where ã
k ij
denotes the k-th preference of the single decision- maker with respect to the i-th criterion in relation to the j-th
criterion, expressed by a triangular fuzzy number. As an example, the fuzzy value ã
1 13
could represent the preference of the single decision-maker relative to the first
criterion with respect to the third criterion with a judgment defined as:
2 If several decision-makers are involved, the individual
preferences ã
k ij
are averaged and computed as follows: 3
where K denotes the number of decision-makers. According to the averaged preferences a new general matrix
of pairwise comparisons, is assembled; it will show the
averaged triangular coefficients of dominance and it will assume the following form:
4 According to the Buckley’s method, it is computed as the
geometric average of the values obtained from the fuzzy comparisons of each criterion,
ũ
i
. The values ũ
i,
are triangular numbers and are defined by the following formula:
ũ 5
Then, the vector sum of each fuzzy element ũ
i
is computed and it is elevated to exponential value -1 by the following
relation in which the symbol ⨁ denotes the fuzzy sum:
ũ ũ ⨁ũ
⨁ ũ ⨁ ⨁ũ
6 The subsequent step amounts to computing the fuzzy weight
relative to the i-th criterion expressed by the following relation:
ũ ũ
⨁ũ ⨁ ũ
⨁ ⨁ũ 7
where the symbol denotes the fuzzy product and the vector
containing fuzzy weight in ascending order. Since the term
is a triangular number as well, it needs to be de-fuzzified
, by using the “center of gravity” method, i.e. by applying the following equation:
8 Notice that
is not a fuzzy number and it has to be normalized as:
9 These phases need to be executed in sequential steps in order
to compute the normalized weights both of the criteria and of the alternatives. Subsequently, multiplying each weight of the
alternative by the relative criterion, the scores relative to each alternative are computed. On the basis of these results, the
alternative with higher score represents the choice of the decision-maker or of the decision-makers in team.
4. CASE STUDY
The decision-making activity, due to the large number of factors that influence it, is structured on an ordered sequence
of phases, necessary to bring the complex problem to a simplified structure that can be easily decomposed in
hierarchical levels.
The proposed hierarchy of dominance identifies four levels: the first level contains the overall objective O; the second
level contains four criteria C
i
that specify contents and meanings; the third level contains twenty one sub-criteria S
i
that further characterize the higher level criteria. At the base of the hierarchy there is the fourth level where the three
alternatives A
i
to be evaluated are located: Choosing the ideal profile for the Director of a Laboratory of
Geomatics O
Education C
1
o
MSc in Engineering S
1
o
Management of a Research Center S
2
o
MSc in Earth Science S
3
o
MSc in Physics S
4
Achievements C
2
o
Cultural background S
5
o
Exam score average S
6
o
Bachelor Science grade S
7
o
MSc grade S
8
Curriculum C
3
o
Age S
9
o
Publications S
10
o
Peer -reviewed Journals S
11
o
National job experiences S
12
o
Job experiences abroad S
13
o
Projects leader S
14
Skills C
4
o
Leadership S
15
o
Problem solving ability S
16
o
Ability in team working S
17
o
Knowledge of languages S
18
o
Computer skills S
19
o
Experiences in international cooperation S
20
o
Mobility attitude S
21
Profile 1 A
1
Profile 2 A
2
Profile 3 A
3
The first phase of the decision-making process is the composition of the group of expert decision-makers involved
in the strategic solution of the problem. They are in charge to give a personal opinion based on their own experiences in the
context of human resources.
Based on the Saaty semantic fuzzy scale, the decision-makers give their opinions individually, by assigning an intensity of
dominance at each pairwise comparison between elements of the same hierarchical level. Individual results are averaged to
take account of the multidisciplinary nature of the problem.
A first matrix of pairwise comparisons is created: the second level criteria Education C
1
, Achievements C
2
, Curriculum C
3
, Skills C
4
are mutually compared with respect to the
This contribution has been peer-reviewed. doi:10.5194isprsarchives-XL-6-W1-31-2015
34
first level overall objective.
Figure 5. Case study hierarchy of dominance The following example represents the simulation for selecting
the ideal profile for the Director of a Laboratory of Geomatics O. Table 1 shows the matrix of pairwise
comparison between the previous criteria.
