## 3 Methodology--Mathematical Modelling and Heuristic Optimization Method
#### 3.1.1 Extended Logistics Resource Optimization  
The extended optimization of supply resources in matrix production can be divided into two main phases. The first phase is a clustering phase, where the available supply-demands are clustered based on time frame related objective function. The second phase represents a vehicle routing problem, where the clustered supply-demands are scheduled. Clustering of Supply-Demand  
The objective function of the clustering of supply-demands is the minimization of the total time deviance of supply-demands from the average time frame, which can be calculated as the sum of supply-demands assigned to route \(r\) in relations warehouse-matrix cell, tool storage-matrix cell and among matrix cells:  
\[\mathrm{TD}^{WTM}=\mathrm{TD}^{W}+\mathrm{TD}^{T}+\mathrm{TD}^{M}\to min. \tag{1}\]  
where \(\mathrm{TD}^{W}\) is the time deviance of clustered supply-demands from component warehouse to matrix cells, \(\mathrm{TD}^{T}\) is the time deviance of clustered supply-demands from tool storage to matrix cells, and \(\mathrm{TD}^{M}\) is the time deviance of clustered supply-demands among matrix cells. The first part of the objective function includes the minimization of time frame deviance from the average time frame between the component warehouse and matrix cells:  
\[\mathrm{TD}^{W}=\sum_{r=1}^{r_{\mathrm{max}}}\left(\mathrm{x}_{\mathrm{i} \beta r}^{WM}\overline{\mathrm{t}_{\mathrm{i}\beta}^{W}}-\overline{\mathrm{t }_{r}^{W}}\right)\text{ and }\overline{\mathrm{t}_{r}^{W}}=\sum_{i=1}^{m}\sum_{ \alpha=1}^{a_{\mathrm{max}}}\sum_{\beta=1}^{\beta_{\mathrm{max}}}\mathrm{x}_{ \mathrm{i}\beta r}^{WM}\overline{\mathrm{t}_{\mathrm{i}\beta}^{W}} \tag{2}\]  
where, \(\overline{\mathrm{t}_{\mathrm{i}\beta}^{W}}\) is average time frame of supply-demand from the component warehouse to matrix cell \(j\) for job \(\alpha\) in time frame \(\beta\), is the average time frame of route \(r\), \(\mathrm{x}_{\mathrm{i}\beta r}^{WM}\) is the assignment matrix of supply-demands from component warehouse to matrix cells. \(\mathrm{x}_{\mathrm{i}\beta r}^{WM}\) takes value 1 if the supply-demand from component warehouse to the matrix cell \(i\) for job \(\alpha\) in time frame \(\beta\) is assigned to route \(r\), otherwise 0. The second part of the objective function includes the minimization of time frame deviance from the average time frame between tool storage and matrix cells:  
\[\mathrm{TD}^{T}=\sum_{r=1}^{r_{\mathrm{max}}}\left(\mathrm{x}_{\mathrm{i} \beta r}^{TM}\overline{\mathrm{t}_{\mathrm{i}\beta}}-\overline{\mathrm{t}_{ r}^{T}}\right)\text{ and }\overline{\mathrm{t}_{r}^{T}}=\sum_{i=1}^{m}\sum_{\alpha=1}^{a_{ \mathrm{max}}}\sum_{\beta=1}^{\beta_{\mathrm{max}}}\mathrm{x}_{\mathrm{i} \beta r}^{TM}\overline{\mathrm{t}_{\mathrm{i}\beta}^{T}} \tag{3}\]  
where, \(\overline{\mathrm{t}_{\mathrm{i}\alpha\beta}^{T}}\) is average time frame of supply-demand from the tool storage to matrix cell \(j\) for job \(\alpha\) in time frame \(\beta\), \(\overline{\mathrm{t}_{r}^{T}}\) is the average time frame of route \(r\), \(\mathrm{x}_{\mathrm{i}\beta r}^{TM}\) is the assignment matrix of supply-demandsfrom component warehouse to matrix cells. \(x^{TM}_{i\alpha\beta r}\) takes value \(1\) if the supply-demand from the tool storage to the matrix cell \(i\) for job \(\alpha\) in time frame \(\beta\) is assigned to route \(r\), otherwise \(0\).