# MILP-Based Approach for High-Altitude Region Pavement Maintenance Decision Optimization
## 4 Model Transformation
### Transformation of Nonlinear Terms  
The presence of nonlinear terms increases the complexity of solving the model, and linearizing them aids in solving the model. Many scholars also often adopt exact algorithms or solvers to solve linear programming problems. For the initial model, five lemmas are proposed. **Lemma 1**. : _For Constraint (1), which is in the form of a piecewise function, it can be equivalently transformed by introducing an auxiliary variable \(\;v_{i,u}\) and a sufficiently large positive number \(\;\text{M}\) as follows:_  
\[\left\{ \begin{array}{l}s_{i,u}\leq S_{i,0}\times e^{\alpha_{i,1}\times u+ \beta_{i,1}}+\sum\limits_{m\in\textbf{M}}\sum\limits_{u\in\{u\ast\leq u\}} \sum\limits_{j\in\textbf{J}}x_{i,j}\times E_{j}\times y_{i,u\ast,m}\times e^{ \alpha_{i,1}\times(u-u\ast)+\beta_{i,1}},\forall i\in\textbf{I},u\in\textbf{U} \\ s_{i,u}\leq S_{\max},\forall i\in\textbf{I},u\in\textbf{U}\\ s_{i,u}\geq S_{\max}-\text{M}\times(1-v_{i,u}),\forall i\in\textbf{I},u\in \textbf{U}\\ s_{i,u}\geq S_{i,0}\times e^{\alpha_{i,1}\times u+\beta_{i,1}}+\sum\limits_{m \in\textbf{M}}\sum\limits_{u\in\{u\ast\leq u\}}\sum\limits_{j\in\textbf{J}} x_{i,j}\times E_{j}\times y_{i,u\ast,m}\times e^{\alpha_{i,1}\times(u-u\ast)+\beta_{i,1}}- \text{M}\times v_{i,u},\forall i\in\textbf{I},u\in\textbf{U}\\ v_{i,u}\in\{0,1\}\end{array} \right. \tag{20}\]  
**Lemma 2**. : _For Constraint (4), which includes logarithmic terms, both sides can be transformed into a linear form by taking their logarithms (i.e., \(\;\ln(r_{i,u})=\ln(\alpha_{i,2})+\beta_{i,2}\times s_{i,u}\)). Further, let \(\;r^{\prime}_{i,u}=\ln(r_{i,u})\), then we have:_  
\[r^{\prime}_{i,u}=\ln(\alpha_{i,2})+\beta_{i,2}\times s_{i,u} \tag{21}\]  
**Lemma 3**. : _For Objective function (18), combining Lemma 2 and the characteristics of the objective function, it can be transformed into the form of Equation (22)._  
\[\min f^{\prime}_{4}=\sum\limits_{u\in\textbf{U}}\sum\limits_{i\in\textbf{I}}r^ {\prime}_{i,u} \tag{22}\]  
**Lemma 4**. : _For the quadratic terms involving multiple decision variables (i.e.,\(x_{i,j}\times y_{i,u,m}\)), based on the properties of the decision variables, both types of variables are 0-1 variables. Therefore, as shown in Table 3, if we let \(\;z_{i,j,u,m}=x_{i,j}\times y_{i,u,m}\), then the value of \(\;z_{i,j,u,m}\) can be determined accordingly._  
Based on the contents in Table 3, the linear expression, \(z_{i,j,u,m}=x_{i,j}\times y_{i,u,m}\), can be obtained as follows:  
\[\left\{ \begin{array}{l}z_{i,j,u,m}\leq x_{i,j},\forall i\in\textbf{I},j\in \textbf{J},u\in\textbf{U},m\in\textbf{M}\\ z_{i,j,u,m}\leq y_{i,u,m},\forall i\in\textbf{I},j\in\textbf{J},u\in\textbf{U},m\in\textbf{M}\\ z_{i,j,u,m}\geq x_{i,j}+y_{i,u,m}-1,\forall i\in\textbf{I},j\in\textbf{J},u\in \textbf{U},m\in\textbf{M}\\ x_{i,j},y_{i,u,m},z_{i,j,u,m}\in\{0,1\},\forall i\in\textbf{I},j\in\textbf{J}, u\in\textbf{U},m\in\textbf{M}\\ \end{array} \right. \tag{23}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & \(x_{i,j}\) & \(y_{i,u,m}\) & \(z_{i,j,u,m}\) \\ \hline
2 & 0 & 1 & 0 \\
3 & 1 & 0 & 0 \\
4 & 1 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Analysis of the value of \(z_{i,j,u,m}\). **Lemma 5**. : _For the model Objective function (17) and the integer signs involved in Constraint (7), the auxiliary variable \(h_{i,j}\) is introduced to replace \(\left\lceil\frac{G_{j}\times x_{i,j}\times L_{i}\times B_{j}}{Q\times O}\right\rceil\) within higher precision. In this case, the additional constraints are:_  
\[\left\{ \begin{array}{l}h_{i,j}\geq\frac{G_{j}\times x_{i,j}\times L_{i}\times B _{j}}{Q\times O},\forall i\in\mathbf{I},j\in\mathbf{J}\\ h_{i,j}\leq\frac{G_{j}\times x_{i,j}\times L_{i}\times B_{j}}{Q\times O}+0.999,999,\forall i\in\mathbf{I},j\in\mathbf{J}\end{array} \right. \tag{24}\]  
Due to the introduction of \(h_{i,j}\), this paper introduces another situation of the product of decision variables.