Fuzzy-TISM: A Fuzzy Extension of TISM for Group Decision Making

Abstract

This paper proposes Fuzzy-TISM, approach for group decision making process. The proposed approach is a fuzzy extension of TISM, which is a multi-criteria decision making technique. TISM is an effective technique and is applied widely to identify relationships among different criteria by creating a comprehensive systematic model of directly and indirectly related criteria. The proposed Fuzzy-TISM approach consolidates the process of group preference aggregation in the fuzzy environment, which can be easily applied to any real world group decision making problem. The proposed approach is a novel attempt to integrate TISM approach with the fuzzy sets. The integration of TISM with fuzzy sets provides flexibility to decision makers to further understand the level of influences of one criteria over another, which was earlier present only in the form of binary (0,1) numbers. 0 represents no influence and 1 represents influence. Due to this, the decision maker is left with only the option of saying 0 or 1 irrespective of the level of influence whether it is low, high, or very high. The proposed Fuzzy-TISM approach take care of this issue and gives a wider flexibility to express the level of influence using fuzzy numbers. The working methodology of proposed Fuzzy-TISM is demonstrated through an illustrative example based on vendor selection.

Full text

ORIGINAL ARTICLE
Fuzzy-TISM: A Fuzzy Extension of TISM for Group Decision
Making
Gaurav Khatwani •Surya Prakash Singh •
Ashish Trivedi •Ankur Chauhan
Received: 31 August 2014 / Accepted: 19 November 2014
ÓGlobal Institute of Flexible Systems Management 2014
Abstract This paper proposes Fuzzy-TISM, approach for
group decision making process. The proposed approach is a
fuzzy extension of TISM, which is a multi-criteria decision
making technique. TISM is an effective technique and is
applied widely to identify relationships among different cri-
teria by creating a comprehensive systematic model of directly
and indirectly related criteria. The proposed Fuzzy-TISM
approach consolidates the process of group preference
aggregation in the fuzzy environment, which can be easily
applied to any realworld group decision makingproblem. The
proposed approach is a novel attempt to integrate TISM
approach with the fuzzy sets. The integration of TISM with
fuzzy sets provides flexibility to decision makers to further
understand the level of influences of one criteria over another,
which was earlier present only in the form of binary (0,1)
numbers. 0 represents no influence and 1 represents influence.
Due to this, the decision maker is left with only the option of
saying 0 or 1 irrespective of the level of influence whether it is
low, high, or very high. The proposed Fuzzy-TISM approach
take care of this issue and gives a wider flexibility to express
the level of influence using fuzzy numbers. The working
methodology of proposed Fuzzy-TISM is demonstrated
through an illustrative example based on vendor selection.
Keywords Fuzzy systems Group decision making 
Interpretive structural modeling (ISM) Multi-criteria
decision making Total interpretive structural modeling
(TISM) Vendor selection problem
Introduction
The process of selecting the best alternative amidst a set of
suitable alternatives is known as decision making. Multi-
criteria decision-making (MCDM) problems are defined as
decision making problems consisting of several criteria
(Bellman and Zadeh 1970; Keeney and Raiffa 1976; Jain
1977; Ching-Lai and Yoon 1981; Nurmi 1981; Tanino
1984; Boender et al. 1989; Chen et al. 1992; Kacprzyk
et al. 1992; Hsu and Chen 1996,1997; Feng and Wang
2000; Chang and Yeh 2002; Lee 2002; Tsaur et al. 2002).
MCDM problems can be divided into two categories. The
first one is the classical MCDM problem, in which indi-
vidual preferences are measured in crisp numbers (Keeney
and Raiffa 1976; Chen et al. 1992; Feng and Wang 2000),
and the second one is the Fuzzy MCDM (Bellman and
Zadeh 1970; Jain 1977; Nurmi 1981; Tanino 1984; Chen
et al. 1992; Kacprzyk et al. 1992; Hsu and Chen 1996,
1997; Chang and Yeh 2002; Lee 2002; Tsaur et al. 2002;
Cebeci and Ruan 2007) problems, in which individual
preferences are evaluated on vagueness, subjectivity and
imprecision and expressed into linguistic terms, which are
further mapped into fuzzy numbers (Zadeh 1965; Zim-
mermann 1987; Zimmerman 1991). ISM is one of the
recognized methods for classical MCDM. It is a method-
ology that helps in discovering relationships among spe-
cific criteria that constitute problem related to the system
(Jharkharia and Shankar 2005). The collective study of
criteria helps us to develop understanding of direct and
G. Khatwani A. Trivedi A. Chauhan
Indian Institute of Management Rohtak, Rohtak, India
e-mail: g_khatwani@yahoo.co.in
A. Trivedi
e-mail: at.iccmrt@gmail.com
A. Chauhan
e-mail: Chauhan.ankur2903@gmail.com
S. P. Singh (&)
Indian Institute of Technology Delhi, Delhi, India
e-mail: surya.singh@gmail.com
123
Global Journal of Flexible Systems Management
DOI 10.1007/s40171-014-0087-4

