Development of Fuzzy TOPSIS-Based Software Tool (FTST)
Fadlelmoula M. Baloul
Ahmed A. Al-Amayrah
D.R Prince Williams
E-mail: [email protected]
E-mail: [email protected]
E-mail: [email protected],
[email protected]
Department of Information Technology,
College of Applied Sciences - Sohar,
Sohar, Sultanate of Oman.
Abstract: The aim of this paper is to describe FTST software tool
for selecting the optimum alternative among many alternatives
according to multi specified criteria. FTST tool is designed based
on the fuzzy TOPSIS method as one of the most popular methods
in multi-criteria decision analysis (MCDA). FTST, as an
automated system, could be used as decision support system
software for any case where the goal is to select the right and the
optimum alternative from many alternatives with intended
criteria. At the end of the paper, the FTST tool will be
demonstrated on a real fuzzy TOPSIS-based system.
Keywords: Decision Support software, TOPSIS, Academic
Advising, Multi-criteria Decision Making.
I. INTRODUCTION
A Decision Support System (DSS) is a computer-based
information system that could be used to eases and supports
the decision makers in any organization to take unbiased
decisions [11]. A well-designed DSS is an interactive software
system designed mainly to help the decisions makers to
compile multi-criteria about anything to take the right
decision. Any DSS has four components which are:
 Input - criteria, variables, weight and rates
 User Knowledge - the user has be a decision maker or an
expert person who has the knowledge of the decision
makers.
 Output - generated tables from which the decision will be
taken.
 Decisions – usually produced by DSS based on the set of
input that used as criteria.
Decision-making problems are an important process for any
organization and it helps the management to identify the most
suitable option from the available feasible alternatives. All the
feasible or possible alternatives are derived from the multiple
criteria. This type of problems is known as multiple criteria
decision making (MCDM). The processes in MCDM problems
are very algorithmic processes that based on well-defined
complicated formulas that could be used in a similar way in
many application areas. Many authors have studied and applied
in many situations like location selection, service quality of
domestic airlines selection, bus company selection, etc.[1-3,59]. In most of the literature, technique for order performance
by similarity to ideal solution (TOPSIS) method is used to
solve the MCMD problem. Due to the uncertainty in MCDM,
fuzzy set [10] plays a vital role in solving MCDA problem.
Chen [4], have combined both the TOPSIS and fuzzy sets to
solve the MCDM problem.
In this paper, we show and describe our software tool
which developed based on the extended concept of the fuzzy
TOPSIS method which was developed by Chen [4], for solving
multi-criteria alternative selection problems in many
application areas and environments. According to the fuzzy
nature of the raw data that used as a core for taking any
decision and the process of the decision itself, some linguistic
variables have been used to rate the weights of all criteria and
rates the different intended alternatives with respect to each
criterion. The fuzzy positive-ideal solution (FPIS) and the
fuzzy negative-ideal solution (FNIS) have been defined in [4].
The fuzzy TOPSIS method involves many complicated
calculations in each of its steps. For example, in step 7 of the
method, a specific vertex method is used to find the distance
between any two triangular fuzzy ratings points and then the
distance of each alternative from FPIS and FNIS, respectively.
Following these calculation formulas manually could be a very
difficult process and some automation is needed.
Our FTST tool which we describe in this paper affords a
great help in applying the fuzzy TOPSIS method in specific
and MCDM in general. FTST tool performs all the complicated
calculations automatically behind the scene and shows the
result of the application within a few clicks. FTST tool could
be implemented and used in many MCDM application areas.
II. PRELIMINARIES
In the following paragraphs, some basic definitions of the
fuzzy sets will be reviewed. These basic definitions will be
used in the framework of this paper.
~
Definition 1. As defined in [10], a fuzzy set A in a universe of
discourse X is characterized by the membership function
𝜇𝐴̃ (𝑥)which associates with each element x in X a real number
in the interval [0,1]. The function value 𝜇𝐴̃ (𝑥) is termed the
~
grade of the membership of x in A [10].
Definition 2. As defined in [4], a triangular fuzzy number n~
can be defined by a triplet n1 , n2 , n3  shown in Fig.1
n~ x
1
0
n1
X
n3
n2
Fig. 1. A triangular fuzzy number
if
if
wij   wi1 , wi 2 , wi 3 
Based on the above fuzzy environment we can define a
following fuzzy decision matrix and weight matrix as
 x11

 x21

D
.
.

