Page 213 - Contributed Paper Session (CPS) - Volume 4

CPS2192 Laurent D. et al.
how useful is the signed distance measure in a, e.g. defuzzification process,
and in computing these evaluations (see Berkachy and Donzé (2015, 2016a,b)).
In the following, we propose a fuzzy individual and global evaluations of a
subset of the Swiss survey SILC (Swiss Federal Statistical Office (2014)). This
survey is conducted every year and aims the Swiss income and living
conditions. A significant part of the questionnaire is written in linguistic terms,
and as such, it is well suited to this kind of approach. Furthermore, we test by
a fuzzy anova study for two attributes of poverty the difference between Swiss
citizens and foreigners. Bourquin (2016)’s Master thesis was at our knowledge
the first one to apply a fuzzy approach to analyse the SILC survey. She
measured by different categories of interest, and for several attributes,
individual and global fuzzy poverty. Based on the same data, we intend to
complete her study. However, our analysis will differ concerning two points.
First, as we will show below, our fuzzy measures will be defuzzified by the
signed distance measure. Second, we will adopt a different weighting scheme.
Let us shortly in sections 2 and 3 define and present the individual and
global assessments, as well as the fuzzy ANOVA (FANOVA). These theoretical
results will be applied in section 4, the empirical part of the study. For more
explanations, one can fruitfully read, e.g. Berkachy and Donzé (2015, 2016a,b,
2018).

2. Individual and global assessments
Let us assume a linguistic questionnaire divided in main and sub-items,
denoted respectively by and , = 1, … , , = 1, … , . We denote

respectively by and the associated weights with the constraints: 0 ≤

≤ 1, ∑ =1 = 1 , 0 ≤ ≤ 1 and ∑ = 1. A sub-item list of linguistic

=1
terms, , = 1, … , , which we suppose fuzzy. The sampling weight is

denoted by , = 1, … , , where is the size of the sample. Finally, we define

the following indicator function:

1 if the observation i has an answer for the linguistic

(1) =
0 otherwise
Assuming that there are no missing values – thus latter assumption could
be easily relaxed -, the global evaluation P of the linguistic questionnaire
is given by:
(2)

∑ =1

̃
̃
= ∑ ∑ ∑ ∑ ( , 0),

=1 =1 =1 =1
where ( , 0) is the signed distance of measured from the fuzzy origin 0̃.

If a fuzzy linguistic term is characterised by a triangular isosceles membership
function, i.e. = ( −1 , , +1 ), = 1, … , , the signed distance ( , 0) is

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