A BI-OBJECTIVE MODEL FOR SITING PARK-AND-RIDE FACILITIES WITH SPATIAL EQUITY

A bi-objective programming model (BP) with spatial equity constraints is proposed to site park-and-ride (P&R) facilities in traffic networks. Both the number and locations of P&R facilities are determined. The maximal coverage and minimal resource utilization criteria, which are generally conflicting, are simultaneously considered to reveal the trade-off between the quality and cost of coverage. Furthermore, the concept of passenger flow volume per cost is defined and several properties of the model solutions are analyzed. Finally, this model is applied to site P&R facilities in Anaheim, California. Application results show the trade-offs associated with passenger flow volume, cost and passenger flow volume per cost, and the effects of spatial equity constraints on the spatial deployment of P&R facilities.


INTRODUCTION
Continued reliance on private automobiles is not likely to be sustainable for urban development, due to limited natural resources, environmental pollution and road congestion issues.Providing reliable and efficient public transportation is an effective way to overcome car dependency and alleviate urban congestion.One of the options considered entails the use of park-andride (P&R), a combinational mode of private cars and public transportations, allowing commuters to use their own cars in the beginning of their trip but switch to public transit at some point later in the same trip.P&R facilities can help alleviate traffic congestion and other adverse external effects of private vehicle travel.
Studies in the United Kingdom and North America have concluded that P&R facilities are effective in reducing congestion [1,2].This importance is reflected as well in transportation planning of other cities like Beijing, where 26 large-sized P&R facilities are planned to be built in the vicinity of the subway lines during Beijing's Twelfth Five-Year Plan period [3].
There is no doubt that the P&R facilities should be well planned to continue to play a role in effective transit systems.One essential aspect when planning for P&R services is finding appropriate geographical locations for the facilities to provide services.Planning and locating P&R facilities is a complex activity that must take multiple factors into account, including potential demands, characteristics of potential users, impacts on the surrounding neighborhoods, and the overall economic performance in terms of benefits and costs [4].From an economic perspective, there are at least two important concerns when siting P&R facilities, i.e., covering as many potential users as possible, and spending as little as possible.The former concern is important as this provides another travel alternative to private automobiles.Generally, a user is regarded as covered if they are within a maximum acceptable distance from a facility, but certainly demand decreases as the distance from services increases.Thus, covering as many users as possible within a coverage distance of a facility is a major concern.In addition, due to the heavy investment of transport infrastructure projects and the limited budget, the cost becomes another major concern.
Existing literature related to the P&R facility location problem can be generally categorized into three classes, i.e., decision support system [5,6], equilibrium analysis [7,8] and mathematical programming [9][10][11][12][13].Horner and Grubesic [5] and Faghri et al. [6] A BI-OBJECTIVE MODEL FOR SITING PARK-AND-RIDE FACILITIES WITH SPATIAL EQUITY CONSTRAINTS X. Lu, R. Guo: A Bi-Objective Model for Siting Park-and-Ride Facilities with Spatial Equity Constraints developed various hybrid knowledge-based expert system and geographic information system tools to determine the optimal location of a P&R facility.These systems provide multiple criteria to investigate alternative solutions, such as close to the central business district (CBD), the competition around the parking lot, and near the subway station.Their studies focus on the system development and do not explicitly use the optimization model to describe the problem.Through the equilibrium analyses, Wang et al. [7] optimized the location and parking charge of a P&R facility in a linear city under the assumption that the traffic demand is uniformly distributed along the corridor.Liu et al. [8] developed a competitive railway or highway system with multiple P&R services in a corridor in which commuters choose between the drive only alternative and the P&R facilities located continuously along the corridor to characterize the equilibrium mode choice.Equilibrium analysis is limited to a particular travel corridor.Sargious and Janarthanan [9] proposed a method for locating P&R facilities such that the total cost of the system is minimized.Honer and Groves [10] proposed a network flow-based model for location analyses of rail P&R.Farhana and Murray [11] proposed a multi-objective spatial optimization model to account for P&R application-specific objectives.These objectives cover as much potential demand as possible, using the existing P&R facilities as much as possible and as close as possible to major roadways.Fan et al. [12] used a bi-level programming model to locate P&R facilities considering the interaction between decision makers and commuters.Commuters are assumed to follow a stochastic user equilibrium behaviour.Aros-Vera et al. [13] proposed a mixed linear programming formulation to determine the location of a certain number of P&R facilities so that their usage is maximized.The facilities are modelled as hubs.Nevertheless, these works, mentioned above, do not address one important issue, i.e., spatial equity.
Bröcker et al. [14] pointed out that the goal of all transport infrastructure projects should be evaluated from social welfare perspective, which is reflected in two aspects, efficiency and social equity.Although the demand coverage issue has been studied a lot in the previous traffic literature [15], so far as we know, most models cannot address the trade-off associated with differing levels of importance for coverage and cost objectives, while taking spatial equity into consideration.It may cause some P&R facilities clustering with spatial equity constraint missing, and the spatial distribution of public resources becomes unfair.Recently, Lu and Huang [16] studied the P&R location problem with spatial equity constraints, but their earlier work ignored several important properties of the model solution, which will be developed further in this paper.
From cost-benefit perspective of public transport infrastructure investment, incorporating the spatial equity factor, this paper proposes a bi-objective programming model for the P&R facility location problem.In the model, the objectives are to maximize the number of P&R users and to minimize the cost of constructing P&R facilities simultaneously.Using a linear weighted technique, the bi-objective model is transformed into a single objective model.The concept of passenger flow volume per cost is introduced.Some properties of the model are analyzed and investigated.Finally, this model is applied to the transportation network in Anaheim, California.

