Nadir objective vector

In MOO, ideal and nadir objective vectors are occasionally used. The ideal objective vector contains the optimum values of the objectives, when each of them is optimized individually disregarding the other objectives. The ideal objective vector denoted by superscript (i.e., [/i /2 J) is shown in Figure 1.2a along with the nadir objective vector denoted by superscript N (i.e., [/i /2 ]). Here,/i is the value of /i(x) when /2(x) is optimized individually, and is the value of /2(x) when /i(x) is optimized individually. Components of the nadir objective vector are the upper bounds (i.e., most pessimistic values) of objectives in the Pareto-optimal set. In case of two objectives, as shown in Figure 1.2a, they correspond to the value of one objective when the other is optimized individually. This may not be the case if there are more than two objectives (Weistroffer, 1985). The ideal objective vector is not realizable unless the objectives are non-conflicting in which case the MOO problem has only a unique solution, namely, ideal objective vector. However, it tells us the best possible value for each of the... 

The upper bounds of the Pareto optimal set, that is, the components of a nadir objective vector are in practice difficult to obtain. It can... 

In the initialization phase of the NIMBUS method, the ranges in the Pareto optimal set, that is, the ideal and the nadir objective vectors are computed to give the DM some information about the possibilities of the problem. The starting point of the solution process can be specified by the DM or it can be a neutral compromise solution located approximately in the middle of the Pareto optimal set. To get it, we set + z )/2 as a reference point and solve (6.4). 

For the case of two objectives. Fig. 1 shows an objective space with a concave part, the CHIM (which is the discontinuous line joining the individual minima of objectives) and several important points. We have to define the Nadir point, pNadin which is the vector of upper bounds of each objective in the entire Pareto-optimal set (in practice, its coordinates can be estimated from the coordinates of the shadow minimum [5-6]). The line that joins the shadow minimum F and the Nadir point represents the quasi-normal direction used by NBI. This line intersects the CHIM in a point given by the vector p ax = [0-5,0.5]. Thus, the upper bound is defined by the distance from this point to the shadow minimum. Similarly, the lower bound is given by the distance to the Nadir point. It is clear that tj = -In. t is positive if the normal is pointing towards the shadow minimum, and negative in the opposite sense). It should be noted that these bounds are set for a two-objective problem. 

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