alternatives (feeders) and four decision
criteria are determined and then priority theory is used.
Priority theory estimates the weights (priorities) of
decision criteria using pair-wise comparison method.
Alternatives are also compared in pair-wise manner
under each criterion. Adjusted average score of the
normalized row and the column sum is suggested to
confirm the priorities. The final scores of each of the
feeders (alternatives) are evaluated by multiplying
(fuzzy arithmetic) the priorities and then adding them.
In this formulated illustration, Feeder F4 is selected first
for maintenance followed by F3, F2 and F1 in that order
as shown in Fig. 6. Design and decision of fuzzy scales
can only be subjectively clinical, when done in
consultation with several experts drawn from the
nuclear industry.
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