## 3 Methodology--Mathematical Modelling and Heuristic Optimization Method
#### 3.1.1 Extended Logistics Resource Optimization
The third part of the objective function includes the minimization of time frame deviance from the average time frame among matrix cells:  
\[\mathrm{TD}^{M}=\sum_{r=1}^{r_{max}}\Big{(}x^{MM}_{i\alpha\beta r}\overline{t ^{MM}_{i\alpha\beta}}-\overline{t^{MM}_{r}}\Big{)}\text{ and }\overline{t^{MM}_{r}}=\sum_{i=1}^{m}\sum_{j=1}^{n}\sum_{\alpha=1}^{ \alpha_{max}}\sum_{\beta=1}^{\beta_{max}}x^{MM}_{i\beta\beta r}\overline{t^{ MM}_{i\beta\beta}} \tag{4}\]  
where \(\overline{t^{MM}_{i\alpha\beta}}\) is average time frame of supply-demand between matrix cell \(i\) and matrix cell \(j\) for job \(\alpha\) in time frame \(\beta\), \(\overline{t^{MM}_{r}}\) is the average time frame of route \(r\), \(x^{MM}_{i\beta r}\) is the assignment matrix of supply-demands between matrix cell \(i\) and matrix cell \(j\) for job \(\alpha\) in time frame \(\beta\). \(x^{MM}_{i\beta r}\) takes value \(1\) if the supply-demand from matrix cell \(i\) to matrix cell \(j\) for job \(\alpha\) in time frame \(\beta\) is assigned to route \(r\), otherwise \(0\). The solutions of the above-described clustering problem are limited by the following two constraints related to time frame and capacity of AGVs:  
_Constraint 1_: We can define an upper limit for time frame deviance for each route and it is not permitted to exceed the upper limit of time frame deviance within route \(r\):  
\[\forall\;i,\alpha,\beta,r:\;\max_{i\alpha\beta r}\left|\overline{t ^{W}_{i\alpha\beta}}-\overline{t^{W}_{r}}\right|\leq t^{Wmax} \tag{5}\] \[\forall\;i,\alpha,\beta,r:\max_{i\alpha\beta r}\left|\overline{t ^{T}_{i\alpha\beta}}-\overline{t^{T}_{r}}\right|\leq t^{Tmax}\] (6) \[\forall\;i,j,\alpha,\beta:r:\max_{i\alpha\beta r}\left|\overline{ t^{MM}_{i\alpha\beta}}-\overline{t^{MM}_{r}}\right|\leq t^{MMmax} \tag{7}\]  
where \(t^{Wmax}\) is the upper limit of time frame deviance for supply-demands from component warehouse to matrix cells, \(t^{Tmax}\) is the upper limit of time frame deviance for supply-demands from tool storage to matrix cells, and \(t^{MMmax}\) is the upper limit of time frame deviance for supply-demands among matrix cells. _Constraint 2_: We can define the upper limit of AGVs' loading capacity and it is not permitted to exceed this upper limit after assigning suitable supply-demands from component warehouse to matrix cells, from tool storage to matrix cells and among matrix cells:  
\[\forall\;r:\sum_{i=1}^{m}\sum_{\alpha=1}^{\alpha_{max}}\sum_{\beta=1}^{\beta_ {max}}(x^{WMA}_{i\alpha\beta r}q^{WM}_{i\alpha\beta}+x^{TM}_{i\alpha\beta r}q^ {TM}_{i\alpha\beta})+\sum_{i=1}^{m}\sum_{j=1}^{n}\sum_{\alpha=1}^{\alpha_{max} }\sum_{\beta=1}^{\beta_{max}}x^{MM}_{i\beta\beta r}q^{MM}_{i\beta\beta}\leq C ^{max}_{r} \tag{8}\]  
where \(q^{WM}_{i\alpha\beta}\) amount of required component from the component warehouse to the matrix cell \(i\) for job \(\alpha\) in time frame \(\beta\), \(q^{TM}_{i\alpha\beta}\) amount of required component from the tool storage to the matrix cell \(i\) for job \(\alpha\) in time frame \(\beta\), \(q^{MM}_{i\alpha\beta}\) amount of required component from matrix cell \(i\) to the matrix cell \(j\) for job \(\alpha\) in time frame \(\beta\), and \(C^{max}_{r}\) is the upper limit of loading capacity of AGV in route \(r\).