#### Abstract

We derive the exact inefficiency upper bounds of the multiclass C-Logit stochastic user equilibrium (CL-SUE) in a transportation network. All travelers are classified on the basis of different values of time (VOT) into M classes. The multiclass CL-SUE model gives a more realistic path choice probability in comparison with the logit-based stochastic user equilibrium model by considering the overlapping effects between paths. To find efficiency loss upper bounds of the multiclass CL-SUE, two equivalent variational inequalities for the multiclass CL-SUE model, i.e., time-based variational inequality (VI) and monetary-based VI, are formulated. We give four different methods to define the inefficiency of the multiclass CL-SUE, i.e., to compare multiclass CL-SUE with multiclass system optimum, or to compare multiclass CL-SUE with multiclass C-Logit stochastic system optimum (CL-SSO), under the time-based criterion and the monetary-based criterion, respectively. We further investigate the effects of various parameters which include the degree of path overlapping (the commonality factor), the network complexity, degree of traffic congestion, the VOT of user classes, the network familiarity, and the total demand on the inefficiency bounds.

#### 1. Introduction

Concerning the path choice behavior in networks, Wardrop proposed two basic principles: one is the user equilibrium (UE) and the other is the system optimum (SO) [1]. The UE principle shows that all used paths have minimum and equal travel cost (or time), and all unused paths have higher or equal travel cost (or time). In addition, the UE principle assumes that every traveler has a full understanding of the network information and chooses the travel path accordingly to minimize their own travel cost. Later, the researchers have extended the UE model in different aspects, such as boundedly rational UE [2–4], fuzzy UE [5–7], and prospect‐based UE [8–10]. The concept of the boundedly rational user equilibrium (BRUE) was proposed in the 1980s. Lou et al. [3] investigated congestion pricing strategies in static networks with boundedly rational route choice behavior. Xuan et al. [4] studied mathematical formulation and solution sets of BRUE. Miralinaghi et al. [7] proposed an alternative approach for a traffic assignment problem that further extends the applications of the fuzzy theory in route choice behavior and network traffic modelling. Xu et al. [10] proposed a conjecture on travelers’ determination of reference points and encapsulates it into the prospect-based user equilibrium conditions. However, in practice, we cannot always assume that this assumption is true. More realistically, travelers who choose the same route may have different perceived travel times due to all kinds of unmeasured factors. Suppose that the perceived travel times are considered as independent and identical distributed (IID) Gumbel random variables [11], travelers’ path choices for minimizing their perceived travel time will lead to the stochastic user equilibrium (SUE) state [12]. Concerning the path choice problem in the network, two disadvantages of the logit-based SUE model are as follows: (1) it cannot explain overlapping between paths and (2) it cannot explain the perception variance of travels of different lengths [13, 14].

To alleviate the above two disadvantages, some extended logit-based SUE models have been proposed in recent two decades, such as C-Logit SUE (CL-SUE) model [13–16], generalized nested logit SUE (GNL-SUE) model [17], cross-nested logit SUE (CNL-SUE) model [18, 19], and paired combinatorial logit SUE (PCL-SUE) model [20]. Other significant theoretical achievements in [21, 22] have also made a contribution to overcoming the disadvantages of the logit-based SUE model. CL-SUE model proposed by Cascetta et al. [13] can solve the overlapping problem by using a commonality factor reflecting path overlapping to modify the systematic part of the utility function. Zhou et al. [14] further provided equivalent variational inequality (VI) and mathematical programming (MP) formulations for the CL-SUE model.

