## 2 Multi-Agent Supply Chain Model  
Suppose the weighted graph of N subchains is \(G=\{\mathcal{V},E,\mathcal{A}\}\), where each node is a subchain, and the facilities in each subchain can be considered as children of the node. \(\mathcal{V}\) is the set of nodes containing the vertices of all nodes, \(\mathcal{V}=\{1,2,\ldots,N\}\). \(E\subset\mathcal{V}\times\mathcal{V}\) is the set of edges in the graph. When nodes \((i,j)\) belong to the same edge and \((i,j)\) can communicate with each other, \(j\) is said to be a neighboring node of \(i\). Denote the set of neighboring nodes of node \(i\) by \(N_{i}=\{j|j\in\mathcal{V},(j,i)\in E\}\). When a node does not have any neighboring nodes, it is called an isolated node. The weighted adjacency matrix between individual nodes is denoted as \(\mathcal{A}=[\alpha_{ij}]\in\mathbb{R}^{N\times N}\), where \(\alpha_{ij}=1\) if and only if \(i\) and \(j\) are neighboring nodes. The Laplace matrix of the graph \(G\) is defined as \(L=[l_{ij}]\in\mathbb{R}^{N\times N}\). \[l_{ij}=\left\{ \begin{aligned} &-\alpha_{ij},& i \
eq j\\ &\sum_{s\in N_{i}}\alpha_{is},& i=j.\end{aligned} \right. \tag{2}\]  
Based on the discussion above, we consider a multi-agent supply chain system. This system comprises N sub-chains, each treated as an intelligent agent and represented as a node within a graph. Each sub-chain contains n facilities. Figure 1 illustrates the \(i\)-th sub-chain, where \(x_{is1}\) represents the inventory status of the first supplier facility in the \(i\)-th sub-chain. \[x_{i}(k) =[x_{i1}(k),\ldots,x_{in}(k)]^{\mathrm{T}}\in\mathbb{R}^{n}\] \[u_{i}(k) =[u_{i1}(k),\ldots,u_{in}(k)]^{\mathrm{T}}\in\mathbb{R}^{n}\] \[d_{i}(k) =\left[0,\ldots,0,d_{in}(k)\right]^{\mathrm{T}}\in\mathbb{R}^{n} \quad(i=1,\ldots,N)\]  
where \(x_{i}(k)\) represents the inventory status of the \(i\)th subchain at time \(k\), \(u_{i}(k)\) denotes the productivity at time \(k\), \(d_{i}(k)\) indicates the market demand at time \(k\), and \(\rho\) is the loss rate in the warehousing process. The production and supply process for each subchain commences with the supplier providing the means of production, progresses through a sequence of processing facilities, and concludes with the customer purchasing the finished product. Here, \(u_{in}(k)\) represents not only the productivity of the current facility but also the demand of the preceding facility. Market demand directly influences only the final facility, yet it propagates through the supply chain, impacting all nodes. From a control perspective, \(x\) can be viewed as the inventory state at time \(k\), \(u\) as the control input, and \(d\) as the external perturbation. To model the supply chain system, we must establish the following assumptions. **Assumption 1**. : _Goods in the warehouse will have a certain amount of wear and tear with storage time, and different goods in different warehouses have different rates of wear and tear._  
**Assumption 2**. : _The user's demand is divided into a known regular demand \(\overline{d}_{in}\), and an unknown time-varying demand \(\omega_{in}\in l_{2}[0,\infty)\), so that the demand is represented as_  
\[d_{in}(k)=\overline{d}_{in}+\omega_{in}(k) \tag{3}\]In Assumptions 1 and 2, the \(i\)th subchain is modeled as  
\[x_{i}(k+1)=(I-\rho_{i})x_{i}(k)+B_{i}u_{i}(k)-d_{i}(k) \tag{4}\]  
where  
\[\rho_{i}=\operatorname{diag}\{\rho_{1},\rho_{2},...,\rho_{n}\}\] \[B_{i}= \begin{bmatrix}1&-1&0&\cdots&0&0\\ 0&1&-1&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\cdots&1&-1\\ 0&0&0&\cdots&0&1\end{bmatrix} .\]  
In a supply chain system with N sub-chains, it is necessary to compare inventory levels between neighboring nodes to adjust productivity and achieve system consensus.