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Abstract—Criterias for the selection of the multimedia
applications for learning are based on Bloom’s digital taxonomy.
Since there are more then one criteria for the selection of
multimedia application for learning, multiple criteria decision
making (MCDM) methods are needed to be used. In this paper
FAHP and TOPSIS methods are proposed to be integrated for the
selection of the best multimedia application for learning and
teaching. First the FAHP method is used for determining the
weights of each criteria and priority values of multimedia
applications. Triangular fuzzy numbers are used in FAHP method
for determining the benefits of one criteria to another. Then the
TOPSIS method is used to determine the final ranking of the
multimedia applications. The best multimedia application for
teaching and learning would be the one that is farthest from the
negative ideal solution and nearest to the positive ideal solution.
The integration of FAHP and TOPSIS methods enables teacher to
efficiently select a more suitable multimedia applications for
learning.
Keywords—Bloom's digital taxonomy, FAHP, fuzzy logic, learning,
MCDM, multimedia applications, TOPSIS.
I. INTRODUCTION
ROCESS of learning and teaching has considerable
changed in last few decades. Along with the large
expansion of ICT technologies, use of new modern tools and
applications in education is required. Besides the classical
learning, eLearing is becoming very popular and widely used
way of learning in todays digital enviroment. Multimedia
applications are taking there place in the classical education,
as well in the eLearning. Using multimedia applications
process of learning and teaching can be optimized and
improved [1], [2]. Since there is an extremely large volume of
various multimedia applications for learning, the need for the
T. Volarić is with the Faculty of Science and Education, University of
Mostar, Matice Hrvatske b.b., 88000 Mostar, BOSNIA AND HERZEGOVINA
(phone: +387 (0) 36 335 446; e-mail: volaric.tomislav@gmail.com).
E. Brajković is with the Faculty of Science and Education, University of
Mostar, Matice Hrvatske b.b., 88000 Mostar, BOSNIA AND HERZEGOVINA
(e-mail: emilbrajko@gmail.com).
T. Sjekavica is with the Department of Electrical Engineering and
Computing, University of Dubrovnik, Cira Carica 4, 20000 Dubrovnik,
CROATIA (e-mail: tomo.sjekavica@unidu.hr).
selection of the most appropriate multimedia application for
learning is inevitable.
For the generation of learning objectives from elementary
to higher education Bloom’s taxonomy [3] is widely used.
The taxonomy seperates forms of learning in three domains:
cognitive, affective and psychomotor domain. Hiearchy of
cognitive domain of learning is divided into six levels from
knowledge to evaluation. The higher the level, the more
complex and more useful it is. To adjust Bloom’s taxonomy
for requirements of 21st century students and teachers revised
Bloom’s taxonomy was published in 2001 [4]. The most
important change in revised Bloom’s taxonomy is that they
have expanded cognitive domain of learning to include
affective and psychomotor domains. After the revised
Bloom’s taxonomy Churches defined Bloom’s digital
taxonomy [5] in which he suplemented levels of taxonomy
with new active verbs and included new opportunities for
learning using advantages of new Web 2.0 technologies
through the proposal of some specific digital tools and
applications.
This paper discusses the selection of most suitable
multimedia application for learning based on Bloom’s digital
taxonomy. As we are talking about more than one criteria for
multimedia application selection, most suitable approach is
using multiple criteria decision making (MCDM) methods.
Different MCDM methods like AHP, FAHP, TOPSIS,
ELECTRE, PROMETHEE, etc. can be used in the decision
making process. Usually two or more MCDM methods are
combined in order to improve the decision making proccess.
In [6] the best strategy for non-formal ways of learning is
selected using FAHP method and SWOT analysis. Three
MCDM methods: AHP, Fuzzy PreRa and incomplete
linguistic preference relations methods are used in [7] for
selection of the multimedia authoring system. FAHP and
TOPSIS methods can be used together in complex decision
problems [8], [9]. Suitable multimedia applications for
learning and teaching can be selected using FAHP and
TOPSIS methods.