The CL-SUE model can overcome the disadvantages of the logit-based SUE model by considering the overlapping effects between paths and give a more realistic path choice probability. The SO state has the minimum inefficiency loss by definition. Therefore, the CL-SUE model is still inefficient compared with the SO model due to the users’ selfish routing. Recently, quantifying and bounding the inefficiency of the equilibrium assignment in the transportation environment has aroused great research interest, while less attention has been paid to multiclass CL-SUE. Koutsoupias and Papadimitriou first gave the concept of “price of anarchy (POA)” [23]. After that, the POA was first used in the traffic network by Roughgarden and Tardos [24]. In the next few years, other great theoretical achievements of POA proposed by Roughgarden [25–27] have greatly promoted the development of inefficiency in the transportation network. In recent years, the above works have been extensively investigated in many aspects [28–36]. Guo et al. [37] gave two efficiency loss bounds of the SUE against SO and stochastic system optimum (SSO) in a stochastic circumstance, respectively. Considering a transportation network with multiple classes of users, Yu et al. [38] studied the inefficiency of the multiclass SUE, which is the extension of research achievements in [37]. Based on the above main work, some researchers have also studied the inefficiency of the extended logit-based SUE. Yong et al. provided several efficiency loss upper bounds of the CL-SUE by considering the overlapping effects between paths [39], which is also the extension of research achievements in [37]. Zeng and Wang initially explored inefficiency upper bound of CNL-SUE in the taxed stochastic transportation network and further investigated inefficiency upper bounds for the low-degree travel time function [40]. For a transportation network with heterogeneous users who have different values of time (VOT), the network optimization usually has two objectives, i.e., minimizing total system travel time (TSTT) and minimizing total system travel cost (TSTC). Guo and Yang further measured the system optimal performance difference by the two different criteria, i.e., time-based criteria and monetary-based criteria [41]. Yu and Wang [42] derived the accurate inefficiency bounds of multiclass UE with elastic travel demand by making full use of equivalent VI under time-based criterion and cost-based criterion, respectively. Han and Yang [43] have given several bounds for the inefficiency of the multiclass traffic equilibrium assignment problem in the tolled network under the two different criteria, respectively. Huang et al. [44] further discussed the efficiency loss of the SUE where all travelers are classified into two main categories, one equipped with advanced traveler information systems (ATIS) and another unequipped. Yu et al. studied the inefficiency of the mixed equilibrium with heterogeneous users [45–47].

Different from the existing studies, this study extends the logit-based SUE to the multiclass CL-SUE. The purpose of our study was to derive four efficiency loss upper bounds of the multiclass CL-SUE. The two corresponding models, i.e., multiclass SO model and multiclass C-Logit stochastic system optimum (CL-SSO) model, should be mentioned before exploring the efficiency loss of the multiclass CL-SUE. The multiclass SO model that minimizes the TSTT has been widely used in the literature. In this article, the multiclass CL-SSO model that minimizes the total perceived travel time (TPTT) is established based on the work in [39, 48, 49]. Therefore, there are four kinds of ways to define the efficiency loss of the multiclass CL-SUE, i.e., comparing multiclass CL-SUE with multiclass SO, or comparing multiclass CL-SUE with multiclass CL-SSO, under the time-based criterion and the monetary-based criterion, respectively. The highlights of our research focus on these four situations. In recent years, the effects of various parameters (e.g., degree of traffic congestion and total traffic demand of the network) on the inefficiency upper bound has been widely studied, while paying less attention to commonality factor reflecting path overlapping. This paper studies the effects of commonality factor reflecting path overlapping on the efficiency loss of the multiclass CL-SUE. The result shows that the commonality factor reflecting path overlapping has a significant impact on the inefficiency upper bound.

The rest of this article is organized below. In Section 2, we have a brief review of multiclass SUE and CL-SUE models, and then formulate the equivalent VI formulations of the multiclass CL-SUE model under time-based criterion and monetary-based criterion, respectively. In Sections 3 and 4, we derive four inefficiency upper bounds by using the equivalent VI formulations of the multiclass CL-SUE. Section 5 discusses the effects of various parameters on the efficiency loss bounds. In Section 6, we summarize the main research findings of this paper in tabular form. We give a numerical example to illustrate our conclusions in Section 7. Finally, Section 8 provides some conclusions and analyses the further research directions.

#### 2. Review of Multiclass SUE and CL-SUE Models

##### 2.1. Multiclass SUE Model

Let be a directed transportation network defined by a set of nodes and a set of directed links. All travelers are classified in terms of different VOT into classes. Table 1 provides the notations used in the paper:

According to Table 1, the following relationships and constraints hold:

In a multiclass SUE model, assume that all users are utility-maximizers. Let denote the travel utility perceived by user class on path . Then, is given bywhere is the perceived travel time of user class on path , is a positive unit scaling parameter, is the measured utility, and is a random term representing the user’s perception error. Let be the probability of user class choosing path , then the utility maximization theory shows that

has the two properties as follows:

All random terms in (2) are supposed to be IID Gumbel random variables. Then, the path choice probability for user class is given byand can be obtained by

Similar to that done in Fisk [12], the multiclass logit-based SUE model can be formulated as an equivalent MP problem below.