indirect relationships among them rather than studying
individuals in isolation. The process begins with identify-
ing criteria relevant to a problem and subsequently, a
structural self-interaction matrix (SSIM) of criteria is
developed using pairwise comparison approach. SSIM is
then converted into reachability matrix and its transitivity
is verified. Finally, elements are partitioned and structural
model is extracted, which is known as ISM (Agarwal et al.
2007). Since ISM is well known classical MCDM
approach, many researchers and practitioners have used it
to solve MCDM problems. However, the inadequacy of
crisp values to model real life situations in uncertain and
fuzzy scenarios that are frequent in expert’s judgments
highlights the need to integrate fuzzy group decision
making in ISM. Further, recent studies have also empha-
sized on uncertainty in linguistic terms under group deci-
sion environment (Xu 2004,2006; Xu and Da 2008; Wei
2009; Fan and Liu 2010). In ISM, the reachability matrix is
constructed by replacing relationship symbols of SSIM by
1 and 0. However, the true maximum and minimum values
are far from these extreme values of 1 and 0. Therefore, the
extreme values are not appropriate to represent relationship
between elements. Previous studies have attempted to
upgrade ISM to TISM in order to make elucidation of
structural model fully interpretive. Further, the importance
of implementing fuzzy TISM has been highlighted (Sushil
2012b) but it fails to deliver the procedural model for
implementation in real world decision making. Moreover,
there have been theoretical contributions in area of flexible
systems management (Sushil 1997) and confluence of
continuity and change management (Sushil 2012a), which
inspired us to develop the fuzzy TISM model. Therefore,
we propose a model for fuzzy TISM to analyze group
preferences and interpret hierarchical relationship of ele-
ments in a complex system under fuzzy environment.
The rest of the paper is organized as follows. In ‘‘Pre-
liminaries section, past relevant work on fuzzy theory and
TISM methodology is reviewed. In ’’Literature Review‘‘
section, group decision making situation where application
of ISM and TISM is involved is discussed. In ’’Fuzzy-
TISM: A Fuzzy Extension of TISM‘‘ section, Fuzzy-TISM
for group decision making is proposed and detailed
description is provided. To demonstrate the application of
Fuzzy-TISM an illustration based on vendor selection
problem is given in ’’Illustrative Example‘‘ section fol-
lowed by conclusions.
Preliminaries
Fuzzy theory plays significant role in dealing with vague-
ness and uncertainty in human language and thoughts in
decision making. The assessment of decision makers
depends upon their past knowledge and experiences and
often their estimations are articulated in equivocal linguistic
terms. However, in order to integrate various opinions,
experiences, ideas and motivations of individual experts it
becomes important to translate the linguistic judgments into
fuzzy numbers. Thus, the problems discussed in group
decision making environment highlights the need to imple-
ment fuzzy logic. Some of the essential definitions of fuzzy
logic and its theory can be referred from (Zadeh 1965;
Laarhoven and Pedrycz 1983; Zimmermann 1987;Kauff-
man and Gupta 1991;Zimmerman1991;Li1999).
In the fuzzy theory, the fuzzy set ~
Bis the subset of
universal set X, which can be characterized by membership
function l~
BxðÞrepresenting a mapping l~
B:X!½0;1. The
function value of l~
BxðÞ
~
Bis called the membership value
representing degree of truth that x is an element of ~
B.
Assuming that l~
B2½0;1, where l~
BxðÞ¼0 denotes that x
belongs completely to fuzzy set ~
B, while l~
BxðÞ¼1 denotes
that x does not belong to fuzzy set ~
B. The membership
function l~
BðxÞof triangular fuzzy number can be defined as
l~
BðxÞ¼
0;x\l
xl
mllxm;
ux
ummxu;
0;otherwise
8
>
>
>
<
>
>
>
:
which can be represented by triplet (l,m,u) and
lBmBu. See Fig. 1.
Following are the theorems which are used in this paper
are briefly discussed.
Theorem 1 Let ~
B1¼ðl1;m1;u1Þand ~
B2¼ðl2;m2;u2Þbe
two triangular fuzzy numbers. The addition operation of ~
B1
and ~
B2is denoted by ~
B1~
B2which results into another
triangular fuzzy number and that can be represented by
~
B1~
B2¼ðl1þl2;m1þm2;u1þu2Þð1Þ
Theorem 2 CFCS (Converting Fuzzy data into Crisp
Scores) defuzzification method (Opricovic and Tzeng,
2003). Let ~
Bk¼lk;mk;uk
ðÞ;k¼1;2;...::nbe the positive
triangular fuzzy number and ~
Bcrisp
kdenotes the crisp value.
~
1
0 X
Fig. 1 A triangular fuzzy number ~
B
Global Journal of Flexible Systems Management
123

The crisp value of i-th criteria can be determined by
following four steps
Step 1: Computing L =min(l
k
); R =maxðukÞ;k¼
1;2;...::nand D¼RL, then Compute for each alterna-
tives using Eq. (5).
xlk ¼ðlkLÞ=D;xmk ¼ðmkLÞ=D;xuk ¼ðukLÞ=D
ð2Þ
Step 2: Compute left score (ls) and right score (rs)
normalized values using Eq. (3).
xls
k¼xmk=ð1þxmk xlk Þand xrs
k¼xuk=ð1þxuk xmk Þð3Þ
Step 3: Compute total normalized crisp value using
Eq. (4).
xcrisp
k¼xls
k1xls
k