Analytic Hierarchy Process (AHP) method is one of the
most famous and in recent years most used method for
deciding, when the decision-making process or the choice of
some of the available alternatives and their ranking is based
Integration of FAHP and TOPSIS Methods for
the Selection of Appropriate Multimedia
Application for Learning and Teaching
Tomislav Volarić, Emil Brajković, and Tomo Sjekavica
P
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
Volume 8, 2014
ISSN: 1998-0140
224
on several attributes that have different importance and that
are expressed using different scales. AHP method allows
flexibility of the decision making process and helps decision
makers to set priorities and make good decisions, taking into
account both qualitative and quantitative aspects of the
decisions [10]. AHP is based on the motto divide and
conquer. Problems that require MCDM techniques are
complex and, as result, it is advantageous to break them down
and solve one ‘sub-problem’ at a time. This breakdown is
done in two phases of the decision process during: the
problem structuring and the alicitation of priorities through
pairwise comparisons [11].
Fuzzy Analytic Hierarchy Process (FAHP) is an extension
of AHP method that uses fuzzy logic, fuzzy sets and fuzzy
numbers. It facilitates determining the ranking of certain
criterias using fuzzy numbers instead of specific numerical
values [12]. Understanding and managing with quantitative
and qualitative data used in MCDM problems is much easier
with FAHP method. In this approach triangular fuzzy
numbers are used to determine the benefits of single criteria
to another.
Technique for Order Preference by Similarity to Ideal
Solution (TOPSIS) method is used for the final ranking of
multimedia applications on mulitple criteria, whose
importance is determined using a FAHP method. Best
alternative multimedia application is the one that is closest to
the positive ideal solution and the farthest from the negative
ideal solution by this method [13].
This paper is structured as follows: in the next section
Bloom’s taxonomy, revised Bloom’s taxonomy and digital
Bloom’s taxonomy are shown and compared. Third section
gives an overview on criterias of the multimedia applications
for learning, while the fourth section presents fuzzy logic,
fuzzy sets and fuzzy numbers. FAHP method that can be used
for determining the weight criteria in MCDM problems is
shown in fifth section. Then the TOPSIS method used for
final ranking is described in sixth section. In seventh section,
last one before conclusion empirical study for selecting
appropriate multimedia application for learning and teaching
is illustrated.
II. BLOOM’S TAXONOMIES
A. Bloom’s taxonomy
According to the American psychologist Benjamin Samuel
Bloom forms of learning can be divided into three categories
[3]:
1. cognitive (knowledge),
2. affective (attitudes), and
3. psychomotor (skills).
Whitin the cognitive category Bloom differ six diferent
hierachial levels of learning. These are, from the simplest
level to the most complex cognitive domain level: i)
knowledge, ii) comprehension, iii) application, iv) analysis,
v) synthesis and vi) evaluation. Bloom's taxonomy is a useful
tool that can help teachers in directing cognitive activities of
students in all categories of thinking, especially those
associated with higher mental operations.
B. Revised Bloom’s taxonomy
Revised Bloom’s taxonomy [4] was published after six year
work of numeros team of experts among whom were Bloom's
student Lorin Anderson and his associate David Krathwohl.
Nouns that marked levels they replaced by verbs. Then they
extended the synthesis to creation and changed the order of
the two highest levels. Levels of cognitive domain of learning
from the Lower Order Thinking Skills (LOTS) to the Higher
Order Thinking Skills (HOTS) in revised Bloom’s taxonomy
are: i) remember, ii) understand, iii) apply, iv) analyze, v)
evaluate and vi) create. Comparision of original Bloom’s
taxonomy and revised Bloom’s taxonomy is shown in Fig. 1.
Finally and most important, they expanded the cognitive
domain of learning (knowledge) to include both affective
(attitude) and pyschomotor domain of learning (skills). Their
intention was to adjust Bloom's taxonomy for the 21st century
teachers and students.
C. Bloom’s digital taxonomy
Teacher and enthusiast Andrew Churches went one step
further when he tried to accommodate taxonomy in the digital
enviroment of 21st century, including additional learn ing
opportunities which are provided with new Web 2.0
technologies [5].
