## 4 Model Transformation and Solution Approaches
### Model Transformation
: _For \(i,i-1\in\mathbf{N}^{A}\), \(j\in\mathbf{S}_{l}^{A}\), \(m\in\mathbf{T}\), constraints (38) and (39) can be transformed into the linear form, as shown in Equation (65):_  
\[v_{i,j}^{ndd}=\quad\sum_{m=1}^{|\mathbf{T}|}\big{(}\lambda_{i,j,m}-\lambda_{i- 1,j,m}\big{)}\times o_{j,m}-\sum_{m=1}^{|\mathbf{T}|}\Big{(}\lambda_{l,j,m}^{c }\times z_{l,i-1}^{d}-\lambda_{i-1,j,m}\times z_{l,i-1}^{d}\Big{)}\times o_{j,m}, \tag{65}\]  
_where the meaning of constraints (37) and (38) is: for \(i,i-1\in\mathbf{N}^{A}\), \(j\in\mathbf{S}_{l}^{A}\), if \(z_{l,i-1}^{a}=0\), then_  
\[v_{i,j}^{ndd}=\sum_{m=1}^{|\mathbf{T}|}\big{(}\lambda_{i,j,m}-\lambda_{i-1,j,m }\big{)}\times o_{j,m}\text{, and if }z_{l,i-1}^{a}=1\text{, then }v_{i,j}^{ndd}=\sum_{m=1}^{|\mathbf{T}|}\Big{(} \lambda_{i,j,m}-\lambda_{l,j,m}^{c}\Big{)}\times o_{j,m}.\]  
_Based on this, we obtain:_  
\[v_{i,j}^{add} =\Big{(}1-z_{l,i-1}^{a}\Big{)}\times\sum_{m=1}^{|\mathbf{T}|} \big{(}\lambda_{i,j,m}-\lambda_{i-1,j,m}\big{)}\times o_{j,m}+z_{l,i-1}^{a} \times\sum_{m=1}^{|\mathbf{T}|}\Big{(}\lambda_{i,j,m}-\lambda_{l,j,m}^{c}\Big{)} \times o_{j,m}\] \[=\sum_{m=1}^{|\mathbf{T}|}\big{(}\lambda_{i,j,m}-\lambda_{i-1,j,m }\big{)}\times o_{j,m}-\sum_{m=1}^{|\mathbf{T}|}\big{(}\lambda_{i,j,m}-\lambda _{i-1,j,m}\big{)}\times z_{l,i-1}^{a}\times o_{j,m}+\sum_{m=1}^{|\mathbf{T}|} \Big{(}\lambda_{i,j,m}-\lambda_{l,j,m}^{c}\Big{)}\times z_{l,i-1}^{a}\times o_ {j,m}\] \[=\sum_{m=1}^{|\mathbf{T}|}\big{(}\lambda_{i,j,m}-\lambda_{i-1,j,m }\big{)}\times o_{j,m}-z_{l,i-1}^{a}\times\sum_{m=1}^{|\mathbf{T}|}\Big{(} \lambda_{l,j,m}^{c}-\lambda_{i-1,j,m}\Big{)}\times o_{j,m}\] \[=\sum_{m=1}^{|\mathbf{T}|}\big{(}\lambda_{i,j,m}-\lambda_{i-1,j,m }\big{)}\times o_{j,m}-\sum_{m=1}^{|\mathbf{T}|}\Big{(}\lambda_{l,j,m}^{c} \times z_{l,i-1}^{a}\times o_{j,m}-\lambda_{i-1,j,m}\times z_{l,i-1}^{a}\times o _{j,m}\Big{)}\] \[=\sum_{m=1}^{|\mathbf{T}|}\big{(}\lambda_{i,j,m}-\lambda_{i-1,j,m }\big{)}\times o_{j,m}-\sum_{m=1}^{|\mathbf{T}|}\Big{(}\lambda_{l,j,m}^{c} \times z_{l,i-1}^{a}-\lambda_{i-1,j,m}\times z_{l,i-1}^{a}\Big{)}\times o_{j,m}\]  
**Lemma 7**. : _For \(i,i-1\in\mathbf{N}^{A}\), \(j\in\mathbf{S}_{l}^{A}\), constraint (40) can be transformed into the linear form, as shown in Equation (66) [22]:_  
\[\left\{ \begin{array}{l}v_{l,j}^{ndd}\leq\Big{(}1-z_{l,i-1}^{a}\Big{)}\times \mathrm{M}+\sum_{m=1}^{|\mathbf{T}|}\Big{(}\lambda_{l,j,m}^{c}-\lambda_{i-1,j, m}\Big{)}\times o_{j,m}\\ v_{l,j}^{ndd}\geq\sum_{m=1}^{|\mathbf{T}|}\Big{(}\lambda_{l,j,m}^{c}-\lambda_{i- 1,j,m}\Big{)}\times o_{j,m}-\Big{(}1-z_{l,i-1}^{a}\Big{)}\times\mathrm{M} \end{array} \right.. \tag{66}\]  
**Lemma 8**. : _For \(i,i-1\in\mathbf{N}^{A}\), \(j\in\mathbf{S}_{l}^{A}\), constraints (42) and (43) can be transformed into the linear form, as shown in Equation (67):_  
\[v_{i,j}^{r}=\Big{(}v_{i-1,j}^{r}+v_{i,j}^{ndd}-L_{i,j}^{n}\Big{)}+z_{l,i-1}^{a} \times\Big{(}v_{l,j}^{r}-v_{i-1,j}^{r}\Big{)}, \tag{67}\]  
**Lemma 13**. : _For \(j\in\mathbf{S}_{l}^{A}\), constraint (57) can be transformed into the linear form, as shown in Equation (72):_  
\[\left\{ \begin{array}{l}L_{l,j}^{n}\geq L_{l,j}^{n1}-\mathrm{M}\times(1-y_{l,j} )\\ L_{l,j}^{n}\geq L_{l,j}^{n2}-\mathrm{M}\times y_{l,j}\\ L_{l,j}^{n}\leq L_{l,j}^{n1}\\ L_{l,j}^{n}\leq L_{l,j}^{n2}\\ y_{l,j}\in\{0,1\}\end{array} \right.. \tag{72}\]  
_where \(y_{l,j}\) is the auxiliary variable, and the derivation of the lemma is consistent with Lemma 11._  
Combining lemmas above, Model I can be further transformed into the form shown as Equation (73) (**Model II**, for short), which is a typical mixed integer programming (MIP) model with quadratic terms  
\[