# Supply Chain Design for Waste Vaporization Through High-Energy-Density Pellet Production in Chile
## 2 Methods
By defining these possible future application contexts, the proposed supply chain design optimization model for HTC can be tailored to meet specific needs, enhancing its applicability and effectiveness. This approach ensures that the model is theoretically sound and practically viable, capable of addressing real-world challenges in diverse settings. The flexibility and adaptability of the model make it a valuable tool for advancing the implementation and commercialization of HTC technology across various sectors. The mathematical model establishes the basis for the practical application of the supply chain design model. The sets, decision variables, constraints, objective function, and parameters were defined, which allowed an accurate representation of the supply chain:  
* Sets: These include supplier locations (i \(\in\) I), plant locations (k \(\in\) K), and customers to satisfy (j \(\in\) J), among others. * Decision variables: These variables encompass the selection of plants and the flow of materials and products through the supply chain. * Restrictions: These consider mass balances, the availability of raw materials, production capacities, and the satisfaction of demand, among others. * Objective functions: These focus on minimizing transportation distance to reduce costs and emissions, reflecting the project's sustainability approach regarding economic and environmental aspects. * Parameters: These parameters include the raw materials available to suppliers, production capacity, rates of transformation of raw materials to final products, demand for final products, distances between suppliers and plants, transportation costs, and emissions, among others. These sets and parameters are shown in Tables 1 and 2. The solution methodology used the Microsoft Excel add-in solver [32], which was selected for its accessibility and ability to handle the number of variables involved in the model for the specific case study. This complement applies the SIMPLEX LP solving method to solve the linear programming model, ensuring an optimal configuration [33; 34]. \begin{table}
\begin{tabular}{c l} \hline \hline \multicolumn{1}{c}{**Set**} & \multicolumn{1}{c}{**Description**} \\ \hline \(n=\{1,\ \ldots,\ |N|\}\) & Raw material types correspond to the type of raw material available for transformation. \\ \(i=\{1,\ \ldots,\ |I|\}\) & Supplier locations correspond to each possible raw material supplier considered. \\ \(j=\{1,\ \ldots,\ |J|\}\) & Demand locations correspond to each potential customer considered. \\ \(k=\{1,\ \ldots,\ |K|\}\) & Production plant locations, corresponding to each possible production plant considered. \\ \(a=\{1,\ \ldots,\ |A|\}\) & Final product types, corresponding to each possible type of final product considered \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variable sets of the model. The objective function was to minimize the transported tons and the traveled kilometers associated with distributing raw materials and final products to their respective locations, according to Equation (1). This consequently reduces transportation costs and environmental emissions [35]. Lower transportation costs are achieved through shorter routes and optimized logistics, reducing fuel consumption and emissions. This approach indirectly aligns with environmental goals by focusing on cost efficiency, which limits environmental impact [25; 35; 36]. \[Min\ Z=\sum_{n}\sum_{i}\sum_{k}N_{n,i,k}*DistPP_{i,k}+\sum_{a}\sum_{k}\sum_{j}P _{a,k,j}*DistPC_{k,j}, \tag{1}\]  
where \(N_{n,i,k}\) is the flow of tons of raw material of type \(n\) from the supplier located at \(i\) to the main production plant located at \(k\), and \(P_{a,k,j}\) is the flow of tons of final products of type \(a\) from the main production plant located at \(k\) to the customer located at \(j\). The constraints associated with the objective function are detailed in Table 3, where \(W_{k}\) is equal to 1 if the main production plant is built at location k and 0 if not.