# Improved Genetic Algorithm for Solving Green Path Models of Concrete Trucks
## 2 Fuel Consumption Models
### GVRP Model  
#### 2.3.1 Objective Function  
In order to better describe the objective function, the concrete-mixing plant, concrete demand site, distribution vehicles, and other related symbols are explained as follows:  
\(H_{i}\): Concrete-mixing plant \(i\), \(1\leq i\leq n\),  
\(N_{j}\): Concrete demand site \(j\), \(1\leq j\leq m\),  
\(Q_{j}\): Concrete order quantity of site \(j\) (m\({}^{3}\)),  
\(V_{k}\): Concrete truck \(k\), \(1\leq k\leq r\),  
\(V_{kij}\): Concrete truck \(k\) from mixing plant \(i\) to site \(j\),  
\(M_{kij}\): The total weight of the \(V_{kij}\) (t),  
\(M_{2 kij}\): Rated load of the \(V_{kij}\) (t),  
\(M_{8k}\): Self-weight of the \(V_{kij}\) (t),  
\(L_{ij}\): The distance traveled by \(V_{kij}\) (km),  
\(T_{ij}\): The time required from mixing plant \(i\) to site \(j\) (h),  
\(P_{k}\): Rated power of \(V_{k}\) hydraulic motor,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Parameter** & **Symbol** & **Parameter** \\ \hline M & The total mass of the vehicle & d\({}_{a}\) & Air resistance workmanship \\ g & Gravitational constant & \(\mathrm{\eta_{motor}}\) & Engine efficiency \\ L & Travel distance (km) & A\({}_{f}\) & Frontal area (m\({}^{2}\)) \\ \(\beta\) & Driving & C\({}_{d}\) & Air resistance coefficient \\ C\({}_{r}\) & Rolling resistance & v\({}_{r}\) & Vehicle speed considering the \\ \(\phi_{t}\) & Effect of temperature (\({}^{\circ}\)F) & e\({}_{v}\) & Wind influence factor \\ d\({}_{r}\) & Rolling resistance work distance & \(\rho_{f}\) & Fuel density (kg/L) \\ \(\mathrm{\Lambda_{s}}\) & Type of pavement & \(\rho\) & Air density (kg/\({}_{m^{3}}\)) \\ \(\mathrm{C_{f}}\) & Fuel calorific value (MJ/kg) & T & Time (h) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Description of symbols.  
\(k\in i\): Vehicle number \(k\) belongs to the vehicle set of mixing plant \(i\),  
\(k\
otin i\): Vehicle number \(k\) doesn't belongs to the vehicle set of mixing plant \(i\),  
\(X_{ik}\): Decision variable.  
\(X_{ik}=\left\{\begin{array}{ll}1\;\mathrm{k}\in\mathrm{i}\\ 0\;\mathrm{k}\
otin\mathrm{i}\end{array}\right.\)  
The objective function is established by combining the fuel consumption models for truck driving and stirring the tank, mentioned previously.  
When multiple mixing plants co-operate to transport concrete to multiple sites, the total fuel consumption of all trucks driving is as follows:  
\(\sum\mathrm{F_{x}}=\sum_{i=1}^{n}\sum_{j=1}^{m}\sum_{k=1}^{r}\mathrm{L_{ij}} \cdot X_{ik}\cdot\mathrm{F_{x}}\) (6)  
Integrate the known factors affecting the fuel consumption of the truck driving as \(f_{1}\), and \(f_{2}\); the final truck driving fuel consumption as follows:  
\(\mathrm{f_{1}}=(C_{r}\phi_{t}\lambda_{s}\mathrm{g}\mathrm{e_{v}}\mathrm{n_{ motor}})/C_{t}\rho_{f}\)  
\(\mathrm{f_{2}}=0.5\mathrm{\mu}\mathrm{A_{f}}C_{d}\mathrm{\beta}\mathrm{e_{v}} \mathrm{n_{motor}}/C_{t}\rho_{f}\)  
\(\sum\mathrm{F_{x}}=\sum_{i=1}^{n}\sum_{j=1}^{m}\sum_{k=1}^{r}\mathrm{L_{ij}} X_{ik}[M_{kij}f_{1}+(5\mathrm{L_{ij}}/18T_{ij})^{2}f_{2}]\) (7)  
When multiple mixing plants co-operate to transport concrete to multiple sites, the total fuel consumption of all trucks stirring the tank is shown in Equation (8).  
\(\sum\mathrm{F_{k}}=\sum_{i=1}^{n}\sum_{j=1}^{m}\sum_{k=1}^{r}X_{ik}(P_{k}T_{ij }/C_{t}\rho_{f}\gamma_{f})\) (8)  
The objective function consists of two parts, which are calculated as follows:  
\(Z=\sum\mathrm{F_{x}}+\sum\mathrm{F_{k}}\) (9)