The equivalent VI of problem (7) is presented in the lemma below.

Lemma 1. *If the link time function, , is differentiable, convex, separable, and monotonically increasing with link flow , a multiclass logit-based SUE model with fixed OD demand is equivalent to the following VI, i.e., find , such that*

The disadvantages of the multiclass SUE model have been found by some researchers. One of the disadvantages is that it cannot explain the overlapping effects between different paths, which means that the unrealistic path choice probability is given.

##### 2.2. Multiclass CL-SUE Model

Cascetta et al. [13] proposed a CL-SUE model, which can overcome the disadvantages of the logit-based SUE by using a commonality factor reflecting path overlapping to modify the systematic part of the utility function. Similar to the work of Cascetta et al. [13], the multiclass CL-SUE model can be established. Then, the path choice probability of the multiclass CL-SUE model is provided bywhere is a commonality factor of path . In this paper, the form of the commonality factor [13] is used as follows:where and are the parameters; denotes the common length of paths and between the OD pair ; and and denote the lengths of paths and between the OD pair , respectively. The length can be regarded as the free-flow travel time [14]. And is provided by

Similar to that done in Zhou et al. [14], the multiclass CL-SUE model can be formulated as an equivalent MP problem below.

To find efficiency loss upper bounds of the multiclass CL-SUE, the equivalent VI of problem (12) is presented in the lemma below.

Lemma 2. *If the link time function, , is differentiable, convex, separable, and monotonically increasing with link flow , a multiclass CL-SUE model with fixed OD demand is equivalent to the following VI, i.e., find , such that*

*Proof of Lemma 2. *Since is monotonically increasing and is compact, the objective function (12) is strictly convex. Hence, the path flow solution of the problem (12) is unique. The necessary and sufficient condition, which can guarantee that is the unique optimal solution of problem (12), is provided as follows:Using and substitutinginto (14), VI (13) can be obtained according to equation .

Under the time-based criterion, is given byBy contrast, under the monetary-based criterion, can be given bySubstituting (16) and (17) into (13), respectively, and using (1), we can get the following time-based VI and monetary-based VI, respectively.

#### 3. Bounding the Inefficiency of the Multiclass CL-SUE Compared with the Multiclass SO

##### 3.1. Time Units

The multiclass SO problem in time units that minimizes the TSTT is presented as follows:

Let be the link flow solution of problem (19), the corresponding path flow solution is denoted by . Let be the link flow solution of the VI (18), the corresponding path flow solution is denoted by . The efficiency loss of the multiclass CL-SUE compared with the multiclass SO under time-based criterion is defined aswhere is the TSTT at the multiclass CL-SUE and is the TSTT at the multiclass SO. Clearly, .

Setting and in VI (18), we can obtain inequality as follows:

This leads to

Now, we begin to find the upper bound of the term in (22). To achieve this, we give the same way which has been used in [37]. For completeness, a brief introduction is given here. Because is increasing, we have for . Hence, we just need to pay attention to for . As shown in Figure 1, let denote the area of the large rectangle and denote the area of the blue rectangle.

Next, an upper bound of the ratio needs to be determined. To complete this, for each link time function and nonnegative link flow , we let

Since when and when , we can obtain . Let denote a class of link time functions, we define

With definitions (23) and (24), the lemma is given as follows:

Lemma 3. *Let be link flow at the multiclass CL-SUE and denote an arbitrary nonnegative link flow. Then,*

*Proof of Lemma 3. *Setting in (23) and (24), we then obtainThis completes the proof.

Let , we haveWe now begin to seek the upper bound on the third term of the right-hand side of (22). To achieve this, we give the lemma which has been proved in [39] below.

