## 1 Introduction  
In the transportation system of the future, connected and automated vehicles (CAVs) might replace human-driven vehicles (HVs) [1], and drivers' travel behaviors will change dramatically in the future when CAVs become more common [2]. As CAVs will reduce human driving errors, travel safety could be significantly enhanced [3] and travel time will be improved due to an increased capacity by reducing headway. [PERSON] et al. [4] suggested that highway capacity might be doubled if 60% of traffic is CAVs and improved four to five times if all vehicles were changed to CAVs. [PERSON] and [PERSON] [5] demonstrated that CAVs can increase the average speed by 1.27 times. Besides these two primary benefits, there are countless more, such as improved trip quality and reduced fuel use. However, before the period of fully CAVs appears, it will encounter a situation in which both HVs and CAVs are present, and the traffic flow pattern of the mixed modes will be different when compared to that of only a single class (e.g., HV or CAV). In the literature, in terms of mixed flows, [PERSON] and [PERSON] [6] focused on the value of time when people drive using CAVs, whereas [PERSON] and [PERSON] [7] and [PERSON] et al. [8] investigated the optimal control of the mixed traffic flows based on the road level. [PERSON] et al. [9,10] studied the network design problem for allocating dedicated lanes and zones for CAVs. Recently, [PERSON] et al. [11] and [PERSON] and [PERSON] [12] used mixed behavior equilibrium theories to simulate CAV flows. [PERSON] and [PERSON] [7] demonstrated that even if a small part of a HV is replaced by a CAV, the network flows may significantly lower travel time.  
To summarize, estimating network flows is critical for transportation planning during the period when both CAVs and HVs are present. As CAVs can follow the car in front of them more closely than HVs, link capacity should increase. As a result, the travel time is dependent on the proportion of mixed flows, and each mode necessitates an asymmetric interaction cost function.  
This study proposed a variational inequality (VI) formulation and the route-based gradient projection (GP) algorithm to solve the multi-class traffic assignment problem (TAP) while explicitly considering mixed modes (e.g., HV and CAV). Due to the special structure of the TAP, GP only requires a simple projection on the non-negative orthant in each iteration; this is thanks to a clever method that makes use of the structure of the TAP. Hence, the necessary computational time is minimal. Furthermore, as a scaling matrix, GP employs the diagonal inverse Hessian approximation, and the \"one-at-a-time\" flow updating approach is used to equilibrate route flows. These characteristics make the GP approach more computationally efficient (see [PERSON] et al. [13] for a detailed computational comparison). From the literature, the GP algorithm has been shown to be an effective algorithm for solving various TAPs:  
* User equilibrium model [13; 14; 15; 16];
* Nonadditive traffic equilibrium problem [17];
* Stochastic user equilibrium problem [18; 19];
* Elastic demand traffic assignment problem [20].  
However, in real networks, the use of GP to solve the multi-class TAP with asymmetric vehicle interactions is quite limited. In this paper, the GP technique was extended to solve a multi-class TAP that explicitly considered mixed modes (e.g., HV and CAV). The remainder of the paper is organized as follows. After the introduction, the next section provides the modeling issues, which were vehicle interactions between HVs and CAVs and the multi-class TAP. Section 3 describes a GP method intended specifically to solve the proposed multi-class TAP. Two numerical experiments were adopted to demonstrate the proposed solution algorithm and model formulation in Section 4. Section 5 presents concluding remarks.