## 3 The Establishment of an Optimization and Scheduling Model for Automobile PBS
### Objective Function  
According to the specific quantification logic of the optimization objectives in the problem, an initial score is set for each optimization objective. The scores are updated based on the following logic, multiplied by the corresponding coefficients, and then combined to obtain the final weighted total score (with a theoretical maximum score of 100 points). Optimization Objective 1 (weight coefficient 0.4, initial score 100 points): Examine the vehicle dispatch sequence to identify all hybrid car bodies. Calculate the number of non-hybrid car bodies between every consecutive pair of hybrid car bodies in the sequence, in the order of their appearance. Deduct 1 point for each pair that does not have exactly two non-hybrid car bodies. Optimization Objective 2 (weight coefficient 0.3, initial score 100 points): Divide the vehicle dispatch sequence into segments to determine if the ratio of four-wheel drive to two-wheel drive vehicles in each segment is 1:1. Deduct 1 point for each segment where the ratio is not met. The basis for dividing the sequence into segments is as follows: if the sequence starts with 4, divide it whenever the value changes from 2 to 4; if the sequence starts with 2, divide it whenever the value changes from 4 to 2. For example, if the dispatch sequence is 442242444224 (4 represents four-wheel drive vehicles, and 2 represents two-wheel drive vehicles), the segmentation result is 4422, 42, 44422 and 4. The last two segments, 44422 and 4, do not satisfy the 1:1 ratio, so we deduct 2 points. Optimization Objective 3 (weight coefficient 0.2, initial score 100 points): Count the number of times the return path is used and deduct 1 point for each use. Optimization Objective4 (weight coefficient 0.1, initial score 100 points): Reduce the number of times the return path is used, with the final objective score being 100 minus the time penalty value. * Hybrid vehicles with a spacing of two non-hybrid vehicles are preferred. Let \(L_{0}\) be the hybrid vehicle binary sequence of the original sequence (with dimensions \(C\times 1\) and \(C\) as the number of vehicles), \(L_{d}\) be the hybrid vehicle binary sequence after adjustment (with dimensions \(C\times 1\)), and \(L_{d}(i)\) be the value of the \(i\)th element in sequence \(L_{d}\). Let \(\boldsymbol{l}(i)=\left[L_{d}(i)L_{d}(i+1)L_{d}(i+2)L_{d}(i+3)\right]^{T}\) be a vector, and \(\boldsymbol{l}=\left[1\quad 0\quad 0\right]^{T}\) be the target sequence. The adjusted hybrid vehicle binary sequence \(L_{d}\) (with dimensions \(C\times 1\)) is partitioned into block vectors starting with 1, resulting in \(l_{i}(i=1,2,3,\cdots,C)\) and \(l_{i}\) with the format \"1 0 \", where \"\(*\)\" indicates that the preceding element 0 may or may not appear, or may appear one or more times. Therefore, the highest score for Optimization Objective 1 is as follows. \[S_{1}=\sum_{i=1}^{C}equals(l_{i},l)-1 \tag{1}\]  
\[equals(l_{i},l)=\left\{ \begin{array}{cc}1&l_{i}\text{ is identical to }l\\ 0&l_{i}\text{ is not identical to }l\end{array} \right. \tag{2}\]
* The sequence of deploying four-wheel drive and two-wheel drive models tends to be 1:1. Let \(D\) be the adjusted two-wheel/four-wheel drive sequence (with dimensions \(C\times 1\)), partitioned into \(h\) sub-sequences \(D_{i}(i=1,2,3,\cdots,h)\) based on the type of drive. Let \(D_{i}\) be the vector (with dimensions \(d_{i}\times 1\)). Therefore, the highest score for Optimization Objective 2 is as follows:  
\[S_{2}=\sum_{i=1}^{h\setminus 2}(2\delta(d_{2i}-d_{2i-1})-2)-\text{mod}(h,2) \tag{3}\]  
where \(\setminus\) denotes integer division; \(\delta(\cdot)\) represents the pulse function; and \(\text{mod}(h,n)\) represents the remainder of \(m\) divided by \(n\). * The frequency of returning paths tends towards 0. By combining the use of the vehicle transfer machine and the car transfer machine according to the first-in-first-out (FIFO) principle, we can infer that using the returning path is the only way to change the vehicle sequence.