## 4 Materials and Methods
### Model Formulation
Constraint set Equation (3) determines a farmer's probability to select a hub or a demand point according to the [PERSON]'s gravity model. Constraint set Equation (4) establishes the quantities of product \(k\) delivered by farmer \(i\) to hub \(h\). Constraint set Equation (5) corresponds to the balance of quantities of product \(k\) traded by farmer \(i\). Constraints set Equation (6) is the hub capacity constraint. Constraint set Equation (7) guarantees that the quantities delivered from hubs to the demand point are not greater than the quantities received by the hub. Constraint setsEquations (8) and (9) ensure the minimum and maximum occupancy levels per vehicle type. Constraint set Equation (10) determines the quantities sent from hub \(h\) to each demand point \(j\) according to the gravity model's probability. Constraint sets Equations (11) and (12) refer to the non-negativity and binary constraints of the decision variables. \[y_{ih}=\frac{\frac{\frac{\omega_{h}^{j}}{d_{h}^{j}}}{d_{h}^{j}}x_{h}}{\sum_{h^{ \prime}\in H}\frac{\omega_{h^{\prime}}^{j^{\prime}}}{d_{h^{\prime}}^{j^{ \prime}}}x_{h^{\prime}}+\sum_{j\in J}\frac{\omega_{j}^{j}}{d_{h^{\prime}}^{j}}} \forall i\in I;h\in H \tag{3}\]  
\[ \begin{split}\sigma_{ki}y_{ih}=s_{kih}\quad\quad\forall k\in K; i\in I;h\in H\\ \sum_{h\in H}s_{kih}+\hat{s}_{ki}=o_{ki}\quad\forall i\in I;k\in K \end{split}  \tag{4}\]  
\[\sum_{k\in K}\sum_{i\in I}s_{kih}\leq q_{h}x_{h}\quad\quad\forall h\in H \tag{6}\]  
\[\sum_{i\in I}s_{kih}\geq\sum_{j\in J}\sum_{v\in V}s^{\prime}_{khjv}\quad\quad \forall k\in K;h\in H \tag{7}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Sets** & \\ \hline \(I\) & Farmers’ locations. \\ \(H\) & Potential location of hubs. \\ \(J\) & Demand points’ locations. \\ \(K\) & Products. \\ \(L\) & Agricultural chains’ classification. \\ \(V\) & Type of vehicle. \\ \hline \hline
**Parameters** & \\ \hline \(c_{h}\) & Operation fixed cost of hub \(h\). \\ \(q_{h}\) & Capacity of hub \(h\). \\ \(margin_{k}\) & Unit profit margin of product \(k\). \\ \(c_{p}\) & Operation fixed cost for each vehicle type \(v\). \\ \(c_{p}\) & Variable cost per kilometer and ton of each vehicle type \(v\). \\ \(e_{p}\) & Average carbon dioxide emission in tons per kilometer for vehicle \(v\). \\ \(c_{p}\) & Unit cost of carbon dioxide emission per kilometer by vehicle \(v\). \\ \(q_{v}\) & Capacity of vehicle \(v\). \\ \(min_{v}\) & Minimum occupancy of vehicle \(v\) (\%). \\ \(\bar{w}_{h}\) & Population of hubs’ location \(h\). \\ \(w_{i}^{\prime}\) & Population of demand point \(j\). \\ \(d_{ih}^{j}\) & Distance between farmer \(i\) and hub \(h\). \\ \(d_{ij}\) & Distance between farmer \(i\) and demand point \(j\). \\ \(d_{hj}\) & Distance between hub \(h\) and demand point \(j\). \\ \(o_{ki}\) & Weekly quantities of product \(k\) by sold by farmer \(i\). \\ \(a_{kl}\) & Binary parameter that establishes if product \(k\) belongs to agricultural chain \(l\). \\ \(\gamma\) & Sensitivity exponent for the distance between farmers and hubs. \\ \(\hat{\gamma}\) & Sensitivity exponent for the distance between hubs and cities. \\ \(\delta\) & Sensitivity exponent for population factor. \\ \(p_{hj}\) & Probability of hub \(h\) of selecting demand point \(j\). \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Definition of indices and parameters. \begin{table}
\begin{tabular}{l c} \hline \hline  & **Decision Variables** \\ \hline \(x_{h}\) & Binary variable \(i\) hub location \(h\) is open \\ \(\hat{s}_{ki}\) & Weekly quantities (tors) of product \(k\) sent from farmer \(i\) to demand points \\ \(y_{ih}\) & Probability of farmer \(i\) of choosing to go to hub \(h\) \\ \(s_{ihi}\) & Weekly quantities (tors) of product \(k\) sent from farmer \(i\) to hub \(h\) \\ \(s_{kijv}\) & Weekly quantities (tors) of product \(k\) sent from hub \(h\) to demand point \(j\) in vehicle \(v\) \\ \(trtck_{kij}\) & Number of vehicles \(v\) with products of agricultural chain \(l\) sent from hub \(h\) to demand point \(j\) \\ \hline \hline \end{tabular}
\end{table}