O C
1
C
2
C
3
C
4
C
1
1;1;1 3;4;5
13;12;1 3;4;5
C
2
15;14;13 1;1;1
14;13;12 1;2;3
C
3
1;2;3 2;3;4
1;1;1 2;3;4
C
4
15;14;13 13;12;1
14;13;12 1;1;1
Table 1. Matrix of pairwise comparisons of the criteria in relation to the overall objective
The values ũ
i
are first calculated as geometric mean of the fuzzy relations:
The vector sum ũ is calculated and raised to -1:
⨁ ⨁
⨁
The fuzzy weight is calculated for each criterion:
The subsequent step is the calculation of the crisp weights Mi by the center of gravity de-fuzzification method:
Normalized weights N
i
are now calculated:
The second matrix of pairwise comparisons is created:third level sub-criteria MSc in Engineering S
1
, Management of a Research Center S
2
, MSc in Earth science S
3
, MSc in Physics S
4
are mutually compared in relation to the second level criterion: Education C
1
.
C
1
S
1
S
2
S
3
S
4
S
1
1;1;1 2;3;4
2;3;4 1;2;3
S
2
14;13;12 1;1;1
2;3;4 1;2;3
S
3
14;13;12 14;13;12
1;1;1 14;13;12
S
4
13;12;1 13;12;1
2;3;4 1;1;1
Table 2. Matrix of pairwise comparisons of the sub-criteria in relation to the criterion C
1
The third matrix of pairwise comparisons is created: third
This contribution has been peer-reviewed. doi:10.5194isprsarchives-XL-6-W1-31-2015
35
level sub-criteria Cultural background S
5
, Exam score average S
6
, B.Sc. grade S
7
, MSc. grade S
8
are mutually compared in relation to the second level criterion: Score C
2
.
C
2
S
5
S
6
S
7
S
8
S
5
1;1;1 3;4;5
3;4;5 2;3;4
S
6
15;14;13 1;1;1
1;2;3 1;2;3
S
7
15;14;13 13;12;1
1;1;1 15;14;13
S
8
14;13;12 13;12;1
3;4;5 1;1;1
Table 3. Matrix of pairwise comparisons of the sub-criteria in relation to the criterion C
2
The fourth matrix of pairwise comparisons is created: third level sub-criteria Age S
9
, Publications S
10
, Peer reviewed journals S
11
, National job experiences S
12
, Job experiences abroad S
13
, Projects leader S
14
are mutually compared in relation to the third level criterion: Curriculum C
3
.
C
3
S
9
S
10
S
11
S
12
S
13
S
14
S
9
1;1;1 15;14;
13 1;2;3
2;3;4 14;13;
12 15;14
;13 S
10
3;4;5 1;1;1
2;3;4 2;3;4
13;12; 1
14;13 ;12
S
11
13;12; 1
14;13; 12
1;1;1 14;13;
12 15;14;
13 15;14
;13 S
12
14;13; 12
14;13; 12
2;3;4 1;1;1
15;14; 13
14;13 ;12
S
13
2;3;4 1;2;3
3;4;5 3;4;5
1;1;1 1;2;3
S
14
3;4;5 2;3;4
2;3;4 2;3;4
13;12; 1
1;1;1
Table 4. Matrix of pairwise comparisons of the sub-criteria in relation to the criterion C
3
The fifth matrix of pairwise comparisons is created: third level sub-criteria Leadership S
15
, Problem solving ability, S
16
, Ability in team working S
17
, Knowledge of languages S
18
, Computer skills S
19
, Experiences in international cooperation S
20
, Mobility attitude S
21
are mutually compared in relation to the fourth level criterion: Skills C
4
.
