1
Introduction
In this study, fuzzy logic (FL), multiple criteria
decision making (MCDM) and maintenance
management (MM) are integrated into one
subject.
2
Introduction
Every system has a life and needs maintenance during its life
cycle.
Maintenance is the key factor to maintain a system under
specified conditions.
Since most of the systems are very complex and are affected
by a lot of mutually exclusive criteria and parameters, selection
of an appropriate maintenance strategy is an important problem
of the maintenance management.
Having a lot of parameters affecting the system on hand, it is
necessary to use a multiple criteria evaluation technique.
Also, if we take account the vague and fuzzy characteristics of
the parameters, it is important to add fuzzy logic approach to the
system
3
Maintenance Management
Maintenance includes all the activities carried out for retaining a
system in a desired operational state.
Maintenance management refers to the application of the
appropriate planning, organization and staffing, program
implementation and control methods to a maintenance activity.
Maintenance management contains all activities including
defining works to be done, planning, resource allocation,
performing maintenance and reporting.
4
Fuzzy Logic
Fuzzy logic (FL) was developed by Zadeh in 1965 to the
problems involving vagueness. Fuzziness is explained in terms
of vagueness. Linguistic variables are used to explain
vagueness in FL. If one can not define the boundaries of
information precisely, then vagueness occurs.Characteristics of
FL are defined as follows:
 Fuzzy cluster is defined as membership function which takes
values in the interval of [0,1].
Information is given by linguistic variables.
FL is suitable for the systems which are difficult to model
mathematically.
5
Fuzzy Logic
Fuzzy numbers are used in FL. In case of ease-of-use, appropriateness of
decision making approaches, and widely usage, fuzzy triangular numbers are
used in this study.
 A~ ( x)
 xL
M  L , L  x  M