MODEL DESCRIPTION
In this section, the proposed model is described in detail.Before doing this, the notations involved in the model are given as follows.Let i and I be the index and index set of areas, respectively, J is the index set of potential facility sites, a i the demand in area i, c j the cost of locating a P&R facility on site j and a non-negative integer, ˆij a the demand in area i anticipated to use P&R facility j based on relative proximity, d ij the distance between area i and potential facility site j, E the maximum acceptable service standard (distance), and α the minimum acceptable distance between two P&R facilities.Location variable x j is defined such that x j =1 if a facility is located at site j and x j =0 otherwise.Allocation variable y ij is defined such that y ij =1 if the demand in zone i is served by a facility at site j and y ij =0 otherwise.
Based on the assumption of user rationality, Holguín-Veras et al. [17] studied the catchment area of a given P&R site and the optimal P&R facility location in a linear and two-dimensional cities, respectively.They found that an ellipse could approximate the catchment area of a P&R facility in a two-dimensional city.Here, the following piecewise distance decay function is used to modify the demand attracted , , ˆ0, , where β is the distance decay parameter, and β>0.
The larger parameter β is, the more quickly the attracted demand ˆij a decays with distance d ij .Other types of functions can also be utilized to represent the distance decay, such as the power function [18].It is worth mentioning that our model is different from the classical set covering model and maximal covering model.In the two classical models, although there exists a criterion of coverage, namely the distance threshold, E, it is supposed that all the residents within the threshold distance will patronize the facility.That is to say, the coverage function in the two classical models is binary: either the demand is covered or not covered.Thus, the two models cannot express other distance decay forms, for example, the decrease of the demand with distance in a piecewise or non-linear manner.
From the perspective of P&R facility investors, they hope to make as much profit as possible, while paying as little as possible.Profit is composed of two parts: parking fee and transit fare.Suppose the parking fee and the transit fare are fixed.Let the sum of the parking fee and transit fare be p.Then the total revenue is ∑∑ .Since parameter p is a constant, the ob- jective of maximizing the revenue could be simplified setting p=1 and maximizing the coverage With the above notations, this proposed model is formulated as: x y In model (BP), objective (2) maximizes the passenger flow volume using P&R facilities and objective (3) minimizes the cost of constructing the P&R facilities.If the construction cost at each potential site is identical, objective (3) could be simplified setting c j =1 and minimizing the number of P&R facilities selected, i.e.

min ( )
Constraints (4) require the demand is assigned to a facility only if it is open.Constraints (5) ensure the demand of each area is allocated to no more than one site.This implies the user rationality assumption, namely, if the demand is inside the threshold distances of multiple facilities, they will choose the nearest one.Constraints (6) are the spatial equity constraints.Here, we define the spatial equity issue in the sense that several P&R facilities may cluster if there are no spatial distance constraints.The constraints ensure the distance of any two P&R facilities is not less than a certain value.In other words, the spatial clustering phenomenon could be avoided and the spatial distribution of P&R facilities in a city could reach a balance to some extent.Note that it is different from that in the context of road pricing.The spatial equity in road pricing literature is defined as the difference of the generalized travel costs of drivers travelling between several origin-destination (O-D) pairs.The spatial equity may be significantly different when tolls are charged at some selected links [19].Constraints (7) are the integrality constraints.

