## 3 Algorithmic Framework
### GA--Chromosome Representation  
Example of Chromosome Representation: a random mixture made up of two ingredients, with percentages (\(\text{w}_{1}\), \(\text{w}_{2}\)) always adding up to 100%. For instance, if Diesel is 78.22% and then Biodiesel is 21.78%. The definitions and abbreviations of the main concepts that will be used hereunder are presented in Table 1. \begin{table}
\begin{tabular}{c c} \hline \hline
**Abbreviation** & **Concept Definition** \\ \hline Min (Max) Ingredient Percentage & Minimum (Maximum) Ingredient Percentage value. \\ \hline InPer & Ingredient Percentage. \\ \hline IPLS & Ingredient Percentage Local Search (in\%). \\ \hline I\({}_{1}\) & Ingredient Percentage Local Search Bound created by minimum ingredient percentage. \\ \hline I\({}_{2}\) & Ingredient percentage Local Search Bound created by maximum ingredient percentage. \\ \hline Max IPLS & Maximum IPLS value. \\ \hline Min IPLS & Minimum IPLS value. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Abbreviations /Definition of Main Concepts. Figure 1: Genetic Algorithm Operation. ### GA--Chromosome Representation  
The process of generating generations involves two distinct phases. Initially, the first generation is established using the random creation of viable blends. Subsequently, in the subsequent phases, the formation of chromosomes for the generations follows a three-part process (also illustrated in Figure 2):  
(a) Phase 1: A selection of the top-performing 10% of chromosomes from the current generation, known as the \"TOP Mixtures,\" is carried over to the next generation. (b) Phase 2: The next 60% of results are derived from the application of the Crossover operator, which produces new chromosomes. (c) Phase 3: The remaining 30% of chromosomes are generated via mutation, utilizing the same method employed for the initial population's formation. The creation of fuel mixtures involves determining the diesel and biodiesel proportions within each chromosome. Subsequently, a fitness function assesses each mixture based on criteria such as cost and density, thereby ordering all mixtures accordingly. The solutions generated via crossover and mutation adhere to specified ingredient percentage ranges (minimum%-maximum%), ensuring that the sum of ingredient percentages always equals 100%. This approach guarantees the production of viable solutions, with no exclusion of feasible options during the evaluation process. Following the establishment of the initial population, a framework is implemented, encompassing a range of \(\pm\)IPLS (Incremental Percentage of the Last Solution) from the best chromosome percentages. This framework, suggested by [PERSON] et al. (2017), facilitates the optimization process by focusing on the optimal solution around the best chromosome from the previous generation. Each generation randomly selects a new IPLS value within a defined range (e.g., Generation 1, IPLS: 6%; Generation 2, IPLS: 9%;... Generation 100, IPLS: 10%). This method targets the prime solution domain from the preceding generation, acquiring values within the range of:  
\[\{\text{min}IPLS=1\%:\text{max}\text{IPLS}=\text{max}InPer-(\text{min}InPer+1)\} \tag{3}\]  
The decision-maker sets IPLS boundaries that meet the requirements of minimum and maximum ingredient percentages:  
\[ \begin{array}{c}for\text{ }the\text{ }\text{IPLSB}_{1}:\text{IPLSB}_{1}\geq\text{min}InPer\\ and\text{ }for\text{ }the\text{ }\text{IPLSB}_{2}:\text{IPLSB}_{2}\leq\text{max}InPer \end{array}  \tag{4}\]  
The IPLSB1 and IPLSB2 limits derived from the process outlined above serve as the updated constraints for the new Biodiesel production. Enforcing these bounds ensures the creation of viable mixtures. The present fuel mixture approach provides the following advantages:  
Figure 2: New generation mixtures production. * Focusing on last-generation best solutions (10% N) and leveraging them in next-generation production. * The frame \(\pm\) I, which leads to a faster optimal solution convergence, implements the local search method.