U x
 A~ ( x)  
,M  x U
U  M
0, Other


1
L
M
U
X
6
Multiple Criteria Decision Making (MCDM)
Decision making is the selection of the best activities which simultaneously
satisfy goals and constraints. Decision making may be characterized as a
process of choosing or selecting 'sufficiently good' alternative(s) or course(s)
of action, from a set of alternatives, to attain a goal or goals. Decision making,
which includes uncertainties, is a subjective process changing from one
person to another. Fuzzy decision making models can be used under
uncertainties and vagueness since classical decision making can not be used
in those situations.
MCDM consists of a finite set of alternatives among which a decision-maker
has to select or rank; a finite set of criteria weighted according to their
importance. In addition a decision matrix consisting of the rating of each
alternative with respect to each criterion using a suitable measure is formed.
The evaluation ratings are, then, aggregated taking into account the weights
of the criteria, to get a global evaluation for each alternative and a total
ranking of the alternatives.
7
Analytic Hierarchy Process (AHP)
AHP which was developed by Saaty in 1980 is became one of the most
widely used methods to solve MCDM problems practically. AHP solves a
problem by structuring it in hierarchic orders. AHP uses those steps below to
solve a problem:
Decomposition: First problem is divided into small parts and structured as
hierarchically. Saaty and Vargas (1991) stated that a decision maker can not
simultaneously compare more than 7 ± 2 elements, and offer hierarchical
decomposition for solving MCDM problems. We construct the structure of the
problem according to its main components: goal/objective set of criteria for
evaluation, and the decision alternatives.
Pairwise comparison: The relative importance of criteria is established
through pairwise comparisons using a square matrix. Hence, judgments are to
be made on the importance of criteria which is done with the aid of Saaty's
nine-point scale .
Synthesis of priorities: Criteria weightings are calculated by using decision
matrixes. Finally, relative weightings are synthesized by adding each other to
select /sort alternatives.
8
Analytic Hierarchy Process (AHP)
Objective
Criterion1
Alternative 1
Criterion 2
Alternative 2
..........
Alternative 3
Criterion n
.........
Numerical Value
Explanation
1
Equally Important
3
Slightly Important
5
Reasonably Important
7
Highly Important
9
Definitely Important
2,4,6,8
Intermediate Values
Alternative n
9
TOPSIS
TOPSIS (Technique for Order Preferences by Similarity to an Ideal Solution)
which was developed by Hwang and Yoon (1981) is used to order alternatives.
TOPSIS sorts alternatives by calculating the distances between ideal solution and
alternatives. For this, first positive and negative ideal solutions are defined
separately. Positive ideal solution is called as maximum benefit solution and
includes the best values of the criteria. Negative ideal solution is known as
minimum benefit solution and includes the worst values of the criteria. Solutions
are defined as points which are the nearest to the positive ideal solution and the
farthest to the negative ideal solution at the same time in TOPSIS. Optimum
alternative is the one which is the nearest to the positive ideal solution and the
farthest to the negative ideal solution. TOPSIS calculation process is given below:
Obtaining normalized decision matrix for alternatives,
Obtaining weighted decision matrix for alternatives,
Calculating positive and negative ideal solutions,
Calculating distances to the positive and negative ideal solution for each
alternative,
Calculating the relative closeness to the ideal solution for each alternative,
Sorting the alternatives.
10
~
91x
Fuzzy AHP
To overcome the difficulties faced in classic MCDM, FAHP is offered
FAHP expands AHP by using the fuzzy cluster theory. Deciding the relative
importance of the criteria and making the fuzzy decision matrix, a fuzzy ratio
scale is used in FAHP.
Fuzzy Number
Explanation
~
1
(1,1,3)
~x
(x-2,x,x+2), x=3,5,7 için
~
9
(7,9,9)
11
Application Model
Criteria comparisons, normalized decision matrix for alternatives, and
weighted decision matrix are prepared using the FAHP. Fuzzy TOPSIS is
used to order alternatives. Fuzzy comparison scale (FCS) is used to compare
alternatives and criteria.
Fuzzy AHP-Fuzzy TOPSIS method which is developed in this study is used
for maintenance strategy selection problem. Developed method is used for
Istanbul Metro maintenance applications including fixed installations for
electronics and electro mechanic systems comprised of signaling, SCADA,
telecommunications, public announces, CCTV, escalators, elevators, fire
detection and extinguishing. Currently, corrective and periodic maintenance
