## 4 Mathematical Model
### Solution Expression  
In Section 3.3, we propose an approach to transform optimal collaborative scheduling for the SACOP into the optimal AGV scheduling for the MMATSP. In this paper, the SACOP optimal collaborative scheduling scheme is referred to as the executable solution, and the MMATSP optimal AGV scheduling scheme is referred to as the direct solution. According to the previous description, the executable solution can be obtained through simulation calculation of the direct solution. For a clearer expression, we use \((\textbf{X},\textbf{Y})\) to represent the direct solution, use \(\textbf{Z}\) to represent the executable solution, and use \(\textbf{Z}=SL((\textbf{X},\textbf{Y}))\) to represent the process of obtaining the executable solution through simulation calculation of the direct solution, where the function \(SL\) represents the simulation calculation method.  
As shown in Equation (1), \((\textbf{X},\textbf{Y})\) is a direct solution expressed jointly by two \(M\times N\) matrices **X** and **Y**, where \(M\) represents the number of available AGVs and \(N\) represents the total number of order tasks. Corresponding to the intuitive expression of AGV task sequences, the order task sequences of \(M\) AGVs are obtained by removing zero elements from \(M\) row vectors extracted from matrices.  
\[\textbf{X}=\begin{pmatrix}X_{11}&\dots&X_{1N}\\ \vdots&\ddots&\vdots\\ X_{M1}&\cdots&X_{MN}\end{pmatrix},\ \textbf{Y}=\begin{pmatrix}Y_{11}&\dots&Y_{1N}\\ \vdots&\ddots&\vdots\\ Y_{M1}&\cdots&Y_{MN}\end{pmatrix} \tag{1}\]Among them: \(\bullet\)\(X_{ij}=\sum_{n=1}^{N}nx_{nij}\) represents the order number of task \(j\) of AGV \(i\); \(\bullet\)\(Y_{ij}=\sum_{w=1}^{W}wy_{x_{ij}w}\) represents the picking station number of task \(j\) of AGV \(i\); \(\bullet\) The decision variable \(x_{mmk}\) is equal to one, iff order \(n\) is assigned as task \(k\) of the AGV \(m\); otherwise, it is equal to zero; \(\bullet\) The decision variable \(y_{mw}\) is equal to one, iff order \(n\) is assigned to picking station \(w\); otherwise, it is equal to zero. The converted \(\mathbf{Z}\) through function \(SL\) is divided into three parts: stacker scheduling scheme \(\mathbf{Z_{1}}\), AGV scheduling scheme \(\mathbf{Z_{2}}\), and picking station scheduling scheme \(\mathbf{Z_{3}}\). Among them: \(\bullet\)\(\mathbf{Z_{1}}\) includes a list of retrieval tasks for each stacker, and each individual instruction of retrieval task \(j\) for stacker \(i\) includes three parts: container number \(C_{ij}\), AGV pickup time \(T_{ij}\), and stacker driving speed \(V_{ij}\), as shown in Equation (2). \[\mathbf{Z_{1}}=\begin{pmatrix}\{C_{11},T_{11},V_{11}\}&\dots&\{C_{1N},T_{1N},V_{1N} \}\\ \vdots&\ddots&\vdots\\ \{C_{S1},T_{S1},V_{S1}\}&\dots&\{C_{SN},T_{SN},V_{SN}\}\end{pmatrix}\] (2) \(\bullet\)\(\mathbf{Z_{2}}\) is the AGV running trajectory list that includes order numbers, node location information, and node entry-departure time, as shown in Equation (3). \[\mathbf{Z_{2}}=(\mathbf{L_{1}},\mathbf{L_{2}},\cdots,\mathbf{L_{M}})^{T},\;\mathbf{L_{m}}=\begin{pmatrix} O_{m1}&O_{m2}&\cdots&O_{mU}\\ x_{m1}&x_{m2}&\cdots&x_{mU}\\ y_{m1}&y_{m2}&\cdots&y_{mU}\\ T_{m1}&T_{m2}&\cdots&T_{mU}\\ T_{m1}^{\prime}&T_{m2}^{\prime}&\cdots&T_{mU}^{\prime}\end{pmatrix}\] (3) \(\bullet\) The picking station scheduling scheme \(\mathbf{Z_{3}}\) is an arrival information table that includes the order number, the pallet number to which the order belongs, and the delivery time of the order, as shown in Equation (4). \[\mathbf{Z_{3}}=(\mathbf{A_{1}},\mathbf{A_{2}},\cdots,\mathbf{A_{W}})^{T},\;\mathbf{A_{w}}=\begin{pmatrix} O_{w1}&O_{w2}&\cdots&O_{wN}\\ P_{w1}&P_{w2}&\cdots&P_{wN}\\ T_{w1}&T_{w2}&\cdots&T_{wN}\end{pmatrix}\] (4)