## 4 Model Transformation and Solution Approaches
### Model Transformation  
For model transformation, we propose the following lemmas using the large m method, formula derivations, auxiliary variables, and auxiliary constraints. **Lemma 1**. : _For \(i,i-1\in\mathbf{N}^{A}\), \(j\in\mathbf{S}^{A}_{l}\), constraints (7)-(12) can be transformed into the linear form shown in Equation (60) [22]:_  
\[\left\{ \begin{array}{l}a_{l,j}-a_{i-1,j}\geq I^{h}_{a}-(1-z^{a}_{l,i-1})\times \text{M}\\ a_{i,j}-a_{l,j}\geq I^{h}_{a}-(1-z^{a}_{l,i-1})\times\text{M}\\ d_{l,j}-d_{i-1,j}\geq I^{h}_{a}-(1-z^{a}_{l,i-1})\times\text{M}\\ d_{i,j}-d_{l,j}\geq I^{h}_{a}-(1-z^{a}_{l,i-1})\times\text{M}\\ a_{l,j}-d_{i-1,j}\geq I^{p}_{a}-(1-z^{a}_{l,i-1})\times\text{M}\\ a_{l,j}-d_{l,j}\geq I^{p}_{a}-(1-z^{a}_{l,i-1})\times\text{M}\\ z^{a}_{l,i-1}\in\{0,1\}\end{array} \right.. \tag{60}\]  
**Lemma 2**. : _For \(i,i-1\in\mathbf{N}^{B}\), \(j\in\mathbf{S}^{B}_{l}\), constraints (13)-(18) can be transformed into the linear form shown in Equation (61) [22]:_  
\[\left\{ \begin{array}{l}a_{l,j}-a_{i-1,j}\geq I^{h}_{b}-(1-z^{b}_{l,i-1})\times \text{M}\\ a_{i,j}-a_{l,j}\geq I^{h}_{b}-(1-z^{b}_{l,i-1})\times\text{M}\\ d_{l,j}-d_{i-1,j}\geq I^{h}_{b}-(1-z^{b}_{l,i-1})\times\text{M}\\ d_{i,j}-d_{l,j}\geq I^{h}_{b}-(1-z^{b}_{l,i-1})\times\text{M}\\ a_{l,j}-d_{i-1,j}\geq I^{p}_{b}-(1-z^{b}_{l,i-1})\times\text{M}\\ a_{i,j}-d_{l,j}\geq I^{p}_{b}-(1-z^{b}_{l,i-1})\times\text{M}\\ z^{b}_{l,i-1}\in\{0,1\}\end{array} \right.. \tag{61}\]  
**Lemma 3**. : _For \(i\in\{\mathbf{N}^{A},\mathbf{N}^{B}\}\), \(j\in\mathbf{S}_{i}\), constraint (27) can be transformed into the linear form shown in Equation (62):_  
\[\left\{ \begin{array}{l}\sum_{i,j}\Bigl{(}b^{a}_{i,j}+b^{d}_{i,j}\Bigr{)}\leq (1+\gamma)\times T^{\text{min\_dev}}\\ A_{i,j}-a_{i,j}\leq b^{a}_{i,j}\\ a_{i,j}-A_{i,j}\leq b^{a}_{i,j}\\ D_{i,j}-d_{i,j}\leq b^{d}_{i,j}\\ d_{i,j}-D_{i,j}\leq b^{d}_{i,j}\\ a_{i,j},d_{i,j},b^{a}_{i,j},b^{d}_{i,j}\in\mathbf{Z}^{+}\end{array} \right.. \tag{62}\]  
_where \(b^{a}_{i,j}\) and \(b^{d}_{i,j}\) are auxiliary variables. Combining the minimization feature of the objective function (Equation (58), conditions \(A_{i,j}-a_{i,j}\leq b^{a}_{i,j}\) and \(a_{i,j}-A_{i,j}\leq b^{a}_{i,j}\) make \(b^{a}_{i,j}\) equal to \(\left|a_{i,j}-A_{i,j}\right|\)[34;35], and constraints \(D_{i,j}-d_{i,j}\leq b^{d}_{i,j}\) and \(d_{i,j}-D_{i,j}\leq b^{d}_{i,j}\) make \(b^{d}_{i,j}\) equal to \(\left|d_{i,j}-D_{i,j}\right|\)._  
**Lemma 4**. : _For \(i,i-1\in\mathbf{N}^{B}\), \(j=\min(\mathbf{S}^{B}_{l})\), constraints (32) and (33) can be transformed into the linear form, as shown in Equation (63) [22]:_  
\[\left\{ \begin{array}{l}d_{l,j}-d_{i-1,j}\geq\max(I^{h}_{b^{\prime}},I^{front }_{b})-(1-z^{b}_{l,i-1})\times\text{M}\\ d_{i,j}-d_{l,j}\geq\max(I^{h}_{b^{\prime}},I^{back}_{b})-(1-z^{b}_{l,i-1}) \times\text{M}\\ z^{b}_{l,i-1}\in\{0,1\}\end{array} \right.. \tag{63}\]  
**Lemma 5**. : _For \(j\in\mathbf{S}_{l}^{A}\), \(m\in\mathbf{T}\), constraints (35) and (36) can be transformed into the linear form, as shown in Equation (64) [34]:_  
\[\left\{ \begin{array}{l}t_{m}-d_{l,j}\leq\mathrm{M}\times(1-\lambda_{l,j,m}^{c} )\\ d_{l,j}-t_{m}\leq\mathrm{M}\times\lambda_{l,j,m}^{c}-1\\ t_{m}-d_{i,j}\leq\mathrm{M}\times(1-\lambda_{i,j,m})\\ d_{i,j}-t_{m}\leq\mathrm{M}\times\lambda_{i,j,m}-1\\ \lambda_{l,j,m}^{c}\in\{0,1\}\\ \lambda_{i,j,m}^{c}\in\{0,1\}\end{array} \right.. \tag{64}\]  
**Lemma 6**.