The following stage is to overlay the two layers of information generated, creating a third layer that contains the municipal districts that fulfill both constraints: to be crossed by paved
federal highways, in a good or regular state of conservation, AND to present at least two of the constituent indicators of HDI greater than the average indexes for the State. The result of
this overlay operation displayed in figure 8see original paper produced a set of 26 alternatives that, in the 2nd stage, are appraised by multi-criteria analysis.
It can seem that we loose information in the next steps, not considering the municipalities excluded by the exclusion criteria. One should notice that these in formations are used in
different phases that have their usefulness in the global solution. The information about infrastructure highways, health longevity, education and work income are considered in
Model 2 section 2.2.3 through other variables. This would be the case of the longevity index that evaluates the life expectance, used to preselect some alternatives, and the
infant mortality rate in Model 2.
2. Choice of multi-criteria evaluation method
The Multi-criteria Decision Aid can be divided into multi-attribute and multi-objective problems. The former deal with discrete alternatives and the latter with a continuous space
of alternatives. Among the multi-attribute problems there is commonly a classification of the methods used as either belonging to the American or to the French Schools of Decision
Aid.
The multi-objective problems are, as a rule, mathematically more difficult, although they demand the decision-makers constant presence. In a multi-objective context, the notion of
optimal solution gives way to the concept of efficient or Pareto optimal solution it is a possible solution if and only if there is not one other solution that improves the value of one
objective function without worsening the value of at least one other objective function.
According the authors, although we gain mathematical simplicity with this transformation, we lose interactivity and we oblige the decision-maker to explicitly state hisher preferences and
even heshe may not know what they are. If the analyst is not concerned about mathematical complexity but the impossibility of the decision-maker to supply coherent
information, heshe must seek the interactivity of the multi-objective problems, particularly if they are allied to software with a great visual appeal, which enables the decision-maker to
implicitly express preferences and to learn throughout the process. It is also worth pointing out that the multi-objective approach enables global visualization of the feasible solutions
space, as well as the efficient frontier set of all efficient solutions, making it easier to understand the problem.
If the initial problem has a multi-attribute structure, it is necessary to convert the alternatives space into a continuous set that contains it, so that this problem can be solved as a multi-
objective one. There are many manners to carry out this action, and in this paper we favored, for its simplicity, to consider that the alternatives space is the set of vectors whose
co-ordinates are a convex linear combination of the original alternatives co-ordinates. So, the problem is solved as though it had a multi-objective structure, having as a result a
solution that belongs to the new space generated, but with a great probability of not belonging to the original alternatives space. It is thus necessary to have one other phase
that consists of choosing from among the real alternatives the one that most resembles this virtual alternative.
Considering that the space produced is metric, it is enough to find which real alternative has the smallest distance to the virtual one. The metric can be any of the existing ones, the usual
ones being the Euclidean, also called L
2
norm, which has a compensatory characteristic the low performance of an alternative in one criterion is compensated by a high performance of
the same alternative in another criterion; see equation 3, and the Tchebycheff metric, also called L
00
norm, non-compensatory. For each alternative, the Tchebycheff metric considers only the criterion that yields the greatest distance to the reference point, being indifferent to
the other criteria, not allowing compensations.
Universitas Sumatera Utara
3. MOLP problem formulation