Every level of Bloom’s taxonomy he supplemented with
new active verbs: i) remembering, ii) understanding, iii)
applying, iv) analyzing, v) evaluating and iv) creating. He
also proposed approach to specific digital tools for each level
of taxonomy.
Fig. 2 shows various Web 2.0 tools and applications that
can enable or enhance the process of learning and teaching
for each level of cognitive domain of learning in Bloom’s
digital taxonomy that is proposed by Samantha Penney [14].
So for example, if the learning objective for students is to
remember tools like Flickr and Delicious can be used. If goal
for students is the creation, then tools like Prezi and Gimp
Fig. 1 Comparison of Bloom's and revised Bloom's taxonomy
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
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ISSN: 1998-0140
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would be suitable.
III. CRITERIAS OF THE MULTIMEDIA APPLICATIONS FOR
LEARNING
Criterias of the multimedia applications for learning are
relying on the Bloom's digital taxonomy [5]:
1
CRemembering – Key terms are: recognising, listing,
describing, identifying, retrieving, naming, locating
and finding. Digital additions are: bullet pointing,
highlighting, bookmarking or favouriting, social
networking, social bookmarking and searching or
„googling“.
2
CUnderstanding – Key terms are: interpreting,
summarising, inferring, paraphrasing, classifiying,
comparing, explaining and exemplifying. Digital
additions are: advanced and Boolean searching, blog
journalling, categorising, tagging, commenting,
annotating and subscribing.
3
CApplying – Key terms are: carrying out, using,
executing, implementing, showing and exhibiting.
Digital additions are: running and operating, playing,
uploading, sharing, hacking and editing.
4
CAnalysing – Key terms are: comparing, organising,
deconstructing, attributing, outlining, finding,
structuring and integrating. Digital additions are:
smashing, linking, reverse-engineering and cracking.
5
CEvaluating – Key terms are: checking, critiquing,
hypothesising, experimenting, judging, testing,
detecting and monitoring. Digital additions are:
Blog/vlog commenting and reflecting, posting,
moderating, collaborating and networking, testing and
validating.
6
CCreating – Key terms are: designing, constructing,
planning, producing, inventing, devising and making.
Digital additions are: programming, filming,
animating, videocasting, podcasting, mixing,
remixing, directing, producing and publishing.
Multimedia applications for learning are valued on the
basis of the above mentioned criterias marked from 1
C to
6
C. Weight of each criteria is determined with MCDM
methods. Finally applications are ranked on the basis of all
six criterias. The applications in the study are marked as
APP1, APP2 and APP3.
IV. FUZZY SETS AND FUZZY NUMBERS
Fuzzy logic is an extension of classical Boolean logic that
is able to use the concept of partial truth. Standard Boolean
logic supports only two values: 0 (false) and 1 (true), while
fuzzy logic supports a range of values from a complete lie to
the complete truth covering the whole range of values from 0
to 1 [15]. In classic set theory for each element is strictly
determined if it belongs or does not belongs to a particular
set. Fuzzy set is an extension of the classic set. With fuzzy
sets one element may partially belong to the set. Fuzzy
number is a generalization of real numbers. It is specified
with interval of real numbers between 0 and 1. It is possible
to use different fuzzy numbers according to the situation, but
in practice trapezoidal and triangular fuzzy numbers are most
used [16].
A. Triangular fuzzy number
Triangular fuzzy number is shown in Fig. 3.
Triangular fuzzy number is defined by three real numbers,
expressed as ordered triplet
uml ,, . The parameters l,
m
and
u
respectively show the lowest possible value, the most
expected value and the maximum value that describes fuzzy
event. If we define two positive triangular fuzzy numbers
111 ,, uml and
222 ,, uml then:
212121222111 ,,,,,, uummllumluml (1)
212121222111 ,,,,,, uummllumluml (2)
l m u
0.0
1.0
M
Fig. 3 A triangular fuzzy number
Fig. 2 Bloom's digital taxonomy pyramid by Samantha Penney
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
Volume 8, 2014
ISSN: 1998-0140
226
111
1
111 /1,/1,/1,, lmuuml
(3)
For lx
or for
u
x
membership function
Mx /
takes the value 0. For mxl
membership function takes
the value
l
m
lx
, while for
u
x
m
function became
m
u
xu
.