techniques are used for those equipments.
First objective is defined and then related criteria and alternatives are defined
hierarchically to achieve this goal. FAHP method is used for comparisons and
fuzzy TOPSIS is used for alternative ordering.
12
Application Model
Objective definition
Defining criteria and subcriteria
Pairwise comparisons of criteria and
subcriteria importance using fuzzy linguistic
variables by every decision maker
Pairwise comparisons of alternatives and
criteria accordance using fuzzy linguistic
variables by every decision maker
Conversion of linguistic comparisons to fuzzy
triangular numbers
Conversion of linguistic comparisons to
fuzzy triangular numbers
Taking the average of the pairwise
comparisons made by the decision makers
Taking the average of the pairwise
comparisons made by the decision
makers
Calculation of relative fuzzy performance
points for criteria and subcriteria
Calculation of relative fuzzy performance
points for the alternatives
Fuzzy weighted performance measurement
of the criteria
Fuzzy weighted performance
measurement of the alternatives
Fuzzy
AHP
Defining alternatives
Fuzzy weighted performance measurement
of the alternatives criteria accordance
13
Application Model
Calculation of the fuzzy negative
and fuzzy positive ideal solutions
Fuzzy TOPSIS
Calculation of the fuzzy distances
between ideal (negative/positive)
solutions and alternatives
Calculation of the classic distances
between ideal (negative/positive)
solutions and alternatives
Defuzzification
Finding the relative distances of the
alternatives to the ideal solution
and ordering.
14
Mathematical Model
Ai: Alternative i, i=1, 2,…, n
Cj: Criterion j, j=1, 2,...,m
a~i j : Fuzzy accordance point for alternative i to criterion j
~
B : Fuzzy comparison matrix for criteria
~
U : Alternative-criteria accordance matrix obtained by evaluating
alternative i for criterion j.
~
N : Final fuzzy weighted evaluation matrix calculated by evaluating for
alternative i to criterion j.
djep: Linguistic variable that decision maker p makes pairwise
comparison for criteria j and e , e=1, 2, ..., m.
~
d jep : Fuzzy triangular numbers for djep linguistic variable.
(Ljep, Mjep, Rjep): Fuzzy triangular numbers’ left, middle, and right side
values in order for djep linguistic variable.
~
d je : Fuzzy pairwise evaluation point for criteria j and e.
~ : Relative fuzzy weighting for criterion Cj.
w
j
15
Mathematical Model
~
W : Fuzzy weighting vector.
~
 : Fuzzy eigenvalue vector.
(i ,  j , k ) : Left, middle, and right side values for fuzzy eigenvalue vector in order.
~
m
ij : Fuzzy weighted performance point for alternative i to criterion j.
(Lmij , Mmij , Rmij ) : Left, middle, and right side values for the
~ fuzzy weighted performance point.
m
ij
~
k j : Maximum fuzzy performance point which alternative i takes for criterion j (fuzzy positive ideal solution).
~
k j : Minimum fuzzy performance point which alternative i takes for criterion j (fuzzy negative ideal solution).
~
d i : Fuzzy distance to positive ideal solution for i alternative’s performance point.
~
d i : Fuzzy distance to negative ideal solution for i alternative’s performance point.
d i : Distance to positive ideal solution for i alternative’s performance point.
d i : Distance to negative ideal solution for i alternative’s performance point.
(Ldi , Mdi , Rdi ) : Left, middle, and right side values for fuzzy distance
(Ldi , Mdi , Rdi ) : Left, middle, and right side values for fuzzy distance
 : Optimism index.
Ri: Relative closeness to ideal solution for alternative i.
~
d i in order.
~
d i in order.
16
Mathematical Model
~
1. djep linguistic variable is converted to fuzzy triangular numbers ( d jep ) using fuzzy
comparison scale.
~
d jep =(Ljep,Mjep,Rjep),
j=1,2,...,m, e=1,2,...,m, p=1,2,...,t
(2)
2. Fuzzy decision making matrix is calculated as below using the decision makers’
~
d
comparisons which is converted to fuzzy triangular numbers ( jep )
p
Lje=
Mje=
p
L je1  L je2  ...  L jep
, j=1,2,...,m; e=1,2,...,m; p=1,2,...,t
M je1  M je2  ...  M jep , j=1,2,...,m; e=1,2,...,m; p=1,2,...,t
(3)
(4)
p
Uje= U je1  U je2  ...  U jep , j=1,2,...,m; e=1,2,...,m; p=1,2,..,t
(5)
~
d je =( Lje, Mje, Rje), j=1,2,...,m; e=1,2,...,m
(6)
~
d ej =( 1/Rje, 1/Mje, 1/Lje), j=1,2,...,m; e=1,2,...,m
(7)
17
Mathematical Model
Then, fuzzy comparison matrix ( B~ ) is become as shown below:
C1
C1
~ C
B 2
...
Cm
C2 ...
~
 d 11
~
 d 21
 ...
~
d m1
~
d 12
~
d 22
...
~
d m2
Cm
...
...
...
...
~
d 1m 
~ 
d 2m 
... 
~ 
d mm 
(8)
~
~
1. After fuzzy comparison is obtained, relative fuzzy weightings ( w j ) and fuzzy weighted vector ( W )
related to criteria are calculated.
m
~ 
w
j
~
d
 je
e 1
m m
~
d
 je
, j=1,2,...,m; e=1,2,...,m;
(9)
j 1 e 1
~
~ ,w
~ ,...,w
~ )
W  (w
1
2
m
T
(10)
18
Mathematical Model
~
~

