1. Introduction
Park and ride (P&R) facilities are a significant component of the public transportation system in many cities around the world. The facility has been recognized as part of sustainable development for many decades. With globalization, economic growth brings more business activities [
1], and the desire to use public transportation more frequently depends on the ease of car access [
2,
3]. Hence, the selection of P&R facility locations becomes essential in encouraging drivers to transfer from their private vehicles to public transportation to alleviate traffic congestion in the urban areas [
4].
The mathematical programming (MP) approach plays an important role in the P&R facility location problem. The optimal facility location is determined through the interaction of the travel choice behavior and the level of service (LOS) of the P&R locations in a transportation network. The pioneer location model was provided by [
5], which assumes an all-or-nothing (AON) assignment (i.e., all travel demands are entirely assigned to the closest facility). Several studies relaxed the AON assumption using the attractiveness of the facility to determine the assignment [
6,
7]. Most of these studies adopted the gravity-based model [
8,
9] and the random utility maximization (RUM) model. The gravity-based model usually considers the attractiveness through the negative exponential impedance or the negative power impedance of the distance. The RUM model considers the attractiveness through the travelers’ perception. The travelers’ observed utility is measured and incorporated with the unobserved utility [
10]. A market share for the open facility is determined based on the travelers’ choice behavior that maximizes their individual utilities. The well-known multinomial Logit (MNL) model is usually adopted to represent such choice behavior for the parking selection [
11,
12]. Benati and Hansen [
13] provided a MNL model-based MP and a linear reformulation of the nonlinear MNL probability in the facility location problem. Haase [
14] adopted the MNL model with a constant substitution pattern to provide a linear reformulation. Aros-Vera et al. [
15] utilized this method for P&R facility location in a hypernetwork [
16], where the level of service (LOS) or travel cost of each intermodal transport journey is considered through the performance of links, and hence routes, in a one-dimensional transportation network. The study in [
17] employed a similar linear reformulation of Benati and Hansen [
13] to give an alternative formulation. In all these approaches, Haase and Muller [
18] argued that the constant substitution pattern assumption seems to be superior to other formulations. Further, Liu and Meng [
19] and Liu et al. [
20] provided the bus-based P&R facility location. Pineda et al. [
21] integrated the traffic and public transportation systems for the P&R facility location.
The drawbacks of the MNL model include: (1) inability to consider the similarity between the choice alternatives and (2) inability to consider the heterogeneity [
22]. These two drawbacks stem from the independently and identically distributed (IID) assumption embedded in the random error term of the MNL model [
23]. In the hypernetwork, the MNL model cannot account for route similarity (route correlation or route overlapping) [
20,
24,
25,
26] and the heterogeneous perception variance from different trip lengths [
22,
27]. It has been shown that heterogeneity is an important factor of the P&R facility location selection [
28].
In this study, we developed a mathematical programming (MP) formulation to consider the travelers’ heterogeneity in selecting a P&R facility location. The multinomial Weibit (MNW) model [
29] was adopted to account for the heterogeneity among the intermodal journey alternatives. The independence of irrelevant alternatives (IIA) property of MNW was explored and used to provide a linear reformulation of its non-linear choice probability. Application of the proposed mixed integer linear programing (MILP) is demonstrated in a real-size transportation network. The numerical examples indicate a significant impact of heterogeneous perception variance on the optimal P&R facility locations. The MNW route-specific perception variance as a function of trip length is more sensitive to a change in distance-based public transport fare structure, and hence, the P&R facility location selection.
The paper is organized as follows.
Section 2 provides a list of notations used in this study.
Section 3 gives some background of the MNL and MNW models. In
Section 4, the proposed MILP is developed with a rigorous proof.
Section 5 shows two numerical examples to demonstrate features of the proposed model and its applicability in a realistic transportation network.
Section 6 concludes this paper.
4. MNW P-Hub Problem for P&R Facility Location
In this section, we propose a mathematical programming (MP) formulation for the P&R facility location problem based on the MNW choice behavior [
29]. This study considers two choice alternatives for a journey, including (1) private vehicle (or auto) and (2) public transport via a P&R facility [
14,
15,
18]. We assume that congestion is moderate, and travelers using public transport exclusively have no impact on the two choices.
The MNW route choice probability of travelers choosing to use a P&R facility
n on route
k between O-D pair
ij can be expressed as
where
is a set of routes between O-D pair
ij interchanging between private vehicles and public transport at P&R facility
n∈
N;
is a set of routes for private vehicles between O-D pair
ij; and
xn is a binary variable for the P&R facility
n∈
N, which has route
,
n∈
N passing through. If P&R facility
n is selected (open),
xn = 1 and
∈ (0,1]. If P&R facility
n is not selected (close),
xn = 0 and
= 0. On the other hand, the probability of selecting a journey from
i to
j with a private vehicle with the route travel cost of
is
Consider the following MP formulation:
s.t.
and Equations (12) and (13), where
qij is a given travel demand between O-D pair
ij. The objective function in Equation (14) is to maximize the number of P&R users, and hence reduce the overall number of private vehicles in the study area [
14,
15]. Equation (15) constrains the number of P&R facilities to
p. Equation (16) is the route choice probability conservation and Equation (17) declares the P&R facility location decision variables are binary.
To develop a mixed integer linear program (MILP) for the above MP, Equations (12) and (13) were linearized as follows. Like the MNL model, the MNW model also exhibits the IIA property, i.e.,
The probability
is always greater than zero. Meanwhile. the probability
has a value between 0 to 1. This probability will be greater than 0 only if
xn = 1 as presented in
Figure 1. We can state the following condition:
Hence, we can rewrite the probability ratio as follows:
Since the private vehicle route choice probability is greater than zero, the IIA property in Equation (20) is always satisfied. In contrast, the IIA property for the routes using the P&R facility is related to
xn. Equation (21) is satisfied only when the P&R locations
m and
n are both selected. If only one of them is selected, all probabilities will be zero. Similarly, Equation (22) is satisfied only when the P&R location
n is selected. If not,
= 0 and
= 0 cannot simultaneously occur. Therefore, the term (1 −
xn) was added to Equations (21) and (22) [
14,
15,
32], and we have
The maximization of the P&R users in the objective function would work with Equations (24)–(26) to obtain the MNW IIA property in Equations (21) and (22). A number of equations for each O-D pair according to Equations (23)–(26) can be determined by
The second term is according to Equation (23) with an equal sign. Thus, under the same number of routes, the O-D with more private vehicle routes has a fewer number of equations for these constraints.
Proposition 1. The mixed integer linear program (MILP) in Equations (14)–(17) and (19), and Equations (23)–(26) generates the maximum number of P&R facility users under the MNW travel choice behavior.
Proof. Assume that there are at least two routes connecting each O-D pair, one for the private vehicles only and the other for the vehicles using the P&R facility locations. We separate them into two cases: (a) xn = 0 and (b) xn = 1. □
Case (a): When
xn = 0,
,
from Equation (25),
from Equation (24), and
from Equation (26). With the probability conservation for each O-D pair in Equation (16) and the IIA property, we have the MNW travel choice behavior, i.e.,
This equation provides the MNW travel choice model in Equations (12) and (13).
Case (b): When xn = 1, , , and .
This results in the IIA property of the MNW model in Equation (18). From Equation (16),
Similarly, we have the MNW travel choice model in Equations (12) and (13). Hence, the MILP in Equations (14)–(17) and (19), and Equations (23)–(26) gives the optimum number of the P&R facility users under the MNW choice behavior. This completes the proof.