V. FAHP METHOD
The AHP is based on the subdivision of the problem in a
hierarchical form. The traditional AHP method is problematic
in that it uses an exact value to express the decision-maker’s
opinion in a pair-wise comparison of alternatives. Chang [12]
introduced a new approach for handling FAHP, with the use
of triangular fuzzy numbers for pair-wise comparison scale of
FAHP, and the use of the extent analysis method for the
synthetic extent values of the pair-wise comparisons.
Let
n
xxxX ,...,, 21
(4)
be an object set , and
n
uuuU ,...,, 21
(5)
be a goal set.
According to the Chang’s extent analysis method [12],
each object is taken and extent analysis for each goal is
performed respectively:
niMMM m
gigigi ,...,2,1,,...,, 21 (6)
where all the
mjM j
gi ,...,2,1 are triangular fuzzy
numbers. The value of fuzzy synthetic extent with respect to
the th
iobject is defined as:
1
1 1 1
m
j
n
i
m
j
j
gi
j
gii MMS (7)
To obtain
m
j
j
gi
M
1
, perform the fuzzy addition operation of
m
extent analysis values for a particular matrix such that:
m
j
j
m
j
j
m
j
j
m
j
j
gi umlM
1111
,, (8)
And to obtain
1
1 1
n
j
m
j
j
gi
M, perform the fuzzy addition
operation of
m
j
j
gi mJM
1
,...,2,1, values such that:
m
j
j
m
j
j
m
j
j
n
i
m
j
j
gi umlM
1111
,, (9)
As 1
M and 2
M are two triangular fuzzy numbers, the
degree of possibility of 21 MM is defined as:
yMxMSUPMMV yx 2121 ,min
(10)
When a pair
yx, exists such that
y
x
and
yMxM 21
, then we have
1
21 MMV .
Since 1
M and 2
M are convex fuzzy numbers we have
that:
21 MMV is 1 if 21 mm , 0 if 12 ul and
2211
12
lmum
ul
, otherwise.
dMMMhgtMMV 12121
(11)
Where d is the ordinate of the highest intersection point
D
between
1
M
and
2
M
like shown on Fig. 4.
The degree possibility for a convex fuzzy number to be
greater than k convex fuzzy numbers
kiM ,...,2,1
1 can
be defined by:
1 2 1 2
, ,..., and and...and
k k
V M M M M V M M M M M M
kiMMV i,...,3,2,1,min (12)
Assume that
iknkSSVAd kii ;,...,2,1,min ,
Fig. 4 The intersection between two triangular
fuzzy numbers 1
M and 2
M
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
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ISSN: 1998-0140
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then the weight vector is given by:
T
n
AdAdAdW
,...,, 21 (13)
where i
A are
n
elements.
Via normalization, the normalized weight vectors are
T
n
AdAdAdW ,...,, 21
(14)
where W is a non-fuzzy number.
FAHP method is used to determine the weight criteria for
decision-making process. In the FAHP procedure, the
pair-wise comparisons in the judgement matrix are fuzzy
numbers.
VI. TOPSIS METHOD
The TOPSIS method requires only a minimal number of
inputs from the user and its output is easy to understand. The
only subjective parameters are the weights associated with
criteria. The fundamental idea of TOPSIS method is that the
best solution is the one which has the shortest distance to the
ideal solution and the furthest distance from the antiideal
solution [11].
TOPSIS method was firstly proposed by Hwang and Yoon
[13]. According to this technique, the best alternative would
be the one that is nearest to the ideal positive solution and
farthest from the ideal negative solution. The positive ideal
solution is a solution that maximizes the benefit criteria and
minimizes the cost criteria, whereas the negative ideal
solution maximizes the cost criteria and minimizes the
benefit criteria. The method is calculated as follows [17].
Establish a decision matrix for the ranking. The structure
of the matrix can be expressed as follows:
n
FFF ...
21
mnmm
n
n
mccc
ccc
ccc
A
A
A
D
...
............
...
...
...
