# A Multi-Customer Vehicle Scheduling Optimization Method for Coal Intelligent Loading System
## 3 Construction of Vehicle Scheduling Optimization Model for Coal Mine Enterprises
### Establishment of a Multi-Customer Vehicle Scheduling Model
#### 3.2.2 Model Building  
In traditional loading operations, the lack of proper vehicle loading notifications often causes customer vehicle congestion during loading wait times, leading to the failure to deliver the goods on time. This paper introduces a model aimed at minimizing overall costs, including operating expenses, carbon emission expenses, penalty costs, and more. Optimizing customers' loading schedules and locations can enhance loading efficiency, decrease operational expenses, and improve service quality. The symbols used in the mathematical model are detailed in Table 4.  
In the loading process, the optimal number of loading warehouses will be determined according to demand to improve loading efficiency and reduce resource wastage. Additionally, the costs of carbon emissions and penalties need to be taken into account. In this context, the optimization objective is defined as minimizing the total cost \(C\) with the following objective function:  
\[C=C_{1}+C_{2}+C_{3} \tag{8}\]  
where: \(C_{1}\) represents operating costs, including labor costs, equipment costs, energy costs, maintenance costs, and management costs. \(C_{2}\) represents the carbon emission cost, which is the carbon emission cost generated during the waiting period of the customer's vehicle. \(C_{3}\) represents the penalty cost incurred due to the failure to load the vehicle within the specified time range. The operating cost, carbon emission cost, and penalty cost are calculated as follows:  
\[C_{1}=\sum_{k=1}^{K}g_{k}Z_{k} \tag{9}\]  
\[C_{2}=\sum_{i=1}^{N}\sum_{j=1}^{M}\sum_{k=1}^{K}x_{ij}^{k}\varepsilon\gamma \beta p_{1}T_{ij} \tag{10}\]  
\[C_{3}=R_{i}\sum_{k=1}^{K}\sum_{i=1}^{N}\sum_{j=1}^{M}\omega\left(t_{ij}^{k}\right) \tag{11}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbols** & **Definition** \\ \hline \(K\) & Number of loading bunkers available to the enterprise \\ \(k\) & Loading bunker index \(k=1,2,\cdots,K\) \\ \(N\) & Number of customers \\ \(i,n\) & Customer index, \(i,n=1,2,\cdots,N\) \\ \(M\) & Number of vehicles \\ \(j,m\) & Vehicle index, \(j,m=1,2,\cdots,M\) \\ \(g_{k}\) & Operating cost of loading bunker \(k\) \\ \(t_{ij}^{k}\) & Service time for loading vehicle \(j\) to customer \(i\) at loading bunker \(k\) \\ \(T_{ij}\) & Loading delay time after the arrival of vehicle \(j\) to customer \(i\) \\ \(\gamma\) & Carbon tax \\ \(\beta\) & Carbon dioxide emission factor \\ \(\epsilon\) & Unit carbon emission cost factor \\ \(P_{1}\) & Fuel consumption per unit of time the vehicle waits \\ \(\eta_{1}\) & Penalty cost factor for delayed loading \\ \(\left[Et_{ij}^{d},\,Lt_{ij}^{d}\right]\) & Range of time customer vehicles are expected to arrive at the business for loading services \\ \(Q_{ij}\) & Load capacity of vehicle \(j\) to customer \(i\) \\ \(G\) & Maximum capacity of the loading bunker \\ \(R_{i}\) & Prioritization penalty factor for customer \(i\) \\ \(x_{ij}^{k}\) & A value in the range \(\{0,1\},\) 1 if vehicle \(j\) to customer \(i\) is loaded on loading bunker \(k\), \\  & 0 otherwise \\ \(Z_{k}\) & Takes a value in the range of \(\{0,1\},\) where 1 indicates loading bunker \(k\) being loaded, and 0 indicates otherwise \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Description of related symbols. The penalty cost function \(\omega\Big{(}t_{ij}^{k}\Big{)}\) in Formula (11) is expressed as:  
\[\omega\Big{(}t_{ij}^{k}\Big{)}=\left\{ \begin{array}{c}F\;Et_{ij}^{d}>t_{ij}^{k} \\ 0\;Et_{ij}^{d}<t_{ij}^{k}<L_{ij}^{d}\\ \eta_{1}\Big{(}t_{ij}^{k}-L_{ij}^{d}\Big{)}\;L_{ij}^{d}<t_{ij}^{k}\end{array} \right. \tag{12}\]  
\(F\) is a significant constant, indicating that if the vehicle fails to load within the expected arrival loading time window, there will be a substantial penalty cost.