# Large-Scale 3D Multi-Story Enterprise Layout Design in a New Type of Industrial Park in China
## 4 Bi-Level Programming Model Formulation
### Bi-Level Integer Programming Model  
In this subsection, the BLINLP model is defined using Equations (11)-(25).  
\[\begin{split}\Psi_{1}&=\text{Min}\sum\limits_{b\in B} \sum\limits_{f\in F}\sum\limits_{i\in I}R_{fi}*\gamma^{f}\bigg{(}\sum\limits_{ s\in S}x_{bsf}N_{s}-\sum\limits_{s\in S}x_{bsi}N_{s}\bigg{)}*x_{bfi}+\\ &\sum\limits_{b\in B}\sum\limits_{e\in E}\sum\limits_{i\in I}R_{ei}* \gamma^{e}\bigg{(}\sum\limits_{s\in S}x_{bse}N_{s}-\sum\limits_{s\in S}x_{bsi }N_{s}\bigg{)}*x_{bei}+\\ &\sum\limits_{b\in B}\sum\limits_{o\in O}\sum\limits_{i\in I}R_{oi}* \gamma^{o}\bigg{(}\sum\limits_{s\in S}x_{bso}N_{s}-\sum\limits_{s\in S}x_{bsi }N_{s}\bigg{)}*x_{boi}\end{split} \tag{11}\]  
s.t.  
\[\sum\limits_{i\in I}A_{i}*x_{bsi}\leq Q_{bs}\ \ \forall\ b\in B,s\in S \tag{12}\]  
\[\sum\limits_{b\in B}\sum\limits_{s\in S}x_{bsf}=1\ \ \forall\ f\in F \tag{13}\]  
\[\sum\limits_{b\in B}\sum\limits_{s\in S}x_{bse}=1\ \ \forall\ e\in E \tag{14}\]  
\[\sum\limits_{b\in B}\sum\limits_{s\in S}x_{bso}=1\ \ \forall\ o\in O \tag{15}\]  
\[\sum\limits_{b\in B}\sum\limits_{s\in S}x_{bsi}=1\ \ \forall\ i\in I \tag{16}\]  
\[\sum\limits_{s\in S}x_{bsf}=\sum\limits_{s\in S}x_{bsi}\ \ \forall\ b\in B,f\in F,i\in I,f=i \tag{17}\]  
\[\sum\limits_{s\in S}x_{bse}=\sum\limits_{s\in S}x_{bsi}\ \ \forall\ b\in B,e\in E,i\in I,e=i \tag{18}\]  
\[\sum\limits_{s\in S}x_{bso}=\sum\limits_{s\in S}x_{bsi}\ \ \forall\ b\in B,o\in O,i\in I,o=i \tag{19}\]\[x_{bfi} =\text{Min}\left\{\sum_{s\in S}x_{bsf},\sum_{s\in S}x_{bsi}\right\} \ \forall\ b\in B,f\in F,i\in I \tag{20}\] \[x_{bei} =\text{Min}\left\{\sum_{s\in S}x_{bser},\sum_{s\in S}x_{bsi}\right\} \ \forall\ b\in B,e\in E,i\in I\] (21) \[x_{bi} =\text{Min}\left\{\sum_{s\in S}x_{bsor},\sum_{s\in S}x_{bsi}\right\} \ \forall\ b\in B,o\in O,i\in I\] (22) \[\Psi_{2} =\text{Max}\sum_{b\in B}\sum_{s\in S}\sum_{j\in J}Z_{js}*x_{bsj}*A_{j}\] (23) s.t. \[\sum_{j\in J}A_{j}*x_{bsj} \leq Q_{bs}-\sum_{i\in I}A_{i}*x_{bsi}\ \ \forall\ b\in B,s\in S\] (24) \[\sum_{b\in B}\sum_{s\in S}x_{bsj}=1\ \ \forall\ j\in J \tag{25}\]  
In this model, Equations (11)-(22) define the optimization model of the upper-level problem. Equation (11) minimizes the overall association risk of the park. Constraint (12) guarantees that the sum of the enterprise areas of each floor cannot exceed this monolayer area. Constraints (13)-(16) restrict an enterprise to having only one specific location and thus cannot be at multiple locations. Constraints (17)-(19) guarantee that each enterprise can be assigned to only one building. Constraints (20)-(22) represent the relationships between the decision variables. The optimization problem of the lower level is defined by Equations (23)-(25). Specifically, Equation (23) describes the objective function of the lower-level problem as the maximization of rental profit for the park. Constraint (24) prevents enterprises from overlapping. Constraint (25) guarantees that each enterprise is located in one location.  
In addition, the proposed model allows some enterprises' locations that are given in advance, which is more realistic in the industry park. If the current enterprises' locations are provided, constraints can be added to the model to ensure the accuracy of layout results. Consequently, the current layout of the industry park is considered when applying the proposed model for designing the next stage layout.