## 2 Methods and Modeling
Determining item transportation between suppliers and warehouses
7. Determining item transportation between warehouses and factories
8. Determining the number of existing doors in each warehouse  
Considering the decision-making variables, the indices and parameters in this model are found in Table 2.  
The first objective function is Equation (1) which aims at minimizing location-fixed costs, transportation costs, and cross-dock costs. \[ \begin{array}{c}\text{Min}\;Z_{1}=\sum\limits_{i=1}^{1}\text{FS}_{i}\text{Si} _{i}+\sum\limits_{j=1}^{J}\text{FC}_{j}\text{C}_{Tj}+\sum\limits_{k=1}^{K} \text{FM}_{k}\text{PD}_{k}+\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\sum \limits_{v=1}^{V}\text{c}_{iyv}\text{G}_{iyv}\\ +\sum\limits_{j=1}^{J}\sum\limits_{v=1}^{V}\sum\limits_{k=1}^{K}\text{c}^{ \prime}_{iyv}\text{F}_{jkv}+\sum\limits_{j=1}^{J}\text{m}_{j}\text{u}_{j}\end{array}  \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Indices** & **Description** \\ \hline I & Suppliers index \\ J & Cross-dock index \\ K & Factory index \\ V & Vehicle index \\ \hline
**Parameters** & **Description** \\ \hline P\({}_{1}\) & Maximum number of allowed suppliers \\ P\({}_{2}\) & Maximum number of allowed cross-docks \\ P\({}_{3}\) & Maximum number of allowed factories \\ \(\lambda_{i}\) & Demand entrance rate from supplier i \\ \(\text{u}_{j}\) & Service rate for each sever in cross-dock j \\ \(\text{FS}_{i}\) & Fixed cost of locating supplier in potential node i \\ \(\text{FC}_{i}\) & Fixed cost of locating supplier in potential node j \\ \(\text{FM}_{k}\) & Fixed cost of locating factory in potential node k \\ \(\text{c}_{iyv}\) & Shipping cost from supplier i to cross-dock j via vehicle v \\ \(\text{c}_{iyv}^{\prime}\) & Shipping cost from cross-dock j to factory k via vehicle v \\ \(\text{t}_{iyv}^{\prime}\) & Shipping time from supplier i to cross-dock j via vehicle v \\ \(\text{t}_{ikv}^{\prime}\) & Shipping time from cross-dock j to factory k via vehicle v \\ \(\text{u}_{j}\) & Employing cost for each server in cross-dock j \\ \(\text{M}_{j}\) & Maximum number of servers who can be placed in cross-dock j \\ \hline \end{tabular}
\end{table} 
Table 2: Indices and parameters. \begin{table}
\begin{tabular}{c c} \hline \hline \(\text{S}_{i}\). & \(\left\{\begin{array}{ll}1,&\text{if a supplier is located in node i}\\ 0,&\text{otherwise}\end{array}\right.\) \\ \hline \(\text{C}_{\text{Tj}}\) & \(\left\{\begin{array}{ll}1,&\text{if a cross}-\text{dock is located in node j}\\ 0,&\text{otherwise}\end{array}\right.\) \\ \hline \(\text{PD}_{\text{k}}\) & \(\left\{\begin{array}{ll}1,&\text{if a factory is located in node k}\\ 0,&\text{otherwise}\end{array}\right.\) \\ \hline \(\text{A}_{\text{ij}}\) & \(\left\{\begin{array}{ll}1,&\text{if supplier i is allocated to cross}-\text{dock j}\\ 0,&\text{otherwise}\end{array}\right.\) \\ \hline \(\text{B}_{\text{jk}}\) & \(\left\{\begin{array}{ll}1,&\text{if cross}-\text{dock j}\text{ is allocated to factory k}\\ 0,&\text{otherwise}\end{array}\right.\) \\ \hline \(\text{S}_{\text{i}}\) & \(\left\{\begin{array}{ll}1,&\text{if a supplier is located in node i}\\ 0,&\text{otherwise}\end{array}\right.\) \\ \hline \(\text{G}_{\text{iyv}}\) & \(\left\{\begin{array}{ll}1,&\text{if vehicle v is selected for transferring goods from supplier i to cross}-\text{dock j}\\ 0,&\text{otherwise}\end{array}\right.\) \\ \hline \(\text{H}_{\text{jkv}}\) & \(\left\{\begin{array}{ll}1,&\text{if vehicle v is selected for transferring goods from cross}-\text{dock j}\text{ to factory k}\\ 0,&\text{otherwise}\end{array}\right.\) \\ \hline \(\text{m}_{\text{j}}\) & Number of existing servers in cross-dock j \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dependent and independent decision variables. Constraint sets (6)-(8) ensure the maximum number of permitted suppliers, cross-docks, and factories in order to avoid facility overlap. Constraint (9) insures that the number of existing servers in each cross-dock doesn't exceed the predetermined allowed number.