W
4. Having fuzzy weighting vector ( ) is found, fuzzy eigenvalue vector ( ), classic eigenvalue vector (  ),
and maximum eigenvalue ( max ) are calculated.
~
  (i ,  j , k )
~
~
(11)
~ ~
  B W / W
(12)
  3 i   j  k
(13)
max  max( )
(14)
5. Consistency index (CI) and consistency ratio (CR) are calculated. If consistency ratio is smaller than 0.1
then matrix is accepted to be consistent, otherwise evaluations should be revised. Randomness index (RI)
values are shown in Table 4.
 max  n
n 1
CI
CR 
RI
CI 
(15)
(16)
Randomness index (Saaty and Vargas, 1991)
n
Randomness
Index
n: Matrix size
1 2
3
4
5
6
7
8
9
10
11
12
13
14
15
0 0 0,58 0,9 1,12 1,24 1,32 1,41 1,45 1,49 1,51 1,48 1,56 1,57 1,59
19
Mathematical Model
~
~
6. Using equations (2)–(10) alternative-criteria accordance matrix ( U ) is formed similarly. U is subjected to
consistency test defined in equations (11)-(16).
C1
C2 ...
A1  a~11
~
~ A2  a 21
U
...  ...

Am a~m1
a~12
a~
22
...
a~
m2
Cm
... a~1m 
... a~2 m 
... ... 

... a~mm 
(17)
~
7. Final fuzzy evaluation matrix ( N ) is formed by multiplying equations (10) and (17) for every criterion
separately.
~
~
~
A1  a~11  w~1, a~12  w~2 ,...,a~1m  w~m  A1  m11 m12 ... m1m 
~
~
~ 
A2  a~21  w~1 , a~22  w~2 ,...,a~2 m  w~m  A2  m 21 m 22 ... m 2 m 
~
N  ...  .............................................  = ...  ...
(18)
... ... ... 
~