MODEL ANALYSIS
Let S denote the feasible region of the model (BP), namely, S is the set of all (x,y) that satisfy equations ( 4) to (7).The model can be transformed into a single objective problem by the linearly weighted method, i.e., where ω is a constant, and ω∈(0,1].If ω≠0, then the model could be rewritten in another form where x y S ∈ , if there is no ( , ) x y S ∈ , such that z x y z x y ( , ) ( , ) x y is an efficient solution.Here, z x y z x y ( , ) ( , ) x y S ∈ .Definition 2. Given a feasible point of (BP), ( , ) * * x y S ∈ , if there is no ( , ) x y S ∈ , such that z x y z x y ( , ) ( , ) x y is a weakly efficient solution.
Definition 3. Q y C x ( ) ( ) is called passenger flow volume per cost (PFVC), where Q(y) is the total passenger flow volume (PFV) using P&R facilities, and C(x) is the total cost for building the P&R facilities.
We put forward the concept of PFVC, which could be regarded as an index of evaluating the return of investment on transport infrastructure investment, like P&R facilities.Generally speaking, marginal profit is used to evaluate the rate of investment return in economics.Since the profit comes from ticket revenue in this model and ticket revenue is proportional to PFV, we could adopt the PFVC to investigate the level of the rate of investment return.
Theorem 1.Given λ≥0, an optimal solution of the single objective problem (SP λ ) must be a weakly efficient solution of the model (BP).Specially, when λ>0, an optimal solution of the single objective problem (SP λ ) must be an efficient solution of the model (BP).
Proof.Let ( , ) * * x y be an optimal solution of (SP λ ), but not a weakly efficient solution of the model (BP).According to Definition 2, there exists ( , ) ′ ′ ∈ x y S , such that z x y z x y ( , ) ( , ) ( )⋅ ′ ′ > ( )⋅ z x y zx y .This implies that ( , ) * * x y is not an optimal solution of (SP λ ), which leads to a contradiction.Specially, when λ>0, the theorem can be proved similarly.The proof is completed.The model (SP λ ) is closely related with the following problem (F 1 ) Firstly, the relationship of model solutions between models (SP λ ) and (F 1 ) is given by Lemma 1. Furthermore, we investigate the properties of the model (SP λ ).Lemma 1.If λ * is the optimal value of the objective function of the problem (F 1 ), then we have i) ƒ(λ) is a convex, piecewise linear and decreasing function with respect to λ≥0; ii) If ƒ(λ)=0, then λ=λ * ; iii) The optimal solution of (SP λ* ) is identical to that of (F 1 ).For the proof of Lemma 1, readers may refer to Ref. [20].Lemma 1 will be helpful for the proof of Property 1 to 3.
From Property 1 and 2, we know that PFVC first increases, then decreases and then remains unchanged with λ≥0.We wonder whether there are some characteristics of the solution when the highest PFVC is achieved?
Property 3. The optimal solution of (SP λ* ) is given by a single open P&R facility that offers the highest PFVC.
Proof.According to (iii) of Lemma 1, we know that the optimal solution of (SP λ* ) is identical to that of (F 1 ).With only one facility located at j, let the objective function value of problem (F 1 ) be z(j), and z j Q C j j ( ) = , where Q j and C j are the corresponding passenger flow volume and building cost.Repeat for j J = 1,..., , and reorder the indices so that z z z J ( ) . Thus, we have z( 1)≥z (1,2).Using an inductive approach, we could show that max PFVC cannot exceed z(1) for any combination of open sites.The proof is completed.