þxrs
kxrs
k

=½1xls
kþxrs
kð4Þ
Step 4: Compute crisp value for ~
Bkusing Eq. (5).
~
Bcrisp
k¼Lþxcrisp
kDð5Þ
Literature review
The theoretical foundation of TISM proposed by Sushil
(Sushil 2012b) is built on ISM methodology developed by
Warfield (1973,1974). Much details on ISM and its various
applications can be referred from (Harary et al. 1965),
Warfield and Hill (1973), Malone (1975), Hawthorne and
Sage (1975), Warfield (1976,1994,1999), Jedlicka and
Meyer (1980), Waller (1980), Mandal and Deshmukh
(1994), (Sushil 1994), Sharma et al. (1995), Warfield
(2003), Saxena et al. (2006), Agarwal et al. (2007), Lee
(2007), Mohammed (2008), (Haleem and Sushil 2012),
Sagar et al. (2013), Mangala et al. (2014), and Srivastava
and Sushil (2014).
TISM (Sushil 2012b) is derived from ISM facilitates the
graphical representation of complex systems. ISM assists
individuals in developing complex relationships among
multiple elements in complex system. ISM is an interpre-
tive method that helps the group to decide structural rela-
tionship between multiple elements and extract structure,
which is portrayed in digraph model. ISM permits identi-
fication of system structure, which contain elements related
to other in some fashion (Farris and Sage 1975). Previous
studies adopted the enhanced version of ISM which is
called TISM (Nasim 2011; Sushil 2012b; Dubey and Ali
2014). TISM is the method that demonstrates direct as well
as transitive relationship in order to make structural model
fully interpretive. Nasim (2011) has shown the applica-
bility of TISM in modelling continuity and change in
e-government, and similarly Prasad and Suri (2011) has
shown the applicability of TISM in the private higher
technical education. TISM has also been used to study
relationship amongst various strategic performance criteria
for effective strategy execution (Srivastava 2013; Srivast-
ava and Sushil 2014). Following are the main steps
involved in the ISM methodology.
TISM (similar to ISM) begins with defining the ele-
ments and determining contextual relationships within
elements. Further, we develop SSIM, reachability matrix,
Transitivity matrix, lower triangular format of reachability
matrix, digraph for TISM and the interpretive structural
based on contextual relationships (Saxena et al. 2006).
Developing SSIM
The relation between any two elements (i and j) is judged
by experts keeping in view of contextual relationship
within each element. Four types of symbols are used
namely V, A, X, and O to demonstrate the relation between
any two elements under consideration. The SSIM can be
prepared by filling responses of group of experts on pair-
wise interaction matrix.
Developing Reachability Matrix
The reachability matrix is constructed by transforming
information within SSIM into 10s and 00s.
Transitivity Check on Reachability Matrix
A transitivity check has to be performed on reachability
matrix. The transitivity matrix is checked for the transi-
tivity rule and updated till full transitivity is established.
Reachability Matrix Partition
After creating reachability matrix and performing transi-
tivity check, the next step is to create digraph and extract
structural model. The partition of reachability matrix pro-
cess can be carried by using relation partition and level
partition on sets and subsets of elements (Warfield 1974).
Creating Digraph for TISM
After identifying the levels of elements the relationship
between elements are constructed using serial numbers of
elements and directed arrows. The constructed digraph is
complex and has to be examined interactively to eliminate
transitivity. After elimination of transitivity we finalize the
digraph for total interpretive structural model. The digraph
portrays information related to hierarchy of elements.
Global Journal of Flexible Systems Management
123

Final TISM Model
Final TISM model is created after the initial digraph of
ISM where all transitive links along with the direct influ-
encing links are shown. In addition, information is also
mentioned along with all links (direct and transitive links)
to give a proper justification behind the influence of one
criterion to other.
Fuzzy-TISM: A Fuzzy Extension of TISM
The detailed procedure for the proposed model of Fuzzy-
TISM is discussed here. The linguistic terms and its lin-
guistic values are shown in Table 1. The linguistic values
are assumed based on the triangular fuzzy numbers for
linguistic variables shown in Fig. 2. Fuzzy interrelationship
between two factors are shown using symbols which are
given in Fig. 3. Stepwise description of Fuzzy-TISM is as
follows.
Step 1: Start of Decision Making Process
The decision making process begins with defining decision
goals, gathering significant information and identifying pos-
sible range of alternatives. Further alternatives are evaluated,
selected and monitored to ensure decision goals are achieved
(Hess and Siciliano 1996; Opricovic and Tzeng 2003). Thus,
after setting decision goal a committee is formed for gathering
group knowledge, which assists in problem solving.
Step 2: Selection of Criteria
In this step, a set of criteria has to be established. The
criteria have relationships through which either they
influence/impact the other criteria or influenced/impacted
by other criteria and may be both. To deal with the
uncertainty in linguistic judgments of experts, we renounce
crisp method of decision making in TISM and incorporate
fuzzy linguistic scale for group decision making (Li 1999).
The varied degree of influence/impact can be expressed in
five linguistic terms as {Very high, High, Low, Very low,
No}. The corresponding positive fuzzy triangular numbers
are demonstrated in Table 1and Fig. 2.
Step 3: Gathering Responses and Creating SSIM Matrix
The relationship between the criteria C¼fCiji¼
1;2...ngis gathered from group of sexperts and filled in
SSIM matrix. The respondents can use the combination of
symbols V, A, X and O and linguistic terms (mentioned in
Table 1) to demonstrate the relationship between the cri-
teria. The respondents will have following four options.
i. V: To demonstrate the relationship from element i to
element j but not vice versa; the relationship can be
represented as V followed by {Very high (VH), High
(H), Low (L), Very low (VL)}. For example V (VH) or
V(H) or V(L) or V(VL).
ii. A: To demonstrate the relationship from element j to
element i but not vice versa; the relationship can be
represented as A followed by {Very high (VH),
High(H), Low (L), Very low (VL)}.
iii. X: To demonstrate the relationship from element i to j
and j to i; the relationship can be represented as X
followed by {Very high (VH), High (H), Low (L),
Very low (VL)}.
iv. O: To demonstrate no existence of relationship; the
relationship can be represented as O followed by {No
influence (No)}. For example O(No)
In addition to above responses in fuzzy, the respondents
are also informed to justify in few words their assessment
at all levels of influence, i.e. VH, H, L, VL, and No of one
criterion to other.
No VL L H VH
1
0 0.25 0.5 0.75 1 X
Fig. 2 Triangular fuzzy numbers for linguistic variables
Table 1 Linguistic scales for the influence
Linguistic
terms
Linguistic
values
Very high influence (VH) (0.75,1.0,1.0)
High influence (H) (0.5,0.75,1.0)
Low influence (L) (0.25,0.5,0.75)
Very low influence (VL) (0,0.25,0.5)
No influence (No) (0,0,0.25)
Very High Influence
High Influence
Low Influence
Very Low Influence
Fig. 3 Symbols for representation of fuzzy relationship between
criteria
Global Journal of Flexible Systems Management
123