21
22221
11211
2
1
(15)
where i
A denotes the alternatives
i
,
m
i
,...,2,1
. j
F
represents th
j criteria, related to th
i alternative; and ij
c is a
crisp value indicating the performance rating of each
alternative i
Awith respect to each criteria ij
c.
Calculate the normalized decision matrix. The normalized
value ij
r is calculated as:
2
1ij
J
j
ij
ij w
w
r (16)
where
n
i
J
j
,...,2,1;,...,2,1
.
The weighted normalized decision matrix is calculated by
multiplying the normalized decision matrix by its associated
weights. The weighted normalized value ij
v is calculated as:
niJjrwv ijijij ,...,2,1,,...,2,1, (17)
where j
w.represents the weight of the t h
j criteria.
Positive ideal solution (PIS) and negative ideal solution
(NIS) are calculated as follows:
**
2
*
1
*,...,, n
vvvA , maximum values (18)
n
vvvA ,...,, 21 , minimum values (19)
Calculate the separation measures, using the
m-dimensional Euclidean distance [18]. The distance of each
alternative from PIS and NIS are calculated:
Jjvvd
n
j
jiji ,...,2,1,
1
2
**
(20)
Jjvvd
n
j
jiji ,...,2,1,
1
2
(21)
Calculate the relative closeness to the ideal solution and
rank the alternatives in descending order. The closeness
coefficient of each alternative is calculated:
Ji
dd
d
CC
ii
i
i,...,2,1,
*
(22)
where the index value of i
CC lies between 0 and 1. The
larger the index value, the better performance of the
alternatives.
By comparing i
CC values, the ranking of alternatives is
determined.
VII. EMPIRICAL STUDY
A numerical example is illustrated and trial data is used for
selecting the best multimedia application for learning.
Assume that three multimedia applications: APP1, APP2,
APP3 are evaluated under a fuzzy environment. Fig. 5 shows
the all main criteria in hierarchic view. To create pairwise
comparison matrix, linguistic scale [11] is used which is
given in Table I.
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
Volume 8, 2014
ISSN: 1998-0140
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TABLE I.
THE LI NGU ISTI C SCALE AN D CORRESPONDI NG
TRI ANGULARY FU ZZY NUM BERS
Linguistic
scale Explanation TFN Inverse
TFN
Equal
importance
Two activities
contribute equally
to the objective
(1,
1,
1)
(1,
1,
1)
Moderate
importance
Experience and
judgement slightly
favor one activity
over another
(0.33,
0.5,
1)
(1,
2,
3)
Strong
importance
Experience and
judgement strongly
favor one activity
over another
(0.75,
1,
1.25)
(0.8,
1,
1.33)
Very strong
importance
An activity is favored
very strongly over
another, its
dominance
(1,
2,
3)
(1/3,
1/2,
1)
Demonstrated
importance
The evidence
favoring one activity
over another is
highest possible
order of affirmation
(1.33,
2,
4)
(1/4,
1/2,
3/4)
In our study we are ranking three multimedia applications
by FAHP and TOPSIS methods. In step 1 with the help of
improved AHP by fuzzy set theory, the procedure is as
follows: first we should make the hierarchy structure.
Proposed tree is shown in Tables II and III.
TABLE II.
EVALUAT ION MATRI X
1
C 2
C 3
C 4
C 5
C 6
C
1
C 1 2 1 1 2 1
2
C 0.5 1 0.5 0.75 0.75 0.5
3
C 1 2 1 2 2 1
4
C 1 1.33 0.5 1 0.5 0.5
5
C 0.5 1.33 0.5 2 1 2
6
C 1 2 1 2 0.5 1
TABLE III.
