3.2 Crisp convolution schemes

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

$$K\_{\mathcal{C}}(X) = \min \left[ w\_i \left( K\_i(X) - K\_i^0 \right) \right], \qquad 1 \le i \le n, \quad \mathbf{X} \in \mathbf{X}\_P,\tag{9}$$

• Convex models of uncertainty, developed by Ben-Haim and Elishakoff [15] as

• Game-theoretic models of uncertainty deriving from conflict among the different goals (in our case, it is a conflict between thermodynamically

• Verbal models of uncertainty deriving from vagueness and initiated to be

Three models of uncertainty cannot exist one by one in parameter estimation, and mathematical tools should reveal this fact. The conventional single-criterion methods of parameter estimation are examples of lopsided vision of multicriteria decision-making problem where only one set of variables, strongly dependent on decision-maker experience, is recommended. A lack of single meaning of optimality concept, following from the basics of modern game theory, points to impossibility to construct a completely formal algorithm for parameter estimation and futility of eliminating the uncertainty in the search process. It is more correct to consider the trade-off or rational parameters, which adequately describe thermodynamic

The following sequence of decision-making steps in model parameter estimation

• Determination of the Pareto-optimal (or compromise or trade-off) set XP as the formal solution of the multicriteria problem to minimize a conflict source

• Informal selection of convolution scheme to switch over a vector criterion K

The most important stage toward the multicriteria problem solving is an estab-

<sup>1</sup> usually does not correspond to the best data

K Xð Þ<sup>0</sup> ≤ K Xð Þ: (8)

. Thus, all X\* vectors satisfying

<sup>2</sup> are the best or "ideal"

<sup>1</sup> and Kmin

lishment 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

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

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

Eq. (8) may be excluded. It is worth to analyze that only those X\* vectors for which

<sup>2</sup> and vice versa. Here Kmin

• Evaluation of the final decision vector Xopt XP to minimize a vagueness

an extension of interval analysis

Distillation - Modelling, Simulation and Optimization

properties and phase behavior, generated by EoS.

into a scalar combination of the Ki (Х)

inconsistent data)

is proposed:

3.1 Pareto set

of uncertainty

source of uncertainty

of the p–T-x phase equilibria data Kmin

parameters in the van der Waals EoS model.

It is obvious that X0 is preferable than X\*

fit for critical line Kmax

hold the inequality:

76

resolved by Zadeh [14]

where wi are the weight coefficients, K<sup>0</sup> <sup>i</sup> is an infimum of the desired result that is acceptable for decision-maker remaining in the coalition, and KC is a global tradeoff criterion. If it will be possible to come to an agreement about preference (weight) for each criterion, then the final decision can be found as a solution of scalar nonlinear programming problem:

$$K\_{\mathbb{C}}(X) = \min \sum\_{i=1}^{n} \left| w\_i \left( K\_i(X) - K\_i^0 \right) \right|, \quad \mathbf{X} \in \mathbf{X}\_P. \tag{10}$$

relationships to obtain from EoS model, the different derivatives of thermodynamic values, for instance, the prediction of critical lines from the EoS with parameters restored from VLE data, is questionable due to diverse sources of uncertainty in

Azeotrope-Breaking Potential of Binary Mixtures in Phase Equilibria Modeling

For illustration, we consider thermodynamic and phase behavior of water-carbon dioxide system near the critical point of water. Experimental data on p-ρ-T-x properties have been taken from [18]; data on phase equilibria and critical curve have been extracted from [19, 20]. The critical lines have been calculated using algorithm developed in [21]. Phase equilibria calculations have been carried out with Michelsen and Mollerup method [22] for cubic EoS. We performed phase equilibria and critical line calculations for the RK EoS with binary interaction parameters k12 and l12 used in

> b<sup>11</sup> þ b<sup>22</sup> 2 � �

and restored from different crisp convolution schemes. For simplicity, the results of calculation for phase equilibria and critical line are presented in Figures 5 and 6 only for most selected compromise schemes. The challenge of conflict between parameters retrieved from different sections of thermodynamic surface remains important for practically all EoS including sophisticated multiparameter equation of

Critical line of Н2О▬СО<sup>2</sup> system [21]. ● Experimental data ▬ Best fit of VLE data °°°° von Neumann's

<sup>a</sup>11a<sup>22</sup> <sup>p</sup> <sup>þ</sup> <sup>a</sup>22ð Þ <sup>1</sup> � <sup>x</sup>

þ b22ð Þ 1 � x

2

2

(14)

both the used models and experimental data.

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

<sup>b</sup> <sup>¼</sup> <sup>b</sup>11x<sup>2</sup> <sup>þ</sup> <sup>l</sup>12xð Þ <sup>1</sup> � <sup>x</sup>

<sup>a</sup> <sup>¼</sup> <sup>a</sup>11x<sup>2</sup> <sup>þ</sup> <sup>k</sup>12xð Þ <sup>1</sup> � <sup>x</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

standard mixing rules

Figure 5.

79

coalition scheme [16] ─ ─ Best fit of critical curve.

In terms of game-theoretical approach, such coalition is defined as the von Neumann coalition scheme [16].

If no concordance among decision-makers is concerning the weight choice, then arbitration network is preferable. Classical arbitration scheme was derived mathematically rigorously by Nash but very often criticized from common sense [16]:

$$K\_C(X) = \min \prod\_{i=1}^n \left| K\_i(X) - K\_i^0 \right|, \quad \mathbf{X} \in \mathbf{X}\_P. \tag{11}$$

All crisp convolution schemes under discussion try to reduce an uncertainty deriving from conflict among different criteria in the Pareto domain. The next step is extenuation of uncertainty driving from vagueness.