Step 4: Calculation of Aggregated SSIM and Final
Fuzzy Reachability Matrix
Here, mode has been used to aggregate responses of indi-
vidual experts, i.e. the preferences of individual experts
with highest frequencies are pooled together in aggregated
SSIM matrix. Further, aggregated SSIM matrix is trans-
formed into fuzzy reachability matrix. The linguistic terms
in aggregated SSIM matrix are replaced by corresponding
fuzzy triangular linguistic values. The following situations
occur during creation of final fuzzy reachability matrix.
i. If the entry (i,j) is V(VH) : The entry (i,j) can be
denoted by (0.75,1.0,1.0) and entry (j,i) will be 0{No}
which will be denoted by (0,0,0.25)
ii. If the entry (i,j) is V(H) : The entry (i,j) can be denoted
by (0.5,0.75,1.0) and entry (j,i) will be 0{No} which
will be denoted by (0,0,0.25)
iii. If the entry (i,j) is V(L): The entry (i,j) can be denoted
by (0.25,0.5,0.75) and entry (j,i) will be 0{No} which
will be denoted by (0,0,0.25)
iv. If the entry (i,j) is V(VL): The entry (i,j) can be
denoted by (0,0.25,0.5) and entry (j,i) will be 0{No}
which will be denoted by (0,0,0.25)
v. If the entry (i,j) is A(VH): The entry (i,j) will be 0{No}
which will be denoted by (0,0,0.25) and entry (j,i) can
be denoted by (0.75,1.0,1.0)
vi. If the entry (i,j) is A(H): The entry (i,j) will be 0{No}
which will be denoted by (0,0,0.25) and entry (j,i) can
be denoted by (0.5,0.75,1.0)
vii. If the entry (i,j) is A(L): The entry (i,j) will be 0{No}
which will be denoted by (0,0,0.25) and entry (j,i) can
be denoted by (0.25,0.5,0.75)
viii. If the entry (i,j) is A(VL): The entry (i,j) will be
0{No} which will be denoted by (0,0,0.25) and entry
(j,i) can be denoted by (0,0.25,0.5)
ix. If the entry (i,j) is X(VH): The entry (i,j) can be
denoted by (0.75,1.0,1.0) and entry (j,i) can be
denoted by (0.75,1.0,1.0)
x. If the entry (i,j) is X(H): The entry (i,j) can be denoted
by (0.5,0.75,1.0) and entry (j,i) can be denoted by
(0.5,0.75,1.0)
xi. If the entry (i,j) is X(L): The entry (i,j) can be denoted
by (0.25,0.5,0.75) and entry (j,i) can be denoted by
(0.25,0.5,0.75)
xii. If the entry (i,j) is X(VL): The entry (i,j) can be
denoted by (0,0.25,0.5) and entry (j,i) can be denoted
by (0,0.25,0.5)
xiii. If the entry (I,j) is X(VH,H): The entry (i,j) can be
denoted by (0.75,1,1) and entry (j,i) can be denoted
by (0.5,0.75,1). Similar other possible scenarios are–
X(VH,L), X(VH,VL), X(H,VH), X(H,L), X(H,VL),
X(L,VH), X(L,H),X(L,VL), X(VL,VH), X(VL,H),
X(VL,L)
xiv. If the entry (i,j) is 0(No): The entry (i,j) and entry
(j,i) is denoted by (0,0,0.25)
The final fuzzy reachability is denoted as ~
Z
~
Z¼
~z11 ~z12 ... ~z1n
~z21 ~z22 ... ~z2n
.
.
..
.
...
..
.
.
~zn1~zn2... ~znn
2
6
6
6
43
7
7
7
5
where ~
Zij ¼ðlij;mij ;uijÞ
Step 5: Calculation of Driving Power and Dependence
for MICMAC Analysis
Fuzzy reachability matrix is generated from aggregated
fuzzy SSIM matrix in step 4 above. The driving power and
dependence are calculated by summing rows and columns
of fuzzy reachability matrix using Eq. (1). To carry out
MICMAC analysis based on fuzzy reachability matrix,
Eq. (5) is applied for defuzzification.
Step 6: Reachability Matrix Level Partition
Here, reachability matrix is portioned using relation and
level partition. Before the start of level partitioning, the
transitivity of reachability matrix is also checked to iden-
tify the presence of any transitive links.
Step 7: Creating Fuzzy-TISM Digraphs and Defuzzified
TISM Digraphs
Following symbols are proposed in Fig. 3to establish the
fuzzy relationship among criteria.
The Fuzzy-TISM digraphs and defuzzified TISM are
represented by simple directed arrows in order to visualize
the level of influence. TISM digraph is constructed after
defuzzification of fuzzy reachability matrix obtained at step
4 above. For an instance, fuzzy reachability matrix can be
defuzzified by considering H and VH fuzzy linguistics
terms as 1 and rest VL, L, No as 0.
Illustrative Example
To demonstrate the application of Fuzzy-TISM approach,
an illustrative example related to vendor selection problem
is considered here. Finally, in the ‘‘Discussion and Inter-
pretation’’ section, detailed discussion on the proposed
Fuzzy-TISM digraphs and defuzzified TISM digraph are
provided.
Global Journal of Flexible Systems Management
123