FUZZY E VALUATION MATRIX
1
C 2
C 3
C 4
C 5
C 6
C
1
C
(1,
1,
1)
(1,
2,
3)
(0.75,
1,
1.25)
(0.75,
1,
1.25)
(1,
2,
3)
(0.75,
1,
1.25)
2
C
(0.33,
0.5,
1)
(1,
1,
1)
(0.25,
0.5,
0.75)
(0.5,
0.75,
1)
(0.75,
1,
1.25)
(0.33,
0.5,
1)
3
C
(0.8,
1,
1.33)
(1.33,
2,
4)
(1,
1,
1)
(1,
2,
3)
(1,
2,
3)
(0.75,
1,
1.25)
4
C
(0.8,
1,
1.33)
(1,
1.33,
2)
(0.33,
0.5,
1)
(1,
1,
1)
(0.33,
0.5,
1)
(0.25,
0.5,
0.75)
5
C
(0.33,
0.5,
1)
(0.8,
1,
1.33)
(0.33,
0.5,
1)
(1,
2,
3)
(1,
1,
1)
(1,
2,
3)
Fig. 5 Hierarchy of multimedia application selecting problem
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
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ISSN: 1998-0140
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6
C
(0.8,
1,
1.33)
(1,
2,
3)
(0.8,
1,
1.33)
(1.33,
2,
4)
(0.33,
0.5,
1)
(1,
1,
1)
In step 2 below results are obtained and have been brought
in Tables IV and V.
TABLE IV.
THE S UMS OF HORIZONT AL AND VE RTI CAL DI RECTION S
Criteria Row Sums Column Sums
1
C (5.25, 8, 10.75) (4.06, 5, 6.99)
2
C (3.16, 4.25, 6) (6.13, 9.33, 14.33)
3
C (5.88, 9, 13.58) (3.46, 4.5, 6.33)
4
C (3.71, 4.83, 7.08) (5.58, 8.75, 13.25)
5
C (4.46, 7, 10.33) (4.41, 7, 10.25)
6
C (5.26, 7.5, 11.66) (4.08, 6, 8.25)
Sum of row or column sums (27.72, 40.58, 59.4)
After forming the fuzzy pair-wise comparison matrix,
weights of all criteria are determined by the help of FAHP.
According to the FAHP method, firstly synthesis values must
be calculated. From Table V, synthesis values respect to main
goal are calculated like Eq. (7):
3878.0,1971.0,8838.0
4.59,58.40,72.2775.10,8,25.5 1
1
SC
2164.0,1047.0,0531.0
4.59,58.40,72.276,25.4,16.3 1
2
SC
4898.0,2217.0,0989.0
4.59,58.40,72.2758.13,9,88.5 1
3
SC
2554.0,1190.0,0624.0
4.59,58.40,72.2708.7,83.4,71.3 1
4
SC
3726.0,1724.0,0750.0
4.59,58.40,72.2733.10,7,46.4 1
5
SC
4206.0,1848.0,0885.0
4.59,58.40,72.2766.11,5.7,26.5 1
6
SC
TABLE V.
THE FU ZZY SYNT HETI C E XTE NT OF EAC H CRI TERIA
Criteria i
SC
1
C (0.8838, 0.1971, 0.3878)
2
C (0.0531, 0.1047, 0.2164)
3
C (0.0989, 0.2217, 0.4898)
4
C (0.0624, 0.1190, 0.2554)
5
C (0.0750, 0.1724, 0.3726)
6
C (0.0885, 0.1848, 0.4206)
These fuzzy values are compared by using Eq. (11) and
following values are obtained:
1,1,1
,1,9238.0
615141
3121
SCSCVSCSCVSCSCV
SCSCVSCSCV
6149.0
,6759.0,9150.0
,5008.0,5808.0
62
5242
3212
SCSCV
SCSCVSCSCV
SCSCVSCSCV
1,1,1
,1,1
635343
2313
SCSCVSCSCVSCSCV
SCSCVSCSCV
7172.0
,7712.0,6035.0
,1,6813.0
64
5434
2414
SCSCV
SCSCVSCSCV
SCSCVSCSCV
9584.0
,1,8473.0
,1,9202.0
65
4535
2515
SCSCV
SCSCVSCSCV
SCSCVSCSCV
1
,1,8969.0
,1,9642.0
56
4636
2616
SCSCV
SCSCVSCSCV
SCSCVSCSCV
Then priority weights are calculated by using Eq. (13).
