3.1 Pareto set

The most important stage toward the multicriteria problem solving is an establishment of the Pareto domain ХР, i.e., such domain in model parameter space where it is not possible to reach a dominance of selected criteria over others. A geometrical visualization of the Pareto set (AB line) for the bi-criteria case in 2D parameter space is given in Figures 3 and 4. For instance, the best least squares fit of the p–T-x phase equilibria data Kmin <sup>1</sup> usually does not correspond to the best data fit for critical line Kmax <sup>2</sup> and vice versa. Here Kmin <sup>1</sup> and Kmin <sup>2</sup> are the best or "ideal" solutions for each criterion found as a result of single-criterion optimization; for obviousness, parameters X1 and X2 could be interpreted as geometric and energetic parameters in the van der Waals EoS model.

To locate the Pareto set, we assume to compare two solutions, X\* and X0, which hold the inequality:

$$\mathbf{K(X\_{\mathcal{O}})} \le \mathbf{K(X\_{\mathcal{I}})}.\tag{8}$$

there is no such X0 when for all criteria the inequality (8) is satisfied. The set of all values XP = X\* is the Pareto set, and the vector XP is the unimprovable vector of the results, in case if from Ki (X0) ≤ Ki (XP) for any i it follows that

Azeotrope-Breaking Potential of Binary Mixtures in Phase Equilibria Modeling

DOI: http://dx.doi.org/10.5772/intechopen.83769

Isolationistic and cooperative trends are usually considered for aggregation of local criteria vector into global scalar criterion. The isolationistic convolution schemes are additive, i.e., global criterion is represented as a weighted sum of local criteria and entropy as a sum of local criterion logarithms. If behavior of each criterion is complied with common decision to minimize some cooperative

i

, 1≤i≤ n, X ∈ XP, (9)

criterion, then a convolution scheme can be presented in the form

KCð Þ¼ <sup>X</sup> min wi Kið Þ� <sup>X</sup> <sup>K</sup><sup>0</sup>

K (X0) = K (XP).

Figure 4.

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Figure 3.

The Pareto curve in criteria space.

3.2 Crisp convolution schemes

The Pareto curve in parameter space.

It is obvious that X0 is preferable than X\* . Thus, all X\* vectors satisfying Eq. (8) may be excluded. It is worth to analyze that only those X\* vectors for which Azeotrope-Breaking Potential of Binary Mixtures in Phase Equilibria Modeling DOI: http://dx.doi.org/10.5772/intechopen.83769

Figure 4. The Pareto curve in parameter space.

Figure 3.

there is no such X0 when for all criteria the inequality (8) is satisfied. The set of all values XP = X\* is the Pareto set, and the vector XP is the unimprovable vector of the results, in case if from Ki (X0) ≤ Ki (XP) for any i it follows that K (X0) = K (XP).