Developing Inter-relationship of Vendor Selection
Problem Using Fuzzy-TISM
Step 1: Start of Decision Making Process
In this study, the responses of 5 experts are collected for
evaluation of interrelationship among criteria. These
experts are well experienced with the vendor selection
process.
Step 2: Selection of Criteria
The following criteria have been selected that are critical
for selecting vendor or supplier:
1. Quality (C1): The quality of the material/goods
supplied by the vendor.
2. Delivery (C2): Time taken to deliver the materials to
the manufacturer.
3. Production facilities (C3): Production facilities
owned by the vendor.
4. Price(C4): the price charged by the vendor for the
supplied goods or materials.
5. Financial Position (C5): The financial status of the
vendor firm.
6. Technological flexibility (C6): the type of technology
utilized by the vendors to produce their materials or
goods and also having flexibility in upgrading it as
per need of advancement.
7. Top level commitment (C7): The commitment levels
at the top positions of the vendor firms.
8. Transport and communication (C8): Transportation
and communication facilities.
9. Service (C9): After sales services provided by the
vendor.
10. Attitude and willingness (C10): Willingness to keep
long-term relation with the firm.
Step 3: Gathering Responses and Creating SSIM Matrix
In this study, responses of five experts are taken and are
aware about the underlying problem and have good amount
of work experience dealing such problem. After collecting
responses from all five experts on the degree of relationship
between the criteria, the initial SSIM matrices obtained
from five different experts are provided from the Tables 3,
4,5,6,7for each expert and are provided in the Appendix.
Step 4: Calculation of Aggregated SSIM and Fuzzy
Reachability Matrix
Table 2shows the aggregated values of SSIM matrix of
five experts shown from Tables 3,4,5,6,7(refer to
Appendix). The responses are aggregated using mode
(linguistic terms with highest frequency of occurrence).
Further, aggregated SSIM matrix is transformed into
final fuzzy reachability matrix. Table 8indicates fuzzy
reachability matrix ~
Zof five experts and is given in
Appendix.
Step 5: Calculation of Driving Power and Dependence
for MICMAC Analysis
The fuzzy values of driving power and dependence for
criteria are shown in the Table 9, which is given in
Appendix. Table 9, also presents the crisp values of driving
power and dependence. Further, the calculation of crisp
values of driving power and dependence for performing
MICMAC analysis is done using Eq. (5). Below the crisp
value of criteria Quality (C1) is shown using Eq. (5).
Similarly, the crisp value is calculated for all other criteria.
Based on the crisp values, the driving power and depen-
dence matrix (MICMAC analysis) is represented in
Table 10.
Table 2 Aggregated SSIM matrix
C10 C9 C8 C7 C6 C5 C4 C3 C2
C1 O(No) O(No) O(No) A(H) A(VH) A(H) X(H) A(VH) O(No)
C2 A(VH) O(No) A(H) A(VH) A(H) A(H) X(H) O(No)
C3 O(No) O(No) O(No) A(H) A(VH) A(H) V(L)
C4 O(No) X(H) A(H) O(No) A(VH) A(H)
C5 O(No) O(No) V(H) A(VH) V(H)
C6 O(No) V(L) O(No) O(No)
C7 V(VH) V(H) O(No)
C8 A(L) V(L)
C9 A(H)
Global Journal of Flexible Systems Management
123