~
~ 
m
... m
Am a~m1  w~1, a~m 2  w~2 ,...,a~mm  w~m  Am m
m1
m2
mm 
8. After having final fuzzy evaluation matrix is calculated, fuzzy TOPSIS method is used to order alternatives.
~
~
The best ( k j ) and the worst ( k j ) fuzzy performance points are calculated using
 max
  min
~
k j 
~ , i  1,2,.., n; j  1,2,...,m
m
ij
~
k j
~ , i  1,2,..,n; j  1,2,...,m
m
ij


~
N.
(19)
20
(20)
Mathematical Model
9. Since positive and negative ideal solutions are found, fuzzy distances to positive and negative ideal
solutions are calculated using equations (21) and (22).
10.
~
d i 

~  k~ 
m
j
ij

~
d i 

~  k~ 
m
j
ij

m
j 1
m
j 1
2
2
i  1,2,...,n
(21)
i  1,2,...,n
(22)
~
~
di and d i fuzzy numbers are defined as:
~
di  (Ldi , Mdi , Rdi )
~
d i  ( Ldi , Mdi , Rdi )
(23)
(24)
11. Calculating distances from fuzzy negative and fuzzy positive ideal solutions to alternatives, fuzzy
negative and fuzzy positive ideal solutions are defuzzified by using the methods explained below. To
compare alternative ordering obtained from different defuzzification methods more than one
defuzzification method is used.
21
Mathematical Model
i.
Centroid method: Centroid method is one of the most widely used defuzzification methods. A lot of author
is used centroid method for defuzzification (Opricovic and Tzeng, 2003; Kuo et al., 2002; Chen et al.,
2005; Chan et al., 2003; Chiou et al., 2005),
di  (Ldi  Mdi  Rdi ) / 3
(25)
di  (Ldi  Mdi  Rdi ) / 3
(26)
ii. Optimism based defuzzification methods: Optimism/pessimism level of decision makers is considered
while defuzzification is made.
a. Kaufmann and Gupta Method: In this method calculation is made as fuzzy numbers’ middle value is
multiplied by a big coefficient. (Chan et al., 2003).
d i  ( Ld i  2Mdi  Rdi ) / 4
(27)
d i  ( Ld i  2Mdi  Rdi ) / 4
(28)
a. Liou and Wang Method: Calculation is made using  optimism index (Kaptanoglu and Ozok, 2006).
d i  ((1   )Ldi  Mdi  Rdi ) / 2
(29)
d i  ((1   )Ldi  Mdi  Rdi ) / 2
(30)
12. After defuzzification process, relative distances of alternatives to the ideal solution are calculated. The
best alternative is the one which is the farthest from negative ideal solution and the closest to the positive
ideal solution.
d i
Ri  
i  1,2,...,n
d i  d i
(31)
22
Ri shows the final performance point. Alternatives are listed related to their Ri values from smaller to bigger.
Fuzzy Comparison Scale
Linguistic Variable
Linguistic Variable’s
Inverse
LO
MO RO
LT
MT
RT
Slightly Unimportant
Slightly Important
1/3
1
1
1
1
3
Unimportant
Important
1/5
1/3
1
1
3
5
Reasonably Important
1/7
1/5
1/3
3
5
7
Highly Unimportant
Highly Important
1/9
1/7
1/5
5
7
9
Definitely Unimportant
Definitely Important
1/9
1/9
1/7
7
9
9
Equally Important
Equally Important
1
1
1
1
1
1
Slightly Important
Slightly Unimportant
1
1
3
1/3
1
1
Important
Unimportant
1
3
5
1/5
1/3
1
Reasonably Important
Reasonably Unimportant
3
5
7
1/7
1/5
1/3
Highly Important
Highly Unimportant
5
7
9
1/9
1/7
1/5
Definitely Important
Definitely Unimportant
7
9
9
1/9
1/9
1/7
Reasonably
Unimportant
(LO, MO, RO): Values of fuzzy linguistic variables, (LT, MT, RT): Values of inverse fuzzy linguistic variables.
23
Criteria List
A Windows based
software has been
developed for the
application
24
Criteria Comparisons
25
Rank Ordering
Fuzzy Weighted Scores
26
Sensitivity Analysis
Sensitivity analysis has been made to investigate the changes of ranks of the
alternatives according to the changes of criteria’s importance. The main
criteria’s importance is changed assigning different fuzzy linguistic variables at
the sensitivity analysis in fuzzy model and the effect of these changes on the
result has been investigated. Also the changes of the result have been
investigated assigning different values to optimism coefficient at the Liou and
Wang method which uses optimism coefficient.
The effect of changes in the importance of criteria on the changes of the ranks
of alternatives has been investigated at sensitivity analysis.
27
Sensitivity Analysis for Cost
The opinion of the decision makers is that cost main criterion is more
important than other criteria except safety criterion. Cost is a little less
important than safety criterion. Accordingly when cost criterion’s importance is
decreased against other criteria ranking doesn’t change at Centroid and
Kaufmann-Gupta methods; and it doesn’t change when α≥0,29 at Liou and
Wang method. RCM moves up to first rank when α<0,29 and TPM takes the
second place.
Kaufman Liou and
Liou and Liou and
Centroid
Alternatives
and
Gupta
Wang
Wang
Wang
(Alfa=0,29) (Alfa=0,5) (Alfa=0,9)
Ri
CM
0,0044
0,0078
0,00120
0,0078
0,0047
PM
0,1817
0,1828
0,1834
0,1828
0,1824
CBM
0,3219
0,3172
0,3106
0,3172
0,3221
PDM
0,5802
0,5677
0,5506
0,5677
0,5804
RCM
0,7882
0,7905
0,7932
0,7905
0,7884
28
TPM
0,8125
0,8047
0,7932
0,8047
0,8134
Sensitivity Analysis for Cost
When the importance of cost is decreased according to other criteria, TPM
continues to go further from positive ideal solution but ranking doesn’t change.
If the importance of cost is made equal to the others, ranking changes at all
methods RCM moves up to the first rank, and TPM moves down to the
second rank.
Kaufman Liou and
Centroid
and
Gupta
Alternatives
Liou and
Liou and
Wang
Wang
Wang
(Alfa=0,1) (Alfa=0,5) (Alfa=0,9)
Ri
CM
0,0056
0,0097
0,0260
0,0097
0,0058
PM
0,1848
0,1858
0,1862
0,1858
0,1857
CBM
0,3184
0,3130
0,2881
0,3130
0,3188
PDM
0,5692
0,5550
0,4894
0,5550
0,5699
RCM
0,8083
0,8091
0,8113
0,8091
0,8086
TPM
0,8007
0,7924
0,7502
0,7924
0,8022
29
Sensitivity Analysis for Safety
Decision makers believe that safety is much more important than other criteria. When
the importance of safety is increased (making it much more important than all criteria)
an improvement is seen at RCM but there is no change at the rankings of alternatives
in Centroid and Kaufman Gupta methods, in Liou and Wang method there is no
change at ranking when α>0.12, RCM moves up to the first rank and TPM moves
down to the second rank when α≤0,12 (Ri=0,7722).
The rankings of the alternatives don’t change when safety’s importance is turned to
“highly important” at Centroid and Kaufmann-Gupta methods but Ri values of TPM and
RCM alternatives approach each other very much. (Ri=0,7879 for TPM, Ri=0,7799 for
RCM in Kaufmann-Gupta method). In Liou and Wang method ranking doesn’t change
for α>0.4 , RCM moves up to first rank (Ri=0,7850), and TPM moves down to second
(Ri=0,7825) for α≤0.4 .
30
Sensitivity Analysis for Safety
The rankings of the alternatives don’t change when safety’s importance
is turned to “definitely important” at Centroid method, ranking changes
at Kaufmann-Gupta method and RCM moves up to first rank, and TPM
moves down to second. At Liou-Wang method ranking doesn’t change
for α>0.64 , TPM moves down to second rank, and RCM moves up to
first rank for α≤0.64 in Liou-Wang method.
When the importance of safety criterion is reduced, RCM goes further
from positive ideal solution, and TPM approaches to positive ideal
solution continuously. But ranks of the alternatives don’t change at any
method.
31
Sensitivity Analysis
Decision makers believe that applicability is less important than cost and
safety, more important than competitive advantage and working morale. When
the importance of applicability increases PDM approaches to RCM, RCM
approaches to TPM but ranking doesn’t change. When the importance of
applicability is decreased, RCM goes further from positive ideal solution, TPM
approaches to the ideal solution. But the ranking of alternatives don’t change.
Decision makers believe that competitive advantage is more important than
working morale, and less important than the other criteria. There is no change
at the ranking of alternatives when the importance of competitive advantage is
increased or decreased.
Decision makers believe that working morale is less important than all the
other criteria. There is no change at the ranking of alternatives when the
importance of working morale is increased or decreased.
32
Conclusion
A fuzzy multiple criteria decision making model has been developed for
choosing maintenance strategies and a study has been made for choosing
maintenance strategies using fuzzy multiple criteria decision making
approach. The method used in this study offers a systematic approach to
the selection of maintenance strategies. Most used maintenance
strategies at Istanbul Metro are corrective maintenance and periodic
maintenance. But it has been understood that corrective and periodic
maintenance used currently in Istanbul Metro is not suitable because the
system of Istanbul Metro is complex, has high importance about safety
and is directly related with the passengers.
33