The result of the deviation estimate is in Table 3 see appendix. This result is then exported to the GIS software for visualization, as displayed in Figure 9 see appendix. One point
should be addressed: in equation 4, the addition of a district s with very low or high value of x
sk
modifies the results. This is a consequence of Arrows Theorem Arrow and Raynaud, 1986 that assures that there is not a multi-criteria method that satisfies simultaneously to
the conditions of universality, Paretos unanimity, transitivity, totality, relevance of all criteria and independence in relation to irrelevant alternatives. In this case, the latter condition could
not hold.
According to this classification Table 3, see appendix, the best municipal district of Rio de Janeiro State, in terms of the quality of urban life, is Petropolis followed directly by
Teresopolis.
D. Results analysis
In the course of the Pareto Race we noticed that some optimization criteria objective functions restricted the analysis, that is, they prevented the reference direction specified by
the decision-maker from being projected onto the efficient frontier. These critical criteria were the homicide rate, the coefficient of maternal mortality and the immunization cover for
measles.
When trying to minimize the first two and to maximize the third seeking improvement of these objectives, other criteria were strongly influenced by the choice of this direction. For
instance, when trying to minimize the homicides rate simultaneously with the immunization coverage maximization, the criteria of population with sanitation facilities and the school
evasion rate moved rapidly to a direction opposed to that desired by the decision-maker, obtaining extreme values in a negative path for these two former criteria, in the order of 10
-6
and 10
4
, respectively. A criterion that showed little influence by any adopted direction was the rate of occupational accidents.
Result III of Table 2 was not the only MPS to be analyzed; another 6 results were studied and were attained in the manner of the proceedings, terminating the search when the
decision-maker believed that that solution portrayed the MPS, that is, the solution had the values of the objective functions that were in agreement with hisher preferences. A second
result also an MPS considered acceptable was analyzed so that the disposition of the alternatives would be checked.
Comparing these two MPSs, it was noticed that the first six municipal districts on the list remained in the same order, changing the hierarchy as of the 7th alternative. The hierarchy
of other solutions was further analyzed, including the case without the Pareto Race. Three of these cases presented some differences in the hierarchy of the alternatives.
However, the first 4 alternatives remained unaltered, the alternative Petropolis standing out as the alternative in first place in the hierarchy.
We verified that in one other case 5th, the criteria homicide rate and immunization coverage for measles, were the ones with the highest distortion when compared to the
values of the solution III the first negatively and the second positively. This difference may be responsible for a great modification in the hierarchy.
In the 6th case analyzed, the criteria homicide rate and infant mortality rate, presented, respectively, values 44 and 34 higher than the preferred ones, in direction of raising
values that should be minimized. This could significantly contribute to changing the disposition of the alternatives, favoring alternatives with high values in these criteria.
Table 3 shows the municipal district, Petropolis in 1st place in the hierarchy, followed by the municipal district, Teresopolis. The municipal district of Niteroi is in 9th place in this
hierarchy. This municipal district was considered by UNDP making use of the HDI
Universitas Sumatera Utara
classification, as the 4th Brazilian City in terms of the quality of life. This classification uses the longevity, education and income indexes, in the construction of HDI.
The hierarchy built in this case study, uses criteria that go beyond those used in the HDI such as the homicide rate and the occupation rate, which can be considered responsible
for this absence of coincidence between these two methodologies. In relation to the estimation of deviations from the best solution using the Euclidean distance,
the compensatory character of this methodology should be stressed, denoting that the low performance of an alternative in one criterion is compensated by a high performance of the
same alternative in another criterion. For the purpose of comparison, we also used the Tchebycheff metric to calculate the deviations. The results proved to be quite similar to the
former, keeping the three first municipal districts.
VI. Conclusions
The first stage of the GIS-MCDA integration, the preliminary study of the alternatives, which leads to the reduction of the set of feasible alternatives, is an important initial stage, bringing
reflections in the following stages, mainly in the reduction of the computational effort. Besides this, some existing multi-criteria software has constraints regarding the number of
criteria and alternatives that can be used. When there exist physical andor qualitative constraints, which can be implemented in GIS, the integration is shown to be quite effective.
The case study presented and the problem proposed does not fit into the traditional multi- objective linear programming models. In this study, the authoe wish to select the municipal
district with the best quality of urban life. For such purpose, criteria were used that do not exhibit explicit constraints. The multi-objective model of incorporating the decision-makers
preferences was shown to be viable and appropriate to the structure of the proposed case study, which dealt with the quality of public services.
The use of MOLP interactive methods, like the Pareto Race that makes it possible to search for solutions on the efficient frontier that are in agreement with the objectives of the decision-
maker, fits into the concept that the most correct decision is that which best represents the interests of the decision-makers.
Comparing the results achieved with the Euclidean and the Tchebycheff metric, the authors verify that the three best municipal districts considering the decision-maker preference
information belong to the so-called Serrana pertaining to the mountain Region of Rio de Janeiro State, showing that there is a quality of life breakdown in the big urban centers. On
the other hand, our model just considers non-subjective indicators, that is, does not consider subjective preferences. Thus, someone wanting to live by the sea will not agree with the
results of our analysis, and must include other exclusion criteria to be analyzed in GIS for instance, select the municipal districts that are in the boundary of the sea or which are X km
away from it.
The case study presented in this paper proposes a methodology that can be adjusted to support public decision-making. The authors surveyed the existence of the variables that
could represent the quality of urban life particularly the variables that are related to the quality of the public services in the Brazilian agencies of information, and how to use them
to subsidize the decision-making process.
An interesting development is the selection of the worst municipal district in terms of the quality of urban life. This situation can be compared to the decisions that should be taken by
municipal planners when choosing areas for investment, with the objective of investing in precarious areas.
Figure 4 displays a clear example: if regular or good conditions of road network were offered to the areas not shaded on the map, these municipal districts would be candidates feasible
alternatives to the choice in the following stages, placing them in the same conditions as the
Universitas Sumatera Utara