Driving power fuzzy value of C1 =(1.5,1.75,4)
L¼minðlkÞ¼1:5, R ¼maxðukÞ¼7:75, D¼6:25,
l1¼1:5;m1¼1:75 and u1¼4. min lk
ðÞand maxðukÞare
calculated considering driving power fuzzy values of all
criterion
x
l1
=0.0000, x
m1
=0.0400, x
u1
=0.0400xls
1¼0:0385
and xrs
1¼0:2941
xcrisp
1¼0:0983
~
Bcrisp
1¼2:1147
Hence, the driving power crisp value of C1 =2.1147.
Dependence fuzzy value of C1 =(4,5.25,7)
L¼minðlkÞ¼1, R ¼maxðukÞ¼8:25, D¼7:25,
l1¼4;m1¼5:25 and u1¼7. min lk
ðÞand maxðukÞare
calculated considering dependence fuzzy values of all
criterion
x
l1
=0.4138, x
m1
=0.5862, x
u1
=0.8276xls
1¼0:5000
and xrs
1¼0:6667
xcrisp
1¼0:5952
~
Bcrisp
1¼5:3155
Hence, the dependence crisp value of C1 =5.3155.
Step 6: Reachability Matrix Partition Using Relation
and Level Partition
Here, the defuzzified reachability matrix is generated based
on the aggregated fuzzy reachability matrix shown in
Table 8of the Appendix. To generate the defuzzified
reachability matrix, the paper considers the fuzzy linguistic
term consisting of Very High Influence (VH) and High
Influence (H) as 1 and rests others 0. (It is to be noted that
at this step decision maker can choose only VH as 1 and
others 0). Table 11 of Appendix provides the defuzizfied
reachability matrix. Table 11 also contains the transitive
links after checking the transitivity among all criteria.
Based on Table 11, the defuzzified MICMAC analysis is
done and shown in Table 12 of the Appendix. Also, the
level partitioning is conducted based on Table 11 and level
partitions are shown from Tables 13,14,15,16 of the
Appendix. Table 13 shows the first iteration where in
Quality (C1), Delivery (C2), Price (C4) and Service (C9) are
found to be at level 1. For representation purpose, the listing
of criteria from C1 to C10 is shown as 1 to 10 from
Tables 13,14,15,16 (refer Appendix), which are the vari-
ous level partitioning matrices. Table 14 shows the second
iteration where in Production facility (C3) and Transport and
Communication (C8) are found at level 2. Similarly, in
Table 15, Technological flexibility (C6), and Attitude and
Willingness (C10) are found to be at level 4. Finally, in
Table 16, financial position (C5) and Top level commitment
(C7) are found to be at level 4 and level 5, respectively.
Step 7: Creating Fuzzy-TISM Digraphs and Defuzzified
TISM Digraph
Here, the fuzzy interrelationships among all criteria are
shown using proposed Fuzzy-TISM approach. Hence, all
Fuzzy-TISM digraphs are shown in Fig. 4. In Fuzzy-TISM
digraphs, the influence of one criterion over all other cri-
teria at all levels (i.e. VH, H, L, VL, No) are shown. Fig-
ure 4a–j together constitutes Fuzzy-TISM and it represents
individual fuzzy relationship of criteria 1–10 with other
criteria. Based on the defuzzified reachability matrix (refer
Table 11), the defuzzified TISM digraph is constructed and
shown in the Fig. 5. Figure 5represents digraph of TISM
considering only fuzzy linguistic terms VH and H as 1 and
rests other 0. Symbols as shown in the Fig. 3are used to
establish the fuzzy relationships between criteria.
Discussion and Interpretation
As it can be seen that Fig. 4a–j individually show the level
of influence of one criterion over others. At the end, Fig. 5
only represents interrelationship based on TISM among all
criteria having very high influence (VH) and high influence
(H). However, in order to establish interrelationship among
criteria in the presence of all levels of influence then
Figs. 4and 5can be considered simultaneously, which is
obtained using Fuzzy-TISM approach. For an instance,
considering level 5, If the decision maker would like to
know the strength of the relationship of criteria present at
level 5 i.e. C7 (Top level commitment) with respect to
other criteria then the decision maker would refer Fig. 5g.
Criteria C7 (Top level commitment) has very high influ-
ence over C2 (Delivery), C5 (Financial Position) and C10
(Attitude and willingness) and high influence over C1
(Quality), C3 (Production facilities) and C9 (Service).
Similarly, considering level 3, if the decision maker likes to
know the strength of the relationship of criteria present at
level 3, i.e. C6 and C10 with other criteria then he/she can
refer Fig. 4f, j, respectively. Criteria C10 (Attitude and
willingness) has very high influence over criterion C2
(Delivery), high influence over C9 (Service) and low
influence over C8 (Transport and telecommunication).
Another criterion C6 (Technological flexibility) at level 3
has very high influence C1 (Quality), C3 (Production
facilities) and C4 (Price), high influence over C2 (Delivery)
and low influence over C9 (Service). In a similar way,
other interpretations can also be done easily using digraphs
provided by the Fuzzy-TISM approach. From Fig. 5, it can
be seen that the criteria that influence vendor selection
Global Journal of Flexible Systems Management
123

(a)
(b)
(c) Criterion 3 has very high influence on criterion 1 and low influence criterion 4
(d)
C3
C1
C4
(e) Criterion 5 has high influence on criterion 1, 2, 3, 4, 6 and 8.
C5
C8
C6C4
C3
C2
C1
(f)
Criterion 6 has very high influence on criterion 1, 3 and 4, high influence on criterion 2 and low
influence on criterion 9
C1
C2 C3 C4
C6
C9
Criterion 1 has high influence on criterion 4
C1 C4
Criterion 2 has high influence on criterion 4
C2 C4
Criterion 4 has high influence on criterion 1, 2 and 3.
C4
C1
C2
C9
Fig. 4 Fuzzy-TISM digraphs of
all criteria influencing others
containing all fuzzy linguistic
terms (i.e. VH,H, L,VL,No)
Global Journal of Flexible Systems Management
123

problem at very high influence (VH) and high influence
(H) are C7 (top level commitment), C5 (financial position),
C6 (technological flexibility), and C10 (attitude and
willingness).
Conclusions
The paper proposes a Fuzzy-TISM, a fuzzy extension of
TISM, approach for group decision making problems. Due
to the presence of fuzziness through fuzzy approach the
decision makers have the flexibility in terms of assigning the
level of influence one criteria can have on other directly.
Earlier in the pure ISM or TISM approaches this was not
possible due to the presence of binary numbers for influence
(1) or not influences (0). The proposed Fuzzy-TISM
approach takes care of this issue and provide wider flexi-
bility while assessing interrelationship among various cri-
teria. In addition to this, the proposed approach is also user
friendly due to the simplified way of introducing the fuzzy
(g )Criterion 7 has very high influence on criterion 2, 5 and 10 and high influence on criterion 1, 3 and 9
C7
C1 C10
C9
C5
C3
C2
(h) Criterion 8 has high influence on criterion 2 and 4 and low influence on criterion 9
(i) Criterion 9 has high influence on criterion 4.
(j) Criterion 10 has very high influence on criterion 2, high influence on criterion 9 and low influence on
criterion 8.
C4
C9
C8
C9
C4
C2
C2
C8
C9
C10
Fig. 4 continued
Global Journal of Flexible Systems Management
123

