Pareto-optimality global

The direct search for a global optimum may not uncover some of the Pareto-optimal solutions close to the overall optimum, which might be good trade-off solutions of interest to the decision maker. 

No Preference Methods (e.g., global criterion and neutral compromise solution) These methods, as the name indicates, do not require any inputs from the decision maker either before, during or after solving the problem. Global criterion method can find a Pareto-optimal solution, close to the ideal objective vector. 

We can say that interactive methods overcome weaknesses of a priori and a posteriori methods the DM does not need a global preference structure and only interesting Pareto optimal solutions need to be considered. The latter means both savings in computational cost, which in many computationally complicated real problems is a significant advantage, and avoids setting cognitive overload on the DM, which the comparison of many solutions typically implies. 

In optimization studies that include multiobjective optimization problems, the main objective is to find the global Pareto optimal solutions, representing the best possible objective values. However, in practice, users may not always be interested in finding the global best solutions, particularly if these solutions are very sensitive to variable perturbations. In such cases, practitioners are interested in finding robust solutions that are less sensitive to small changes in variables (Deb Gupta 2004 Branke 1998). 

The NBI technique has the advantage that an equally distributed set of p produces an equally distributed set of points on the Pareto surface. The characteristic parameter of the method is an integer number (spac) from which a set of yff uniformly spaced is generated. The points obtained by solving the NBI subproblems are Pareto-optimal solutions if the components of the shadow minimum are the global minima and the Pareto surface is convex. Nevertheless, points on concave parts can be found, but it is not assured that these solutions are non-dominated. 

These curves are somewhat similar eis those obtained with NBI, but we can see that better results were obtained with a low value of the penalty coefficient and a population size of 200. In contrast, the worst points were found with a Rp= l.Oe+5. Low Rp means that the search is directed towards the unconstrained optimum. It should be noted that the penalty function only includes the equality constraints which assure that the solution is on the normal. Thus, although an infeasible point is not penalized enough, the solution is expected to be feasible with respect to the process constraints, and, consequently, it can be a global Pareto-optimal solution. 

Remark 5.1. Pareto preference (Example 5.1 number 1) satisfies Assumption 5.1, so does the preference represented by a linear value function. As indicated in Example 5.1, number 2, one can use LD y) and LP y) to replace D y) and P y) respectively when the assumption does not hold. If we do so, our results stated in this section are still valid, but only in a local sense (local optimal vs. global optimal). For the details of such treatment and conditions for the local results to be valid as the global results refer to Yu (1985, chap. 7). 

A multi-criteria optimization problem generally consists of n decision variables, m constraints, and k evaluation functions. Thereby the evaluation functions can be in conflict with each other, making it difficult to find the global optimum. To find this optimum, the solution space /I C R is created by the decision variables x = (xi, , x ) of the decision space Q with the objective function vector F O A The objective function vector F x) = (/ j(x), , / (x), X e O is optimized considering the constraints / (—x) > 0 = 1,... m x Q (Muschalla 2006). The individual fitness value of each fitness function then can be processed by the evaluation function in different ways. This can be done by a scalarization method or a Pareto dominance-based approach. 

Finally, extending the methodology to all optimal solutions. Fig. 5 shows the tolerance region over the whole Pareto front of the problem. Since every solution in this set has different parameters, the equations o f the ellipses which perform the global tolerance region, as... 

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