9238.01,1,1,9238.0,1min
1
Cd
'
2min 0.5808, 0.5008,0.9150,0.6759,0.6149 0.5
008
d C
11,1,1,1,1min
3
Cd
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
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ISSN: 1998-0140
230
6035.07172.0,7712.0,6035.0,1,6913.0min
4
Cd
8473.09584.0,1,8473.0,1,9202.0min
5
Cd
8969.01,1,8969.0.1,9942.0min
6
Cd
Calculated priority weights form vector:
8969.0,8473.0,6035.0,1,5008.0,9238.0
W
After the normalization of these values priority weights
respect to main goal are calculated using Eq. (14) as:
1879.0,1775.0,1264.0,2095.0,1049.0,1935.0W
Decision matrix for the ranking is then established.
TABLE VI.
DECI SION MAT RIX
1
C 2
C 3
C 4
C 5
C 6
C
APP1 2 3 4 2 3 2
APP2 5 2 3 2 3 4
APP3 4 2 5 3 2 4
Normalized decision matrix is calculated like in Table VII.
TABLE VII.
NORMALI ZED DECI SION MAT RIX
1
C 2
C 3
C 4
C 5
C 6
C
APP1 0.298 0.727 0.565 0.485 0.639 0.333
APP2 0.745 0.485 0.424 0.485 0.639 0.666
APP3 0.596 0.485 0.707 0.727 0.426 0.666
Weighted normalization matrix in Table VIII is formed by
multiplying each value with their weights.
TABLE VIII.
WEIGHT ED NORMALI ZATION M ATRIX
1
C 2
C 3
C 4
C 5
C 6
C
APP1 0.057 0.076 0.118 0.061 0.113 0.062
APP2 0.144 0.050 0.088 0.061 0.113 0.125
APP3 0.115 0.050 0.148 0.092 0.075 0.125
In step 3 positive and negative ideal solutions are
determined by talking the maximum and minimum values for
each criteria using Eq. (18) and Eq. (19):
125.0,113.0,092.0,148.0,076.0,144.0
*A
062.0,075.0,061.0,088.0,050.0,057.0
A
Then the distance of each alternative from PIS and NIS
with to respect to each criteria is calculated with the help of
Eq. (20) and Eq. (21).
222
222
*
1125.0062.0113.0113.0092.0061.0
148.0118.0076.0076.0144.0057.0
d
222
222
1062.0062.0075.0113.0061.0061.0
088.0118.0050.0076.0057.0057.0
d
222
222
*
2125.0125.0113.0113.0092.0061.0
148.0088.0076.0050.0144.0144.0
d
222
222
2062.0125.0075.0113.0061.0061.0
088.00888.0050.0050.0057.0144.0
d
222
222
*
3062.0125.0075.0075.0061.0092.0
148.0148.0076.0050.0144.0115.0
d
222
222
3062.0125.0075.0075.0061.0092.0
088.0148.0050.0050.0057.0115.0
d
At the end closeness coefficient of each multimedia
application is calculated by using Eq. (22).
TABLE IX.
RANKING OF T HE M ULTI ME DIA AP PLI CATION S
*
i
d
i
d i
CC Rank
APP1 0.115 0.0543 0.321 3
APP2 0.071 0.113 0.613 2
APP3 0.053 0.108 0.667 1
Considering the Table IX, prefered multimedia application
for learning is APP3 for decision maker’s preference in this
empirical study. Different rankings can be obtained by using
different decision maker’s preference values.
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
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VIII. CONCLUSION
The proposed approach in this paper is based on FAHP and
TOPSIS methods. We have shown how FAHP method is first
used to determine the weight criteria for decision-making
using triangular fuzzy numbers. Then with TOPSIS method
order of multimedia applications for learning was defined.
Criterias for the selection of multimedia applications for
learning and teaching were defined according to the Bloom's
digital taxonomy. Empirical study for the selection of
multimedia applications for learning was shown and
discussed.
Our ongoing research is directed towards the development
of a fuzzy decision making model for the selection of a
suitable multimedia application for learning using subjective
judgments of decision makers. In future studies other multi-
criteria methods like fuzzy PROMETHEE and ELECTRE can
be used to improve process of selecting multimedia
applications for learning and teaching.
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Volume 8, 2014
ISSN: 1998-0140
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