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C. Valsecchi and R. Todeschini
4 Software
All the algorithms and calculations were carried out in MATLAB (version 2019a)
by means of routines written by the authors. The DRAPE toolbox v. 2 (new version) is available for download at http://www.michem.unimib.it/download/matlabtoolboxes/drape-toolbox-for-matlab/.
5 Results and Discussion
In order to analyse the effect of different weighting schemes, the following two
approaches were adopted:
1. In the first case an equal weight (w k = 1/21) was assigned to each indicator, i.e.
the initial contribution of the indicators is equal to 0.0048. Therefore, henceforth
this first approach will be called equal.
2. Then, we weighted equally the three wellbeing dimensions which have thus a
contribution of one third each. In this case the sum of the weights of the indicators
belonging to a dimension has to be equal to 0.33. Therefore, each indicator of
human wellbeing has a lower weight (w H = 0.0367) than those of environment
wellbeing (w N = 0.0471) and economy wellbeing (w E = 0.066). From now,
this second approach will be called proportional since the initialized weights are
inversely proportional to the number of indicators per dimension.
Furthermore, we assessed the DRAPE ranking method with both equal and
proportional weight initialization approaches considering: (1) all countries together
and (2) countries grouped by continents (Europe, Africa, North, South and Central
America and Asia-Oceania). The CPWR ranking was considered. For Europe alone
we took into account only 18 out of 21 indicators because H1, H2 and H3 were
almost constant. For sake of visualization a resolution of one decimal digit was
used.
5.1 All Countries
The DRAPE method was applied on 154 countries described by the 21 wellbeing
indicators as summarized in Fig. 1. A weight (equal or proportional) is assigned
to each indicator prior to the construction of the basic tournament table and of the
progressively smoothed ones. The different thresholds found in the basic tournament
table after the correlation pruning are 10 (0.5, 0.55, 0.59, 0.6, 0.62, 0.67, 0.71, 0.74,
0.79 and 0.81) using the equal weighting scheme and 35 for the proportional one.
It can be noticed that the higher the thresholds selected the more the ties in the
consensus ranking, i.e. more objects (points) have the same PWR value. This is
C. Valsecchi and R. Todeschini
4 Software
All the algorithms and calculations were carried out in MATLAB (version 2019a)
by means of routines written by the authors. The DRAPE toolbox v. 2 (new version) is available for download at http://www.michem.unimib.it/download/matlabtoolboxes/drape-toolbox-for-matlab/.
5 Results and Discussion
In order to analyse the effect of different weighting schemes, the following two
approaches were adopted:
1. In the first case an equal weight (w k = 1/21) was assigned to each indicator, i.e.
the initial contribution of the indicators is equal to 0.0048. Therefore, henceforth
this first approach will be called equal.
2. Then, we weighted equally the three wellbeing dimensions which have thus a
contribution of one third each. In this case the sum of the weights of the indicators
belonging to a dimension has to be equal to 0.33. Therefore, each indicator of
human wellbeing has a lower weight (w H = 0.0367) than those of environment
wellbeing (w N = 0.0471) and economy wellbeing (w E = 0.066). From now,
this second approach will be called proportional since the initialized weights are
inversely proportional to the number of indicators per dimension.
Furthermore, we assessed the DRAPE ranking method with both equal and
proportional weight initialization approaches considering: (1) all countries together
and (2) countries grouped by continents (Europe, Africa, North, South and Central
America and Asia-Oceania). The CPWR ranking was considered. For Europe alone
we took into account only 18 out of 21 indicators because H1, H2 and H3 were
almost constant. For sake of visualization a resolution of one decimal digit was
used.
5.1 All Countries
The DRAPE method was applied on 154 countries described by the 21 wellbeing
indicators as summarized in Fig. 1. A weight (equal or proportional) is assigned
to each indicator prior to the construction of the basic tournament table and of the
progressively smoothed ones. The different thresholds found in the basic tournament
table after the correlation pruning are 10 (0.5, 0.55, 0.59, 0.6, 0.62, 0.67, 0.71, 0.74,
0.79 and 0.81) using the equal weighting scheme and 35 for the proportional one.
It can be noticed that the higher the thresholds selected the more the ties in the
consensus ranking, i.e. more objects (points) have the same PWR value. This is