## 3 Algorithmic Framework
### Experimental Results
Subsequently, Set 2 underwent testing to assess the effectiveness of the proposed GA, involving 25,000 independent simulations per temperature (ranging from 5 \({}^{\circ}\)C to 25 \({}^{\circ}\)C) and various combinations of weights (w\({}_{1}\) and w\({}_{2}\)). This testing phase lasted for a duration of 3645.34 s (approximately 60.76 min or approximately 1 h). Table 5 presents these experimental results from temperatures 5 \({}^{\circ}\)C to 25 \({}^{\circ}\)C, with columns information: w\({}_{1}\) and w\({}_{2}\) weights combination, diesel ingredient (%) in the mixture, biodiesel ingredient percentage in the mixture, evaluation function value of mixture, mixture cost, and mixture density. Weights w\({}_{1}\) and w\({}_{2}\) always sum up to 100% and w\({}_{2}\)\(\geq\) 50% due to the Emphasis on Density criterion. Diesel ingredient percentage ranges between 74.89% and 75.02%, while biodiesel ingredient percentage amounts to 24.88-25.12%. Evaluation Function values range from \(-\)0.8852 to \(-\)0.0681. The negative evaluation function value emphasizes the density criterion rather than the cost criterion. Figure 5: Set 1: (**a**) Density Statistical Analysis and (**b**) Best Density Solutions Allocation. The Evaluation Function values in all groups are negative but close to 0 value (w\({}_{1}\) = 50% and w\({}_{2}\) = 50%), which results in a neutral criterion assessment. All the other groups offer clear priority to the density criterion. As regards cost criterion values, the new fuel mixtures cost between 1.6979 EUR/L and 1.7386 EUR/L. The last information concerns density criterion values range 0.8340 g/mL-0.8397 g/mL, satisfying the ASTM D1298-99 limits: 0.8200 g/mL-0.8450 g/mL. Each row in Table 5 provides details regarding the best mixture identified from 1000 independent simulations (resulting in a total of 25 optimal mixtures). The evaluation is centered around the minimum Total Mixture Function Value (TMFV). As the value of w\({}_{2}\) rises, the TMFV value also increases, indicating a greater emphasis on the density criterion over the cost criterion. There are slight variations in the percentages of ingredients, optimal mixture cost, and density among the optimal mixtures, as previously discussed. A statistical analysis of the optimal solution in Set 2 follows. Optimal Mixture in Set 2 (temperature 20 \({}^{\circ}\)C, w\({}_{1}\) = 10%, w\({}_{2}\) = 90%):  
* Diesel percentage: 74.88%
* Biodiesel percentage: 25.12%
* TMFV: \(-\)0.8852
* Mixture Cost: 1.7112 EUR/L
* Mixture Density: 0.8397 g/mL  
In a series of 1000 independent trials, the lowest Total Mixture Function Value (TMFV) recorded was \(-\)0.8852, while the highest TMFV reached was \(-\)0.8753. The difference between the minimum and maximum TMFV values is 0.0099. Meanwhile, the average TMFV across all trials stands at \(-\)0.8802, with a standard deviation of 0.0018 (as illustrated in Figure 6a). The effectiveness of the method is illustrated in Figure 6b, where the majority of optimal solutions, totaling 936 mixtures (comprising 93.6% of the total), fall within the range of \(-\)0.8852 to \(-\)0.8793 TMFV (with \(-\)0.8852 being the minimum TMFV). This outcome validates the GA's capability to generate solutions closely aligned with the minimum fitness function within a brief timeframe. Based on best solutions allocation, only 64 (\(64=34+25+5\)) mixtures (6.4%) were between \(-\)0.8792 and \(-\)0.8748 (maximum mixture TMFV \(-\)0.8753).