Solution to the MOLP problem

Min ∑ = 26 1 i i PNSi λ Min ∑ = 26 1 i i IMRi λ Max ∑ = 26 1 i i PDWi λ Min ∑ = 26 1 i i SERi λ Max ∑ = 26 1 i i ICMi λ Min ∑ = 26 1 i i MMCi λ Max ∑ = 26 1 i i ORTi λ Min ∑ = 26 1 i i OARi λ

s.t.

. ,..., 1 , 1 26 1 n i i i i = ≥ = ∑ = λ λ

4. Solution to the MOLP problem

In order to search for the MPS, MOLP interactive methods are the most appropriate, mainly due to their interactive feature and possibility of learning through the process. The VIG software was chosen, which implements the Pareto Race method. This choice was made based upon its interactivity, good graphical interface, permitting the use of a great number of objective functions. The theoretical basis of Pareto Race is in the reference direction approach to MOLP, developed by Korhonen and Laakso 1986. In this approach, any direction specified by the decision-maker is projected onto the efficient frontier. Using a reference direction, a subset of efficient solutions an efficient curve is generated and presented for the decision-makers evaluation. The interface is based on a graphical representation. One picture is produced for each interaction. The decision-maker can move in any direction on the efficient frontier he likes, and no specific assumptions concerning hisher underlying utility function are needed during the search process. As stated by Korhonen and Laakso 1986, the visual representation gives the decision- maker a holistic perception of changes in objective function values as heshe moves to a given direction on the efficient frontier. The reference direction is built from the specification of the aspiration levels for each objective function example, Table 1. Multi-objective optimization initial result, without conducting the Pareto Race, see appendix. Universitas Sumatera Utara Projected over the efficient solutions set, this direction produces a path over the efficient frontier, and this is presented to the decision-maker. The search ends when the decision- maker believes that the values of the objective functions are hisher most preferred values, that is, hisher MPS example, Table 2. some results after the Pareto Race, see appendix. This MOLP models solution supplies the best or ideal alternative that presents the objective functions ideal values. This ideal alternative is the linear combination of the other alternatives. The value of each λi can be interpreted as the contribution of each alternative i to the composition of the ideal alternative. In the case in which the optimization result is λi = 1, and all the others λj= 0, j ≠ i, the ideal municipal district is thus represented exactly by the alternative i.

C. Results

According to the authors, in Table 1 we found the results obtained by the software VIG before the Pareto Race. This is the optimum default solution given by the program, without the intervention of the decision-maker. Table 2 displays some results obtained after the Pareto Race I, II, III, i.e., they are some of the decision maker’s MPS. In these cases, there is interference on the part of the decision-maker in the search for the best decision, which is guided in the direction of his preference. These races were obtained, in most cases, with the intention of improving the values of the critical objectives, the ones that prevented the decision-makers reference direction from changing according to hisher convenience. The municipal districts that have λ different from zero for all the solutions found are: Angra dos Reis, Arraial do Cabo, Barra do Pirai, Itaperuna, Itatiaia, Macae, Mangaratiba, Petropolis, Pirai, Rio de Janeiro, Teresopolis, Tres Rios, Vassouras and Volta Redonda. The Pareto Race solutions are efficient and feasible, resulting from an optimization procedure. The choice of one of the solutions reflects the decision-makers preference for a particular configuration of values, to the detriment of others. However, the plain choice of one of the solutions does not dictate the final solution, given that the main objective, the choice of the best municipal district of Rio de Janeiro State, in terms of the quality of urban life, was not achieved. The best alternative has the best level of the attributes of the alternatives in each criterion, and in this case it is the vector formed by the elements of the column Solutions obtained in Table 2, III. The smaller the distance, the closer to the ideal the alternative is. In case improbable the best alternative is a real one, the smallest distance value is zero, whatever the metric used. To represent the deviation of each alternative from the ideal point, the Euclidean distance was chosen, whose mathematical expression is 3, where aik is the normalized value of the alternative i in the objective k and ak is the normalized ideal value in the objective k. 4 is the normalization equation. L2k = ∑ = − n k ak aik 1 2 Aik = MinjXjk MaxjXjk MinjXjk Xik − − 12 3 4 Universitas Sumatera Utara