## 4 Multi-Objective Optimization of DTS  
The double-transition shift process not only requires appropriate action between each on-coming and off-going clutches, the reasonable allocation of overlapping time between the two sets of clutches is important as well. In this article, we set the group of \(BS\) and \(CS\) shift time unchanged, and take the \(BL\) and \(BM\) shift time as variables. Five control strategies are proposed to sequentially change the overlap time of the two sets of clutches (summarized in Table 2).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Time**} & \multirow{2}{*}{\(BS\) **and \(CS\)**} & \multicolumn{5}{c}{\(BL\) **and \(BM\)**} \\ \cline{3-6}  & & **Strategy 1** & **Strategy 2** & **Strategy 3** & **Strategy 4** & **Strategy 5** \\ \hline \(t_{0}\sim t_{1}\) & & low gear & & & & \\ \(t_{1}\sim t_{2}\) & & torque phase & low gear & & & \\ \(t_{2}\sim t_{3}\) & high gear & inertia phase & torque phase & low gear & & \\ \(t_{3}\sim t_{4}\) & inertia phase & high gear & inertia phase & torque phase & low gear & \\ \(t_{4}\sim t_{5}\) & torque phase & high gear & inertia phase & torque phase & low gear & \\ \(t_{5}\sim t_{6}\) & low gear & & & high gear & inertia phase & torque phase \\ \(t_{6}\sim t_{7}\) & & & & & & high gear & inertia phase \\ \(t_{7}\sim t_{8}\) & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Five control strategies for double-transition shift (DTS).  
The shift jerk \(j\) and the clutch friction loss \(W\) as the performance indicators of the double-transition shift are a pair of paradoxes. The multi-objective function is  
\[\text{min}J_{1}=\frac{1}{p}\sqrt{\frac{1}{t_{f}-t_{0}}\int_{t_{0}}^{t_{f}}j^{2}dt }\text{,min}J_{2}=\frac{r}{q}\int_{t_{0}}^{t_{f}}\sum Wdt, \tag{22}\]  
where \(r\) is the weight coefficient, and \(p=8\) and \(q=40,000\) are the normalizing coefficients between the two indicators.  
The multi-objective genetic algorithm has been used to optimize a nonlinear problem in the vehicle field [25; 26]. Especially, non-dominated sorting genetic algorithm II (NSGA-II) is claimed to have better performance compared with the other optimization methods. A flow diagram of NSGA-II is shown in Figure 3, where all the major steps are shown. A detailed analysis of the algorithm can be found in [27].  
In this paper, the optimal front-end individual coefficient is set to 0.3, the population density is set to 20, the maximum number of iterations is 50, and the fitness function deviation is 1e\(-\)100. The full-loaded flat road of heavy-duty mining trucks is selected as the simulation condition (vehicle mass \(m=72\) T, road slope \(\alpha=0^{\lx@math@degree}\)); then, it changes the weight coefficient \(r\) in turn to obtain the Pareto optimal frontier, as shown in Figure 4. The Pareto optimal result shows the following: (1) Strategy 1 and Strategy 5 are step by step to complete the transition of the two sets of clutches. The shift time is longer, and it is difficult to reduce the clutch friction loss below 35 KJ. (2) Overlapping both the inertia phase in Strategy 2, the average shift jerk is much higher than other strategies, reaching above 8 m/s\({}^{3}\). (3) Strategy 3 and Strategy 4 can achieve better shift quality, with similar shift jerk and clutch friction loss. The clutch friction loss in Strategy 3 is slightly lower than that of Strategy 4 by 2 KJ, and the shift jerk is slightly higher by 0.2 m/s\({}^{3}\).  
Figure 3: Multi-objective genetic algorithm diagram (non-dominated sorting genetic algorithm II, NSGA-II).