Lemma 4. *The maximization problem is considered as follows:subject towhere is a constant. is the maximum value of , where solves equation and .*

From Lemma 4, we havewhere solves the equation with , , and is the number of paths connecting OD pair . Substituting (27) and (30) into (22), it yields

Let denote the total traffic demand, be the average of , and . Then, we can rewrite (31) as

In addition, we further define as the actual average travel time of all travelers at multiclass SO. Then, the theorem is presented as follows:

Theorem 1. *Let denote a class of differentiable, separable, convex, and monotonically increasing link time functions . Let be the TSTT at the multiclass CL-SUE and be the TSTT at the multiclass SO under time-based criterion. Then,*

Furthermore, we can have the following corollary by comparing the efficiency loss upper bounds of the multiclass CL-SUE and the multiclass SUE.

Corollary 1. *In a transportation network, the efficiency loss upper bound against the multiclass SO by the multiclass CL-SUE under time-based criterion is not less than that by the multiclass SUE.*

*Proof of Corollary 1. *Let be link flow solution of the VI (8), the corresponding path flow solution is denoted by . is the TSTT at the multiclass SUE. The inefficiency upper bound of the multiclass SUE against the multiclass SO under time-based criterion was given by Yu et al. [38], i.e.,where is the average of , , and solves the equation .

We can see that the difference between (33) and (34) is only dependent upon the difference between and , or and . is the solution of with , . We always have , when , . Since is a nondecreasing function of , we can obtain , which implies that . Hence, the upper bound (33) is not less than the upper bound (34). This completes the proof.

The above analyses show that the commonality factor reflecting path overlapping has a significant impact on the inefficiency upper bound (33). Therefore, it is meaningful to study the efficiency loss of the multiclass CL-SUE model, which provides a more realistic upper bound than the multiclass SUE model by considering the path overlapping problem.

##### 3.2. Monetary Units

The multiclass SO problem in monetary units that minimizes the TSTC is presented as follows:

Let be link flow solution of problem (35), the corresponding path flow solution is denoted by . Let be link flow solution of the VI (18), the corresponding path flow solution is denoted by . The efficiency loss of the multiclass CL-SUE compared with the multiclass SO under monetary-based criterion is defined aswhere is the TSTC at the multiclass CL-SUE and is the TSTC at the multiclass SO. Clearly, .

Setting and in VI (18), we can obtain inequality as follows:

This leads to

To derive the upper bound of the term in (38), we need to define a new parameter [43]. For each link time function , nonnegative link flow , and the VOT , we define

Here, suppose that always holds. For guaranteeing , we further definewhere .

Define

And

Hence, we can obtain . Furthermore, when and when , .

Let and in definition (39), we then have

Thus, we can obtain

From Lemma 4, the upper bound on the last term of the right-hand side of (38) can be provided by

Substituting (44) and (45) into (38), it yields

In the same way, let denote the total traffic demand, denote the average of , and . Then, we can rewrite (46) as

Furthermore, we define as the actual average travel cost of all travelers at multiclass SO. Then, the theorem is presented as follows.

Theorem 2. *Let denote a class of differentiable, separable, convex, and monotonically increasing link time functions . Let be the TSTC at the multiclass CL-SUE and be the TSTC at the multiclass SO under monetary-based criterion. Then,*

#### 4. Bounding the Inefficiency of the Multiclass CL-SUE Compared with the Multiclass CL-SSO

Before analyzing the inefficiency of the multiclass CL-SUE against the multiclass CL-SSO, we first give a brief introduction to the corresponding multiclass CL-SSO model. Considering commonality factor reflecting path overlapping, we establish the multiclass CL-SSO model, which is the extension of research achievements in [39, 48]. The definition of SSO given by Maher et al. [48] states that the SSO problem is to minimize the TPTT. Therefore, the multiclass CL-SSO problem is also to minimize the TPTT. For the multiclass CL-SUE model, the TPTT in the transportation network can be provided by (the process of proof is shown in Appendix A)

Substituting (16) and (17) into (49), respectively, and in view of (1), the minimization of TPTT can be formulated as the following minimization problem under time-based criterion and monetary-based criterion, respectively.