APPLICATION EXAMPLE
The above bi-objective programming model is applied to site P&R facilities in Anaheim, California, which is shown in Figure 1.The Anaheim network consists of 38 areas, and each of them is represented by the symbol '○' with a number.These positions are also selected as the potential sites for P&R facilities.Travel demand data of each area are obtained from the homepage of traffic network testing problem [21], see Table 1.The distance information could be obtained from Figure 1.The construction cost data of locating P&R facilities assumed in this paper are shown in Table 2.We use the Lingo software to solve the model (SP λ ).The consuming time is so little that it can be negligible.Here, parameter E is set large enough so that the area demand is al-ways covered by P&R facilities, but it declines with the distance.Other parameter values are set as α=3(km) and β=0.2.Note that we set α here to show the effect of spatial equity constraints.Since the change of α does not lead to general conclusions, we will not do sensitivity analysis about parameter α.The location results of P&R facilities are shown in Table 3 when parameter λ takes different values.As can be seen from Table 3, when λ≥0, ƒ(λ) is a decreasing function.In this case, when λ * =134.3,we have ƒ(λ * )=0; when λ=λ * =134.3, the optimal solution ( ( ), ( )) , where λ 0 can be set to 150.When λ>λ 0 , Q(y * ), C(x * ) and the optimal location of (SP λ ) remains unchanged.The upper bound of the PFVC is λ * =134.3 with λ ∈ +∞ [ , ) 0 .These results support Lemma 1, Property 1 and 2 in Section 3. In addition, the optimal location of (SP λ* ) is given by a single open P&R facility 37 offering the highest PFVC, which verifies Property 3.
Figure 2 shows the effective solutions of the model (BP) and the trade-offs related with PFV, PFVC and cost.These effective solutions (Pareto optimal solutions) form the Pareto frontier, represented by the curve marked by a dash line.It reflects the trade-off between the coverage goal and the cost goal.As can be seen from Figure 2, the two goals are conflicting.The rightmost point of Figure 2 yields the largest volume, and also most cost.From the slope of the Pareto frontier, the right two solutions may not be favored by the decision makers, since a large number of construction costs only brings a small increase in volume.
Then, the trade-offs associated with PFV, PFVC and cost should be carefully investigated.Since the PFVC can be used as an index of evaluating the return of investment, it is necessary to take the PFVC into the decision process.The trade-off between PFVC and cost, is represented by the curve marked with a real line, see Figure 2. Along with the increase in coverage and construction cost, the PFVC first increases and then decreases.Although the three points on the left side of the two figures are close to or reach the maximal PFVC, their location results are only one parking lot to build in the Anaheim network, which is impossible in reality.Thus, the decision makers could select from the rest four options, which corresponds to cases λ=3, 6, 12, and 25, respectively.is the result of our model; there are 5 transfer facilities selected, the optimal deployment tends to be more spread-out due to the spatial equity constraints.

CONCLUSION
P&R services are an integral element of public transportation system.Locating P&R facilities is an important step when planning for such services.Coverage and cost issues are two major concerns from economic perspective.This paper develops a bi-objective programming model to account for P&R application specific objectives.The objectives are conflicting but integrated into one model.Moreover, the developed model addresses the issues of broad importance in location analysis in general, such as spatial equity and distance decay.This model is transformed into a single objective problem via the linearly weighted method.The concept of the passenger flow volume per cost is defined.Properties of solutions of the model are studied.Finally, this model is applied to the traffic network in Anaheim, California.
It should be pointed out that spatial equity in this paper is considered from the point of view that it may lead to inequity if the facilities are too close to each other.However, it does not mean that the greater the distance, the more fair the deployment.Further research about spatial equity of P&R facilities should be discussed in the future.In addition, demand allocation is based on the distance decay function in our paper.It is meaningful to take travellers' route choice behaviour into consideration.Next, it is assumed that the capacities of P&R facilities are infinite in our model.In fact, the actual space of P&R facility has great impact on their spatial deployment and the travellers' transferring choice [22].It would be insightful to explore such capacitated P&R facility location models.Finally, the travel demand may fluctuate year by year.It is also interesting to study the P&R facility location problem under demand uncertainty [23].

X
. Lu, R. Guo: A Bi-Objective Model for Siting Park-and-Ride Facilities with Spatial Equity Constraints λ * ≤ λ 1 < λ 2 , the result could be proved similarly.The proof is completed.From Property 1 (ii), we know that the PFVC first increases and then decreases with λ ∈ +∞ [ , ) 0 .When λ=λ * , the PFVC takes the maximum λ * .When λ λ * are the lower bound and upper bound of the PFVC, respectively.When λ λ ∈ +∞ [ , ) * , λ * is the upper bound of the PFVC, what is the lower bound?Property 2 gives the answer as below.

Figure 2 -Figure 3
Figure 2 -Effective solutions of the model (BP) and the trade-offs related with PFV, PFVC and cost

Figure 3 (
Figure3displays the locations for P&R facilities when λ=6.Figure 3(a)  is the result by the model without spatial equity constraints.A total of 10 P&R facilities are located, whereas spatial clustering phenomenon is generated to some degree, as highlighted by the dashed ellipses.Figure 3(b)  is the result of our model; there are 5 transfer facilities selected, the optimal deployment tends to be more spread-out due to the spatial equity constraints.

Figure 3 (
Figure3displays the locations for P&R facilities when λ=6.Figure 3(a)  is the result by the model without spatial equity constraints.A total of 10 P&R facilities are located, whereas spatial clustering phenomenon is generated to some degree, as highlighted by the dashed ellipses.Figure 3(b)  is the result of our model; there are 5 transfer facilities selected, the optimal deployment tends to be more spread-out due to the spatial equity constraints.

Table 1 -
Input data of the area demand

Table 2 -
Input data of the construction cost of P&R facilities

Table 3 -
Effects of Parameter λ on the locations of P&R facilities