triangular values in TISM for better decision making under
fuzzy environment. In the proposed Fuzzy-TISM approach,
mode has been used as the method for aggregation of group
preferences under fuzzy environment, which makes it easily
implementable for any group decision making process of
real business application. Furthermore, this aggregation
method also helps in preserving the fuzzy linguistic values,
which can be subsequently used in establishing fuzzy rela-
tionship of each criterion.
The proposed method is a novel attempt in this direc-
tion. The incorporation of fuzzy in TISM allows the
respondent to judge degree of relationship between criteria.
Here, in this paper, the respondent can select influence
levels namely {Very high, High, Low, Very low, No} of
one criterion over others. The final Fuzzy-TISM model
consists of individual fuzzy relationships between each
criterion with other. Finally, fuzzy TISM model can fully
interpret the structural model, which can help managers in
considering the relationships of significant strength and
discounting of weak strength.
Acknowledgments The communicating author expresses his sin-
cere thanks to Prof. Sushil, Department of Management Studies, IIT
Delhi and anonymous referees for their constructive suggestions,
which has led to significant improvement in the quality of this
manuscript.
Appendix
See Appendix Tables 3,4,5,6,7,8,9,10,11,12,13,14,
15,16
C9
Service
C2
Delivery
C3
Producon facilies
C4 Price
C1
Quality
C8
Transport and communicaon
C10
Atude and Willingness
C6
Technological
flexibility
C5
Financial Posion
C7
Top level commitment
Beer the quality, higher
the price and vice versa
Beer delivery ensures mely
services to firm and achieving
desired service leads to beer
delivery
Facilitates technology
updaon, transfer and
adapon
Ensures advanced
producon facilies
Provide fund for
advance logisc
support
Commitment from
top level ensures
fund availability
Ensures economical
producon
Commitment
drives
willingness
Drives clear and
transparent
communicaon
Logiscs opmizes
the producon
cost
Beer services demands for high price
and price impacts the service quality
Time delivery influence
price and vice versa
Fig. 5 TISM digraph of vendor
selection considering fuzzy
linguistic term very high
influence (VH) and high
influence (H) as 1 and rest as 0
Global Journal of Flexible Systems Management
123

Table 3 SSIM matrix of expert # 1
C10 C9 C8 C7 C6 C5 C4 C3 C2
C1 O(No) O(No) O(No) O(No) A(VH) O(No) V(VH) O(No) O(No)
C2 O(No) O(No) A(H) A(VH) O(No) O(No) X(H,VH) O(No)
C3 O(No) O(No) O(No) A(H) A(VH) A(VH) O(No)
C4 O(No) A(H) A(L) O(No) A(VH) O(No)
C5 O(No) O(No) V(H) A(H) V(H)
C6 O(No) V(H) O(No) O(No)
C7 V(VH) V(H) O(No)
C8 A(H) V(L)
C9 A(H)
Table 4 SSIM matrix of expert # 2
C10 C9 C8 C7 C6 C5 C4 C3 C2
C1 O(No) O(No) O(No) A(H) A(VH) A(H) A(H) A(L) O(No)
C2 A(VH) O(No) A(VH) O(No) O(No) A(H) X(H) O(No)
C3 O(No) O(No) O(No) A(L) A(H) A(L) V(H)
C4 O(No) X(H) O(No) O(No) A(VH) A(H)
C5 O(No) O(No) O(No) A(L) V(H)
C6 O(No) V(L) O(No) O(No)
C7 V(VH) V(H) O(No)
C8 A(L) O(No)
C9 A(H)
Table 5 SSIM matrix of expert # 3
C10 C9 C8 C7 C6 C5 C4 C3 C2
C1 O(No) O(No) O(No) A(H) A(VH) A(L) X(H) O(No) O(No)
C2 A(H) O(No) A(VH) A(VH) A(H) A(H) V(L) O(No)
C3 O(No) O(No) O(No) A(H) O(No) A(H) O(No)
C4 O(No) O(No) A(H) O(No) A(H) A(H)
C5 O(No) O(No) V(L) A(VH) V(H)
C6 O(No) V(L) O(No) O(No)
C7 V(H) V(VH) O(No)
C8 A(L) V(VH)
C9 A(H)
Table 6 SSIM matrix of expert # 4
C10 C9 C8 C7 C6 C5 C4 C3 C2
C1 O(No) O(No) O(No) A(H) A(H) A(H) V(H) A(VH) O(No)
C2 O(No) O(No) A(H) A(L) A(H) A(H) V(L) O(No)
C3 O(No) O(No) O(No) A(H) A(L) A(H) V(L)
C4 O(No) X(L) A(H) O(No) O(No) O(No)
C5 O(No) O(No) V(H) O(No) V(H)
C6 O(No) O(No) O(No) O(No)
C7 V(VH) V(L) O(No)
C8 O(No) O(No)
C9 A(H)
Global Journal of Flexible Systems Management
123