C
4
S
15
S
16
S
17
S
18
S
19
S
20
S
21
S
15
1;1;1 14;13;
12 1;2;3
1;2;3 2;3
;4 2;3;
4 13;1
2;1 S
16
2;3;4 1;1;1
1;2;3 2;3;4
2;3 ;4
2;3; 4
1;2;3 S
17
13;12; 1
13;12; 1
1;1;1 2;3;4
2;3 ;4
1;2; 3
13;1 2;1
S
18
13;12; 1
14;13; 12
14;1 3;12
1;1;1 1;2
;3 13;
12; 1
1;2;3 S
19
14;13; 12
14;13; 12
14;1 3;12
13;1 2;1
1;1 ;1
1;2; 3
1;1;1 S
20
14;13; 12
14;13; 12
13;1 2;1
1;2;3 13
;1 2;1
1;1; 1
1;1;1 S
21
1;2;3 13;12;
1 1;2;3
13;1 2;1
1;1 ;1
1;1; 1
1;1;1
Table 5. Matrix of pairwise comparisons of the sub-criteria in relation to the criterion C
4
Twenty one matrices of pairwise comparisons are now created for the fourth level alternatives Profile 1 A
1
, Profile 2 A
2
, Profile 3 A
3
in relation to the twenty one third level sub-criteria S
i
. An example concerning the sub-criterion Engineering S
1
is addressed in the sequel.
S
1
A
1
A
2
A
3
A
1
1;1;1 3;4;5
1;1;1 A
2
15;14;13 1;1;1
15;14;13 A
3
1;1;1 3;4;5
1;1;1
Table 6. Matrix of pairwise comparisons of the alternatives in relation to the sub-criterion S
1
Once all the weight vectors at each hierarchical level have been obtained, the fourth level alternatives Profile 1 A
1
, Profile 2 A
2
, Profile 3 A
3
are weighted against the second level criterion Education C
1
.
C
1
S
1
S
2
S
3
S
4
w
Ai
w
Si
0,43 0,25
0,10 0,22
A
1
0,44 0,51
0,52 0,17
0,41 A
2
0,12 0,14
0,31 0,52
0,23 A
3
0,44 0,35
0,17 0,31
0,36
Table 7. Table of the weights of the alternatives associated with criterion C
1
The fourth level alternatives Profile 1 A
1
, Profile 2 A
2
, Profile 3 A
3
are weighted with respect to the second level criterion Score C
2
.
C
2
S
5
S
6
S
7
S
8
w
Ai
w
Si
0,52 0,20
0,09 0,19
A
1
0,21 0,33
0,33 0,33
0,27 A
2
0,47 0,33
0,33 0,33
0,40 A
3
0,32 0,33
0,33 0,33
0,33
Table 8. Table of calculation for the weights of the alternatives against the criterion C
2
The fourth level alternatives Profile 1 A
1
, Profile 2 A
2
, Profile 3 A
3
are weighted with respect to the second level criterion Curriculum C
3
.
C
3
S
9
S
10
S
11
S
12
S
13
S
14
w
Ai
w
Si
0,10 0,19
0,06 0,08
0,32 0,27
A
1
0,11 0,54
0,23 0,51
0,20 0,54
0,37 A
2
0,63 0,12
0,39 0,14
0,60 0,12
0,34 A
3
0,26 0,33
0,39 0,35
0,20 0,33
0,29
Table 9. Table of calculation for the weights of the alternatives against the criterion C
3
The fourth level alternatives Profile 1 A
1
, Profile 2 A
2
, Profile 3 A
3
are weighted with respect to the second level criterion Skills C
4
.
C
4
S
15
S
16
S
17
S
18
S
19
S
20
S
21
w
Ai
w
Si
0,17 0,28
0,15 0,10
0,08 0,09
0,13 A
1
0,52 0,14
0,14 0,21
0,20 0,17
0,25 0,23
A
2
0,17 0,35
0,51 0,65
0,20 0,52
0,58 0,41
A
3
0,31 0,51
0,35 0,14
0,60 0,31
0,17 0,36
Table 10. Table of calculation for the weights of the alternatives against the criterion C
4
The fourth level alternatives Profile 1 A
1
, Profile 2 A
2
, Profile 3 A
3
are weighted with respect to the first level overall objective
“Choosing the ideal profile for the Director of a Geomatics Laboratory
” O.
O C
1
C
2
C
3
C
4
w
Ai
w
Si
0,35 0,13
0,42 0,10
A
1
0,41 0,27
0,37 0,23
0,35 A
2
0,23 0,40
0,34 0,41
0,32 A
3
0,36 0,33
0,23 0,36
0,33
Table 11. Table of calculation for the weights of the alternatives against the overall objective O
This contribution has been peer-reviewed. doi:10.5194isprsarchives-XL-6-W1-31-2015
36
The alternative Profile 1 A
1
is finally chosen since it is associated with a greater weight.
5. CONCLUSIONS