 xn1
~ ~ ~
W  w
w
1
x13 ...x1n 

x23 ...x21n 
.
.




xn 3 ...xmn 
x12
x22
.
2
xn 2
~
w
3

(1)
xij and wi for all
where
and j  1, 2,..., n are linguistic variables.
i  1, 2,..., m
The linguistic variables can be defined as triangular fuzzy
n~ .
The membership function is  n~ x  defined as [11]:
 x  n1
n  n ,
 2 1
 x  n2
,
n  x   
 n3  n 2


0,
xij   aij , bij , cij 
numbers,
n1  x  n 2 ,
n 2  x  n3 ,
~
xij  aij , bij , cij  and wi   wi1 , wi 2 , wi 3  .
The weight of each criterion is defined as
𝑟𝑤
̃ 𝑖𝑗 = (
𝑎𝑖𝑗
,
[𝑅𝑖𝑗 ]
𝑚𝑎𝑥
𝑏𝑖𝑗
[𝑅𝑖𝑗 ]
,
𝑚𝑎𝑥
𝑐𝑖𝑗
[𝑅𝑖𝑗 ]
)
(2)
𝑚𝑎𝑥
 i  1, 2,3,...m , j  1, 2,3,...n .
Where
otherwise.
~
Definition 3. As defined in [8,2] D is called a fuzzy matrix,
~
if at least an entry in D is a fuzzy number.
Definition 4. As defined in [4], a linguistic variable is a
variable whose values are linguistic terms.
 Rij   max a11 , b11 , c11 , a21 , b21 , c21 ,..., amj , bmj , cmj 
max
 j  1, 2,3,...n .
As defined in [4], for i=1 to m, j=1 to m, the normalized
decision matrix denoted by 𝑅̂ can be calculated as
𝑎𝑖𝑗
𝑏𝑖𝑗
𝑐𝑖𝑗
𝑅̂ = [ 𝑟̃𝑖𝑗 ]𝑚×𝑛 , where 𝑟̃𝑖𝑗 = (
,
,
)
(3)
max 𝑎𝑖𝑗 max 𝑎𝑖𝑗 max 𝑎𝑖𝑗
The concept of linguistic variable is very useful in dealing
with situations which are too complex to be reasonably
described in conventional quantitative expressions [15]. For
example, "weight" is a linguistic variable; its values could be:
very low, low, medium, high, very high, etc. These linguistic
values can be represented by fuzzy numbers.
Definition 5. As defined in [4], let
~  m , m , m  and
m
1
2
3
Considering the different risk level of each criterion, we can
construct the weighted normalized fuzzy decision matrix as
~
rij . w j
V  v~ij mn , where v~ij  ~

Then the fuzzy positive ideal solution (FPIS, A ) and fuzzy

negative ideal solution (FNIS, A ) will be defined as



n~  n1 , n2 , n3  be two triangular fuzzy numbers, then the
A  v~1 , v~2 ,...v~n and A  v~1 , v~2 ,...v~n
vertex method is defined to calculate the distance between
them as
~, n~   1 n  m 2  n  m 2  n  m 2
d m
1
1
2
2
3
31
3
The decision maker has to determine the importance level
each criterion and the rates the different intended alternatives
as
Where


(4)
v j  1,1,1 and v j  0,0,0



The distance of each alternative from A and A can be
calculated using the following formula:
n