Let be path flow solution of problem (50), the corresponding link flow solution is denoted by . Then, the minimum TPTT is denoted by . Let be path flow solution of problem (51), the corresponding link flow solution is denoted by . Then, the minimum total perceived travel cost (TPTC) is denoted by . They are given by

Correspondingly, let and denote the TPTT and the TPTC at the multiclass CL-SUE, respectively. Then, we have

By definition, we easily have and . However, it can be found from (52)–(54) that , , , and may be negative, which implies that the ratio and may be meaningless. Therefore, instead of using the ratio and , we use the terms and to quantify the absolute inefficiency of the multiclass CL-SUE against the multiclass CL-SSO under time-based criterion and monetary-based criterion, respectively. They are defined as

##### 4.1. Time Units

Setting and in VI (18), we can obtain inequality as follows:

This leads to

Thus,

We can get the upper bound of the term in (58) from Lemma 3. Let , we can obtain inequality below.

The upper bound on the last term of the right-hand side of (58) is equal to zero from Gibbs’ inequality. Therefore, the following inequality holds:

Here, if and only if . Substituting (59) and (60) into (58), the following theorem can be obtained.

Theorem 3. *Let denote a class of differentiable, separable, convex, and monotonically increasing link time functions . Then, the absolute inefficiency of the multiclass CL-SUE against the multiclass CL-SSO under time-based criterion, , is upper bounded, i.e.,*

We now begin to discuss the tightness of the bound given in (61) and provide Corollary 2 (the similar process of proof is shown in Guo et al. [37]).

Corollary 2. * if and only if .*

Corollary 2 It states that the upper bound (61) is tight if and only if (without traffic congestion). Here, we have and . However, traffic congestion is a common phenomenon in a realistic transportation network. As a result, we always have , which means that the upper bound (61) is usually not tight in most cases.

##### 4.2. Monetary Units

Setting and in VI (18), we can obtain the inequality as follows:

This leads to

Thus,

We can get the upper bound of the term in (64) by applying the definitions (39)–(42). Let be and be , then we havewhere is the TSTC at the multiclass CL-SSO.

The upper bound on the last term of the right-hand side of (64) is equal to zero from Gibbs’ inequality. Therefore, the following inequality holds:

Here, if and only if . Substituting (65) and (66) into (64), we can obtain the theorem as follows.

Theorem 4. *Let denote a class of differentiable, separable, convex, and monotonically increasing link time functions . Then, the absolute inefficiency of the multiclass CL-SUE against the multiclass CL-SSO under monetary-based criterion, , is upper bounded, i.e.,*

Next, we will discuss the tightness of the bound given in (67) and provide Corollary 3 as follows.

Corollary 3. * if and only if .*

*Proof of Corollary 3. *Firstly, if , we can obtain from Theorem 4. According to the definition of the multiclass CL-SSO, we have . Hence, we can obtain , which means that . Since , we have . Therefore, if . Secondly, we assume that holds. From the specific derivation process of (67), implies that inequality (66) takes equality; i.e., . Since , we can have , , and . With definitions (39)–(42), if holds, we can obtain . The proof is completed.

Corollary 3 shows that the upper bound (67) is tight if and only if . The expression means that the multiclass CL-SUE state is equivalent to the multiclass CL-SSO state. Here, we have , , , and , which are usually not satisfied in a realistic transportation network. As a result, we always have , which means that the upper bound (67) is usually not tight in most cases.

#### 5. Effects of Parameters on the Inefficiency Bounds

This section discusses the effects of various parameters on the upper bounds (33), (48), (61), and (67), namely, , , , , , , and .

is a dimensionless coefficient defined only by the class of link time functions. Both the upper bounds (33) and (61) are monotonically increasing functions of . Consider a widely used class of link travel time functions, , where denotes free-flow travel time on link , is a constant, and reflects the degree of traffic congestion. Roughgarden [27] provided a specific expression of as follows:

Equation (68) shows that when (without traffic congestion) and when (with severe congestion). When and , and , respectively. So, the upper bounds (33) and (61) are both monotonically increasing functions of . When the value of can be reduced, we will have more space to improve the congestion in the traffic network by driving the multiclass CL-SUE state to the multiclass SO or the multiclass CL-SSO state.

Both