# Thermal Conductivity of Low-GWP Refrigerants Modeling with Multi-Object Optimization
## 2 Methods
### NSGAII Method  
The algorithm chosen for parameter optimization is NSGA 2 [28]. It is a robust multi-objective algorithm that is efficient and highly tested in the literature [35; 36; 37]. Fundamentally, as also shown in Figure 2, it is based on the following steps:  
1. Random initialization of the base population;
2. Selection of the initial population through a process of non-dominance;
3. Application of crowding distance for subsequent selection of individuals;
4. Selection of individuals based on crowding distance;
5. Application of genetic algorithm and application of mutation and crossover;
6. Recombination and population selection for building the next generation. The crowding distance used in the NSGA II algorithm can estimate the density of the solutions surrounding a particular solution. In the NSGA II, the solutions with higher crowding distance are selected for the next population. The crowding distance is fundamental because it prevents the algorithm from searching for local minima instead of global ones. The steps to find the crowding distance are described in [28]. In the case under consideration, to describe the conductivity of low-GWP refrigerants, we started from Equation (1) by optimizing the coefficients \(\lambda_{0}\), \(a\), \(b\), \(c\), \(d\), \(f_{0}\), \(f\), and \(g\) through the NSGA II genetic algorithm. For the initialization part and the i  
Figure 2: Non dominated selection [28]; \(R_{t}=P_{t}\cup Q_{t}\) is the total population, \(F_{i}\) is the non-dominated set, and \(P_{t+1}\) is the chosen subsequent nondominated fronts. algorithm, we used a commercial program called ModeFrontier [38]. In particular, the following objective functions were chosen:  
1. Average Absolute Relative Deviation \[\text{AARD}\%=\frac{100}{N}\sum_{i=1}^{n}\frac{\left|\lambda_{L,i}^{exp}-\lambda _{L,i}^{calc}\right|}{\lambda_{L,i}^{exp}}\] (2)
2. Coefficient of Determination \[\text{R}^{2}=1-\frac{\sum_{i=1}^{n}\left(\lambda_{L,i}^{exp}-\lambda_{L,i}^{ calc}\right)^{2}}{\sum_{i=1}^{n}\left(\lambda_{L,i}^{exp}-\overline{\lambda}_{L} \right)^{2}}\] (3) where \(\lambda_{L}^{exp}\) is the experimental liquid thermal conductivity, \(\lambda_{L}^{calc}\) is the calculated liquid thermal conductivity, \(\overline{\lambda}_{L}\) is the mean of experimental liquid thermal conductivity, and \(n\) is the number of experimental points. The two objectives conflict with each other; the optimization algorithm minimized AARD% and maximized \(\text{R}^{2}\) at the same time. As explained in the introduction, the 2074 points of the dataset defined by [PERSON] et al. [20] were used. The eight coefficients of the equation were optimized with the following bounds: \(\lambda_{0}\in[0,1]\), \(a\in[-0.5,0.5]\), \(b\in[0,0.5]\), \(c\in[0,0.5]\), \(d\in[0,0.5]\), \(g\in[0,1]\), \(f\in[-1,0.1]\), and \(f_{1}\in[0,0.15]\). For the initialization of the optimization process, 80 points chosen as follows were used:  
* 10 points by a random process;
* 10 points via a pseudo-random Sobol sequence [39];
* 60 points using the incremental space filler algorithm. This algorithm is very useful in creating the DoE to generate a uniform distribution of data in the input space [40]. The following operating parameters were used to implement the NSGA2 algorithm:
* Number of generations: 100;
* Crossover probability: 0.9;
* Mutation probability: 1. As a final step, ten algorithms for choosing the optimal solution were compared using the tool available in [31; 32]. To obtain the best results, the objective functions were given weights. The obtained results are reported in Table 1. The first attempt was to assign the same weight to the two objective functions, but then, we noticed that the coefficient of determination varied only to the fourth decimal digit, and it was decided to give more importance to the AARD% by using the following weights: 80% for AARD% and 20% for \(\text{R}^{2}\).