d i   d v~ij , v~i for all i  1,2,..., m
j
and
(5)
n


d i   d v~ij , v~i  for all i  1,2,..., m .
(6)
j
Where d .,. is the distance between two triangular fuzzy
numbers.
As the last step of the method, a closeness coefficient is
defined as a key to identify the ranking order of each
alternative has been calculated as
d i
CCi  *
d i  d i
Just for comparison, we have used an example that have
been used as numerical example in [4], to show how our tool
could be used to choose the best alternative among a set of
possible alternatives automatically as shown in next section.
FTST DEMONSTARTION
A company desires to hire a person for a job. After the
initial evaluation, three candidates are remaining for further
evaluation (A1, A2 and A3).
The decision maker runs a final interview to select the most
appropriate candidate for the job. The following five criteria
are used for evaluation:
This decision problem structure is described in Fig. 2.
LINGUISTIC VARIABLES CHOSEN FOR THE
IMPORTANCE WEIGHT OF EACH CRITERION
(7)
All these steps have been designed and implemented as
integrated software tool using Visual Basic Programming
Language.
(1) Emotional steadiness (C1),
(2) Oral communication skill (C2),
(3) Personality (C3),
(4) Past experience (C4),
(5) Self-confidence (C5).
1- Based on the screen on Fig. 3, the user has to specify the
number of criteria, the number of alternatives and number
of rates/weight scales.
2- The linguistic variables of the importance weight of each
criterion in table I will be entered in FTST tool as shown in
Fig. 4.
TABLE I.
As specified in [4], in total, the algorithm for the fuzzy
TOPSIS-based system will follow the following steps:
1- Set the evaluation for the given criteria.
2- Choose suitable linguistic variables for the importance
weight of the given criteria and the given linguistic
ratings for alternatives.
3- Estimate the weight of criteria to get the estimated fuzzy
weight of each criterion, and puddle the decision makers'
suggestions to get the grouped fuzzy rating of all
alternative under all criterions.
4- Normalize the fuzzy decision matrix.
5- Build the weighted normalized fuzzy decision matrix.
6- Calculate the fuzzy positive ideal solution (FPIS) and the
fuzzy negative ideal solution (FNIS).
7- Find the distance of FPIS and FNIS for each alternative.
8- Find the closeness coefficient (CC) of each alternative.
9- Determine the ordering rank of each alternative according
to its closeness coefficient (CC).
III.
Our software tool (FTST) is used to solve the complicated
computational steps as given in the following algorithm:
VL (Very low)
L (Low)
ML (Medium low)
M (Medium)
MH (Medium high)
H (High)
VH (Very high)
3-
(0.0,0.0,0.1)
(0.0,0.1,0.3)
(0.1,0.3,0.5)
(0.3,0.5,0.7)
(0.5,0.7,0.9)
(0.7,0.9,1.0)
(0.9,1.0;1.0)
Based on the data in table II, the linguistic variables for
alternatives ratings will be assessed and entered into the
system as shown in Fig. 5.
TABLE II. LINGUISTIC VARIABLES CHOSEN FOR
TABLE III. ALTERNATIVE RATINGS
VP (Very poor)
P (Poor)
MP (Medium poor)
F (Fair)
MG (Medium good)
G (Good)
VG (Very good)
(0,0,1)
(0,1,3)
(1,3,5)
(3,5,7)
(5,7,9)
(7,9,10)
(9,10,10)
4- Using the linguistic rating variables, the decision maker
evaluates and rates each criterion as shown in Fig. 6.
5- The decision maker determines the fuzzy weight of each
criterion for each alternative as shown in Fig 7.
6- FTST performs all the complex computations based on the
fuzzy TOPSIS method.
7- FTST generates all the intermediate tables for the user as
shown in Fig. 8.
8- FTST determines and displays the best choice from the
given alternatives as its final result as shown in Fig. 9.
IV.
CONCLUSION AND FUTURE WORK
The paper has discussed developing and demonstrating a
tool for fuzzy TOPSIS-based model to evaluates three
alternatives and determines the best alternative. Five criteria
and three alternatives were used to assess the developed tool.
The developed software tool could be used by the decision
maker to automate choosing the best alternative in any
application area such as medicine, finance, education, human
resource, software development, etc.
In the real life situation there may be two or more decision
makers are possible and each of them may have different views
and ideas for alternatives evaluation. This idea motivates us to
upgrade our tool for fuzzy TOPSIS method for multi decision
makers in the future.
Fig. 2. The
structure of the problem
Fig. 3. The
Basic Input Screen
Fig. 4. The linguistic variables for the importance weight for each criterion
Fig. 5. The linguistic variables and values chosen for alternatives rating
Fig. 6. The selection of the criterion’s importance weight
Fig. 7. The decision maker rating for each alternative/criterion
Fig. 8. The intermediate tables generated by FTST
Fig. 9. The FTST final result screen
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