Table 7 SSIM matrix of expert # 5
C10 C9 C8 C7 C6 C5 C4 C3 C2
C1 O(No) O(No) O(No) O(No) A(VH) O(No) X(H) A(VH) O(No)
C2 A(VH) O(No) A(H) A(VH) A(H) O(No) X(H) O(No)
C3 O(No) O(No) O(No) O(No) A(VH) A(H) V(L)
C4 O(No) X(H) O(No) O(No) A(VH) A(H)
C5 O(No) O(No) V(VH) A(VH) O(No)
C6 O(No) O(No) O(No) O(No)
C7 V(L) V(H) O(No)
C8 O(No) V(L)
C9 O(No)
Table 8 Fuzzy reachability matrix based on Aggregated fuzzy SSIM matrix
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10
C1 No No H No No No No No No
C2 No No H No No No No No No
C3 VH No L No No No No No No
C4 H H No No No No No H No
C5 H H H H H No H No No
C6 VH H VH VH No No No L No
C7 H VH H No VH No No H VH
C8 No H No H No No No L No
C9 No No No H No No No No No
C10 No VH No No No No No L H
Table 9 Final fuzzy reachability matrix ~
Zof 5 experts with fuzzy and crisp values of driving power and dependence of criteria
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 ** #
C1 (1,1,1) (0,0,0.25) (0,0,0.25) (0.5,0.75,1) (0,0,0.25) (0,0,0.25) (0,0,0.25) (0,0,0.25) (0,0,0.25) (0,0,0.25) (1.5,1.75,4) 2.1147
C2 (0,0,0.25) (1,1,1) (0,0,0.25) (0.5,0.75,1) (0,0,0.25) (0,0,0.25) (0,0,0.25) (0,0,0.25) (0,0,0.25) (0,0,0.25) (1.5,1.75,4) 2.1147
C3 (0.75,1,1) (0,0,0.25) (1,1,1) (0.25,0.5,0.75) (0,0,0.25) (0,0,0.25) (0,0,0.25) (0,0,0.25) (0,0,0.25) (0,0,0.25) (2,2.5,4.5) 2.8288
C4 (0.5,0.75,1) (0.5,0.75,1) (0,0,0.25) (1,1,1) (0,0,0.25) (0,0,0.25) (0,0,0.25) (0,0,0.25) (0.5,0.75,1) (0,0,0.25) (2.5,3.25,5.5) 3.5940
C5 (0.5,0.75,1) (0.5,0.75,1) (0.5,0.75,1) (0.5,0.75,1) (1,1,1) (0.5,0.75,1) (0,0,0.25) (0.5,0.75,1) (0,0,0.25) (0,0,0.25) (4,5.5,7.75) 5.5519
C6 (0.75,1,1) (0.5,0.75,1) (0.75,1,1) (0.75,1,1) (0,0,0.25) (1,1,1) (0,0,0.25) (0,0,0.25) (0.25,0.5,0.75) (0,0,0.25) (4,5.25,6.75) 5.2630
C7 (0.5,0.75,1) (0.75,1,1) (0.5,0.75,1) (0,0,0.25) (0.75,1,1) (0,0,0.25) (1,1,1) (0,0,0.25) (0.5,0.75,1) (0.75,1,1) (4.75,6.25,7.75) 6.1480
C8 (0,0,0.25) (0.5,0.75,1) (0,0,0.25) (0.5,0.75,1) (0,0,0.25) (0,0,0.25) (0,0,0.25) (1,1,1) (0.25,0.5,0.75) (0,0,0.25) (2.25,3,5.25) 3.3492
C9 (0,0,0.25) (0,0,0.25) (0,0,0.25) (0.5,0.75,1) (0,0,0.25) (0,0,0.25) (0,0,0.25) (0,0,0.25) (1,1,1) (0,0,0.25) (1.5,1.75,4) 2.1147
C10 (0,0,0.25) (0.75,1,1) (0,0,0.25) (0,0,0.25) (0,0,0.25) (0,0,0.25) (0,0,0.25) (0.25,0.5,0.75) (0.5,0.75,1) (1,1,1) (2.5,3.25,5.25) 3.5449
* (4,5.25,7) (4.5,6,7.75) (2.75,3.5,5.5) (4.5,6.25,8.25) (1.75,2,4) (1.5,1.75,4) (1,1,3.25) (1.75,2.25,4.5) (3,4.25,6.5) (1.75,2,4)
# 5.3155 5.9667 3.7883 6.1780 2.3437 2.1314 1.3288 2.6277 4.4617 2.3437
* Dependence; ** Driving power; # Crisp value
Global Journal of Flexible Systems Management
123

Table 11 Defuzzified reachability matrix with fuzzy linguistic terms Very High Influence (VH) and High Influence (H) as 1 and rest as 0.
Shaded region indicates transitive links
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10
C11101000010
C21101000010
C31111000010
C41101000010
C51111110110
C61111010010
C71111111111
C81101000110
C91101000010
C101101000011
Table 10 Driving power and Dependence Matrix (MICMAC) based on fuzzy reachability matrix of Table 9
Table 12 Driving power- Dependence Matrix (MICMAC) based on defuzzified reachability matrix based on Table 11
Global Journal of Flexible Systems Management
123

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8 8 5,7,8 8 2
10 8,10 7,10 10
Table 15 Third Iteration of final fuzzy reachability matrix partition
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7 5,6,7,10 6,7 6,7
10 10 7,10 10 3
Table 16 Fourth Iteration of final fuzzy reachability matrix partition
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7 5,7 7 7 5
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Key Questions
1. What is the flexibility in the Fuzzy-TISM approach?
2. How the Fuzzy-TISM approach has improved the group
decision making process?
3. How the Fuzzy-TISM approach can be compared with the
Fuzzy-ISM approach?
Gaurav Khatwani is Doctoral student at Indian
Institute of Management Rohtak, India. He holds a
B. Tech from Rajasthan Technical University. His
research interest lies in decision support systems,
digital marketing, group decision making and
optimization techniques. He has 30 months of
corporate experience in Information Technology
industry.
Surya Prakash Singh is an Associate Professor in
the DMS, IIT Delhi, India. He holds a PhD from
IIT Kanpur. He is also a Post-Doctoral Fellow
from NUS Singapore-MIT USA Alliance. His
research interest lies in facility layout problems,
heuristics, and meta-heuristics. His work has been
published in leading international journals such as
IJPR, LNCS, IJAMT, EJM, RBR, IJRTE, and
APMR. He regularly reviews articles for many leading journals. His
biography appeared in Marquis, USA Who’s Who in Science and
Engineering, December 2007, and Who’s Who in the World,
November 2010. Recently, he has been awarded Young Outstanding
Faculty Fellowship from IIT Delhi.
Global Journal of Flexible Systems Management
123

Ashish Trivedi is research scholar at Indian
Institute of Management Rohtak, India. He holds
an MBA degree in international business from
University of Lucknow. His areas of research
include supply chain management and optimiza-
tion techniques. Prior to this he served as a faculty
at Institute of Cooperative and Corporate Man-
agement Research and Training (ICCMRT),
Lucknow.
Ankur Chauhan is Doctoral student at Indian
Institute of Management Rohtak, India. He holds a
M. Tech from National Institute of Technology,
Jalandhar. His research interest lies in reverse
logistics and waste recycling.
.
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