## 2 Problem Model
### Mathematical Model
#### 2.2.2 Constraints  
After ensuring the objective of the VRPCSC, some constraints need to be confirmed in this model to make the VRPCSC fully satisfied with the problem description in Section 2.1. First, the relationship between products and materials satisfies linear constraints:  
\[o_{p}=\frac{{o^{\prime}}_{p1}}{\sigma_{1}}=\frac{{o^{\prime}}_{p2}}{\sigma_{2 }}=\cdots=\frac{{o^{\prime}}_{pl}}{\sigma_{L}} \tag{6}\]  
where \(\sigma_{1},\sigma_{2},\cdots,\sigma_{L}\) are all known real number constants. According to the product demands of the warehouses, the demands for purchased materials can be calculated by Formula (6). However, in the process of purchasing materials, each material provided by the supplier cannot exceed its maximum supply quantity. The constraint is shown in Formula (7):  
\[\sum_{m\in SM}\sum_{p\in SP}g_{smkpl}o^{\prime}{}_{pl}\leq e_{sl},\forall s\in SS,\forall k\in SK,\forall l\in SL \tag{7}\]  
Further, in other to ensure that the supplier delivers the materials to the corresponding manufacturer at most by one vehicle, suppliers' vehicles should meet the following constraints:  
\[g^{\prime}_{sm}=\min\{1,\sum_{k\in SK}\sum_{p\in SP}\sum_{l\in SL}g_{smkpl}\}, \forall s\in SS,\forall m\in SM \tag{8}\]  
When finished, the products will be delivered by the manufacturers' vehicles. In the same way, the weight of the products delivered by the manufacturers' vehicles cannot exceed their respective maximum loading capacities. The constraint is shown in Formula (9):  
\[\sum_{p\in SP}g^{\prime\prime}{}_{mkp}o_{p}\leq u_{mk},\forall m\in SM,\forall l \in SL \tag{9}\]  
In the process of delivering products to the warehouses, the manufacturers' vehicles should also meet the following constraints:  
\[\sum_{m\in SM}\sum_{k\in SK}g^{\prime\prime}{}_{mkp}=1,\forall p\in SP \tag{10}\]  
\[\sum_{p\in SP}y_{mkp}\leq 1,\forall m\in SM,\forall k\in SK \tag{11}\]  
\[\sum_{p\in SP}\sum_{p^{\prime}\in SP}y^{\prime}{}_{mkp^{\prime}}\leq\delta \sum_{p\in SP}y_{mkp},\forall m\in SM,\forall k\in SK,\exists\delta\in R^{+} \tag{12}\]  
\[\sum_{p\in SP}\sum_{p^{\prime}\in SPT}mkpp^{\prime}\leq SPT-\gamma(SPT), \forall SPT\in SP,SPT\
eq\emptyset,\forall m\in SM,\forall k\in SK \tag{13}\]  
where Formula (10) ensures that only one manufacturer delivers products to each warehouse; Formula (11) ensures that each manufacturer's vehicle can deliver products only once; Formula (12) ensures that the manufacturer's vehicle must start from the manufacturer's place to deliver products to the warehouse; Formula (13) ensures the continuous delivery process of one manufacturer's vehicle and \(\gamma(SPT)\) denotes the minimum number of manufacturer's vehicles needed to serve all the warehouse in set \(SPT\).  
In addition, since the materials are related to the products, there are some constraints between the transportation of materials and corresponding products:  
\[\sum_{s\in SS}\sum_{m\in SM}\sum_{k\in SK}\sum_{p\in SP}g_{smkpl}=1,\forall l \in SL \tag{14}\]  
\[\sum_{s\in SS}\sum_{l\in SL}g_{smkpl}=L\times g^{\prime\prime}_{mkp},\forall m \in SM,\forall k\in SK,\forall p\in SP \tag{15}\]  
where Formula (14) ensures that each material corresponding to the products needed by each warehouse is provided by only one supplier, processed by one manufacturer and delivered by one manufacturer's vehicle; Formula (15) indicates the relationship between material delivery and product delivery.  
Above all, the VRPCSC model can be established from Formula (1) to Formula (15). Then the algorithm of this model will be solved in detail.