## 4 Mathematical Model
### Objective Function
#### 4.3.1 Stacker Energy Consumption Objective  
In the context of this paper, different AGV scheduling schemes determine different horizontal driving speeds of stackers. Therefore, before giving the expression for \(F_{1}\), it is necessary to first clarify the relationship between the energy consumption of a single operation of the stacker and the horizontal driving speed of that operation. Firstly, we provide a simple Inference 1 directly without proof:  
**Inference 1**. : _The energy consumption difference of a stacker completing the same task at different speeds is directly proportional to the difference in the square of the horizontal driving speed._  
When we release the constraint of the demand time window for stacker retrieval tasks, there must be a unique minimum operating energy consumption of the stacker subsystem for each direct solution. Therefore, we can use the difference between the actual energy consumption and the theoretical minimum energy consumption to characterize the stacker energy consumption objective. Furthermore, we provide the final expression for \(F_{1}\) based on Inference 1, as shown in Equation (5). \[F_{1}=\text{min}{\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}{\left({V^{2}(i,j)-V_{ 0}^{2}}\right)}}=\text{min}{\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}{f_{1}(i,j)}} \tag{5}\]  
\(V(i,j)\) represents the value of the horizontal driving speed of stacker \(i\) to retrieve task \(j\), and \(f_{1}(i,j)\) represents the unit objective function value of stacker \(i\) to retrieve task \(j\). Since the waiting time for AGV pickup is determined by an external factor, the operating strategy of the stacker is necessary to provide an exact calculation method for \(V(i,j)\) to ensure the computability of function \(SL\). The calculation method for \(V(i,j)\) is shown as follows. Firstly, we let \(C(i,j)\) be the \(j\)-th retrieved container of the stacker \(i\), where \(T(i,j)\) represents the handover completed time of \(C(i,j)\), and \(T^{\prime}(i,j)\) represents the arrival time of the AGV picking up \(C(i,j)\). Meanwhile, \(t(i,j)=T(i,j)-T(i,j-1)\) represents the overall time for retrieving \(C(i,j)\), and \(t^{\prime}(i,j)=T^{\prime}(i,j)-T(i,j-1)\) represents demand time for task \(C(i,j)\). In addition, let \(\Delta t(i,j)=T(i,j)-T^{\prime}(i,j)\) represent the waiting time for AGV to pick up container \(C(i,j)\). For each container \(C(i,j)\), its inherent storage location attribute corresponds to three inherent time attributes, as shown in Equations (6)-(8), in which \(t_{best}(i,j)\) represents theoretical optimal operation time, \(t_{xmin}\) represents the minimum operation time considering the horizontal movement speed limit, \(t_{ymin}\) represents the minimum operation time considering vertical movement speed limit, \(x(i,j)\) represents the horizontal coordinates of the \(C(i,j)\) storage location, and \(y(i,j)\) represents the vertical coordinates of the \(C(i,j)\) storage location. \[t_{best}(i,j)=2\frac{x(i,j)}{V_{x0}} \tag{6}\]  
\[t_{xmin}(i,j)=2\frac{x(i,j)}{V_{xm}} \tag{7}\]  
\[t_{ymin}(i,j)=2\frac{y(i,j)}{V_{y}} \tag{8}\]  
On the premise of fully discussing all possible size relationships among \(t_{best}(i,j)\), \(t_{xmin}\), \(t_{ymin}\), we provide a calculation method for \(f_{1}(i,j)\), as shown in Equation (9), and an update method of \(T(i,j)\) that supports iteration, as shown in Equation (10). \[f_{1}(i,j)=\left\{ \begin{array}{l}{0,t_{best}(i,j)<\text{max}(t_{ymin},t_{xmin },t^{\prime}(i,j))}\\ {4\frac{x^{2}(i,j)}{t^{2}(i,j)}-V_{0}^{2},\text{max}(t_{ymin},t_{xmin})<t^{ \prime}(i,j)<t_{best}(i,j)}\\ {V_{xm}^{2}-V_{0}^{2},t^{\prime}(i,j)<\text{max}(t_{ymin},t_{xmin})<t_{best}(i,j)\delta t_{ymin}<t_{xmin}}\\ {\frac{x^{2}(i,j)}{y^{2}(i,j)}V_{y}^{2}-V_{0}^{2},t^{\prime}(i,j)<\text{max}( t_{ymin},t_{xmin})<t_{best}(i,j)\delta t_{xmin}<t_{ymin}}\\ \end{array} \right. \tag{9}\]  
\[T(i,j)=\left\{