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Table 2: Logic Rules used for resilience estimation.
HHR=Human Health Resources, DL= Development Level,
EO=Emergency Operability, R = Resilience, VH = Very
High, H = High, M = Medium, L= low, VL = Very Low
1.
If (HHR is L) and (DL is L) and (EO is L) then (R is VL)
2. If (HHR is M) and (DL is M) and (EO is M) then (R is M)
3. If (HHR is H) and (DL is H) and (EO is H) then (R is VH)
4. If (HHR is M) and (DL is L) and (EO is L) then (R is L)
5. If (HHR is H) and (DL is H) and (EO is L) then (R is M)
6. If (HHR is L) and (DL is M) and (EO is L) then (R is L)
7. If (HHR is M) and (DL is M) and (EO is L) then (R is M)
8. If (HHR is H) and (DL is M) and (EO is L) then (R is H)
9. If (HHR is L) and (DL is H) and (EO is L) then (R is M)
10. If (HHR is M) and (DL is H) and (EO is L) then (R is M)
11. If (HHR is H) and (DL is H) and (EO is L) then (R is H)
12. If (HHR is L) and (DL is L) and (EO is M) then (R is L)
13. If (HHR is M) and (DL is L) and (EO is M) then (R is M)
14. If (HHR is H) and (DL is L) and (EO is M) then (R is H)
15. If (HHR is L) and (DL is M) and (EO is M) then (R is M)
16. If (HHR is H) and (DL is M) and (EO is M) then (R is H)
17. If (HHR is L) and (DL is H) and (EO is M) then (R is M)
18. If (HHR is M) and (DL is H) and (EO is M) then (R is H)
19. If (HHR is H) and (DL is H) and (EO is M) then (R is H)
20. If (HHR is L) and (DL is L) and (EO is H) then (R is M)
21. If (HHR is M) and (DL is L) and (EO is H) then (R is H)
22. If (HHR is H) and (DL is L) and (EO is L) then (R is H)
23. If (HHR is L) and (DL is M) and (EO is H) then (R is H)
24. If (HHR is M) and (DL is M) and (EO is H) then (R is VH)
25. If (HHR is H) and (DL is M) and (EO is H) then ((R is VH)
26. If (HHR is L) and (DL is H) and (EO is H) then (R is H)
27. If (HHR is M) and (DL is H) and (EO is H) then (R is VH)
27 rules for calculating fragility and resilience val-
ues respectively. The rules were intended to follow
risk management literature which could suggest pos-
sible outcomes when three of these elements interact
to form resilience or fragility. The Mamdani aggrava-
tion model, that has as input variables the resilience
and the fragility, discretized into 5 classes each, is
composed of 25 fuzzy rules. In Table 2 the rules of the
Mamdani resilience model are presented as an exam-
ple. As mentioned before, the use of classical fuzzy
systems, with well established fuzzy inference theory,
allow a high level understandability model, easily un-
derstandable by experts which leads towards a deepest
discussion in the topic of social vulnerability descrip-
tion and casual interrelation.
Let’s describe the inference process by follow-
ing the example of the proposed Resilience FIS. The
fuzzy inference engine combines the fuzzy if-then
rules (see Table 2) into a mapping from fuzzy sets
in the input space U ⊂ R
n
to fuzzy sets in the output
space V ⊂ R, based on fuzzy logic principles.
Let’s U = U
1
x U
2
x U
3
⊂ R
n
and V ⊂ R, where
U
1
, U
2
and U
3
represents the universes of discurse of
Marginal Slums, Social Disparity Index and Popula-
tion Density input variables, respectively, and V the
universe of discourse of Resilience. In hour case each
input variable contains three fuzzy sets and the output
variable is discretized into five fuzzy sets. Then, the
fuzzy rule based shown in Table 2 can be expresed in
a canonical form as shown in Equation 4.
R
(l)
: IFx
1
isA
l
1
and...andx
n
isA
l
n
T HENyisB
l
(4)
where A
l
1
and B
l
are fuzzy sets in U
i
and V , respec-
tively, x = (x
1
, x
2
, x
3
) ∈ U are Marginal Slums, So-
cial Disparity Index and Population Density linguistic
variables, y ∈ V is the Resilience linguistic variable
and l = 1, 2, ..., 27 is the rule number. Consider now
the fuzzy facts: x
1
is A
0
1
, x
2
is A
0
2
, x
3
is A
0
3
, being A
0
1
, A
0
2
and A
0
3
fuzzy sets.
The Generalized Modus Ponens allows the deduc-
tion of the fuzzy fact y is B
0
by using the compo-
sitional rule of inference (CRI), defined trough the
fuzzy relation between x and y, as defined in Equa-
tion 5.
B
0
= A
0
◦ R (5)
where A
0
= (A
0
1
, A
0
2
, A
0
3
). The simplest expression of
the compositional rule of inference can be written as
Equation 6.
µ
B
0i
(y) = I (µ
A
i
(x
0
), µ
B
i
(y)) (6)
when applied to the ith-rule; where:
µ
A
i
(x
o
) = T
µ
A
i
1
(x
1
), µ
A
i
2
(x
2
), µ
A
i
3
(x
3
)
where x
0
= (x
1
, x
2
, x
3
). Here, T is a fuzzy conjuctive
operator and I is a fuzzy implicator operator.
Once the inference is perfromed by means of the
compositional rule of inference scheme, the resulting
individual (one for each rule) output fuzzy sets are
aggregated into an overall fuzzy set by means of a
fuzzy aggregation operator and then a defuzzification
method is employed to transform the fuzzy set into a
crisp output value, i.e. the resilience level following
the example. The defuzzification method used in this
work is the Centre Of Gravity (COG), which slices
the overall fuzzy set obtained in the inference process
into two equal masses. The centre of gravity can be
expressed as Equation 7.
COG =
R
b
a
xµ
B
(x)dx
R
b
a
µ
B
(x)dx
(7)
where B is fuzzy set on the interval [a, b].
In this research, this Mamdani fuzzy inference
process is used for both descriptors, i.e. fragility and
resilience. Once the evaluation has been made for
both descriptors, it’s possible to calculate the aggra-
vation coefficient using the Aggravation fuzzy infer-
ence system (see Figure 2). The antecedents in this
case are the resilience and fragility descriptors and the
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