## 3 Methodology  
### Analysis and of Influential Relationship  
Taking the entrance lanes of two straight-going lanes as an example, four conditions are selected as the number of straight-going vehicles as follows: 600, 750, 900, and 1050 veh/h. With the help of MATLAB, the influence of the different numbers of left-turn vehicles and lengths of weaving sections on the delay of the left-turn lane inside or outside of the entrance lane is fitted and discussed. The relation curve is shown in Figure 2. The three straight-going lanes have similar influential relationships.  
It can be seen from Figure 2 that when the number of straight-going vehicles is constant, the delay changing law of inside or outside left-turn lanes has certain similarities. With the increase in the number of left-turn vehicles and the decrease in the length of the weaving section, the delay will increase significantly, and the delay change of the inside lane is more obvious than that of the outside lane. The more vehicles that need to merge from outside to inside, the more left-turn vehicles there are. The shorter the length of the weaving section, the greater the probability of the above-mentioned lane changing behavior and colliding with straight-going vehicles within a shorter distance. When the number of straight-going vehicles increases, the delay of the left-turn lanes inside or outside will rise, but the inside scheme increases faster than that of the outside. Inside and outside delay surfaces have intersections, as shown through further analysis of Figure 2a-c. As shown in Table 2, The projection curve equation of the intersection line of inside and outside delay surfaces corresponding to different numbers of straight-going vehicles can be obtained by combining the inside and outside fitting surface equations, eliminating the vertical coordinates, and setting it zero. The curve equation expresses the corresponding relationship between the number of left-turn vehicles and the length of the weaving section under the critical state of equal inside and outside delay of the left-turn lane. X represents the number of left-turn vehicles, and y represents the length of the weaving section. Therefore, the advantages and disadvantages of the inside or outside left-turn lane scheme can be determined by the position relationship between the projection curve and the points formed by the number of left-turn vehicles or the length of the weaving section. For a combination of the number of left-turn vehicles and the length of the weaving section (x, y), the calculated value y\({}^{\prime}\) can be obtained by substituting x into the corresponding equation in Table 2, and then y and y\({}^{\prime}\) can be compared. If y > y\({}^{\prime}\), the (x, y) point is located above the intersection projection, and the delay of the outside left-turn lane in the corresponding curved surface is greater than that of the inside; therefore, it is more advantageous to choose an inside left-turn lane.  
Figure 1: Road conditions at the four intersections. (**a**) The intersection of Southwest Rd. and Nansha St. (inside). (**b**) The intersection of Southwest Rd.and Huanghe Rd. (inside). (**c**) The intersection of Xinggong St. and Shenliao Rd. (outside). (**d**) The intersection of Xinlong St. and Shiji Rd. (outside).  
On the contrary, if \(y<y^{\prime}\), the outside left-turn lane is more advantageous. The two curved surfaces in Figure 2d have no intersection. It means that the outside left-turn lane is better than the inside one when the number of straight-going vehicles is above 1050, no matter how the number of left-turn vehicles and length of weaving section are taken within their respective ranges.  
Figure 2: _Cont_.  
It can be seen from Table 2 that the determination method based on the line of intersection is only applicable to the case where the number of straight-going vehicles is exactly 600, 750, or 900 veh/h. For more general cases, a delay model needs to be constructed for analysis.