Exploring the Impact of Commuter’s Residential Location Choice on the Design of a Rail Transit Line Based on Prospect Theory

Abstract

This paper explores the impact of prospect theory based commuter’s residential location choice on the design problem of a rail transit line located in a monocentric city. A closed-form social welfare maximization model is proposed, with special consideration given to prospect theory based commuter’s residential location choice over years. Commuters are assumed to make residential location choice by a trade-off between daily housing rent and generalized travel cost to minimize their prospect values. The solutions properties of the proposed model are explored and compared analytically. It is found that overestimation exists for the optimal solutions of rail line length, headway, and fare based on traditional utility theory, compared with the optimal solutions of the proposed prospect theory based model. A numerical example is given to illustrate the properties of the proposed model.

1. Introduction

Rail transit lines are being launched in many cities of China in recent years, due to the rapid development of economy and the dramatic growth of urban population. For instance, the Shanghai Municipal Government has commenced the project of extending railway line 11 about 5.76 km westwards with a total of four stations recently. In Hong Kong, a new metro line connecting Shatin new town to the central with a total length of 17 km and ten stations are also being built, which starts in 2011 and is expected to finish in 2019.

Rail transit lines can alleviate boring traffic congestion and make life more convenient as regards maneuverability for a specific set of people, namely, those living in the vicinity of the line and new stations to be constructed. Hence, commuters prefer living along the candidate rail transit lines so as to enjoy such advantage of rail service.

In many areas, especially in cities with high population densities like Shanghai and Hong Kong, commuters’ behaviour of making residential location choice and rail travel mode choice simultaneously has been identified [1–3]. In other words, commuters’ behaviour of simultaneous residential location choice and rail travel mode choice can directly affect the performance of the candidate rail transit line. The output results of the above commuter’s simultaneous residential location choice and rail travel mode choice are the population densities in residential locations.

The discrete choice models were largely used to determine the residential location choice with generalized travel costs of various travel mode choices as the determinant factors [4–6]. The discrete choice models can help estimating population densities in residential locations, and explain the trade-offs commuters are faced with. Nevertheless, their use has been criticized in that most of the discrete choice models were proposed based on utility theory.

Although utility theory was applicable in many contexts, it may be inadequate in estimation of population densities over a long-term planning horizon. Before the reach of relative equilibrium of population densities, commuters undergo a relative long-term learning process of residential location choice and rail mode choice. This long-term learning process can be partially attributed to the existence of perception error and uncertainty on housing rent and generalized travel cost. Unfortunately, the long-term learning process cannot be captured by utility theory.

Prospect theory can be seen as an extension of utility theory. Compared with utility theory, which are based on normative preference axioms, prospect theory describes lotteries choices by a two-step process: an initial phase of editing and a subsequent phase of evaluation [7, 8]. Because of the property of two-step process, prospect theory can be used to describe the above long-term learning process of commuters.

Reference point is a key parameter of prospect theory. However, there are no prefect models to predict the value of reference point in transportation models [9]. Some transportation models associated with prospect theory are summarized in Table 1. Katsikopoulos et al. [10] investigated car drivers’ risk preference behaviour on route choice with travel time of the reference route as reference point. Senbil and Kitamura [11] explored commuters’ departure time choice with work start time as reference point in decision frame 1 and preferred arrival time as reference point in decision frame 2. By contrast, Jou et al. [12] examined commuters’ departure time with the reference point of earliest acceptable arrival time and work start time. Xu et al. [9] modelled drivers’ route choice with effective reversed time as reference point.

Table 1 

Modes and reference points in some previous models.

Our goal is to explore the impacts of prospect theory based commuters’ residential location choice on the design of a rail transit line. Commuters are assumed to choose residential locations along the candidate rail transit line by a trade-off between daily housing rent and generalized travel cost. To capture the above long-term learning process of residential location choice and rail mode choice simultaneously, two reference points are adopted: willingness-to-pay on daily housing rent and willingness-to-pay on generalized travel cost.

Commuters’ residential location choice is affected by many design variables of a rail transit line, such as rail line length, rail station locations (spacing), headway, and fare. Specifically, rail line length is closely concerned with the coverage area of rail service; railway station locations (spacing) have a direct effect on the train operating speed, dwelling delays of trains, and in-vehicle time of commuters at stations; headway could be used to determine the waiting time of commuters at stations and fare is a component of generalized travel cost.

Normally, the above four design variables can be distinguished into two types: long-term and short-term decision variables. Long-term decision variables cannot be changed during operation stage, but short-term variables can be updated. For example, rail line length and rail station locations (spacings) should be determined during planning stage and are inflexible to change during operation stage, whereas fare and headway are still flexible to unevenly change in actual operation.

Commuters’ generalized travel cost consists of fare and various travel costs, including access time cost, waiting time cost, and in-vehicle time cost. Specifically, access time cost is closely concerned with rail line length and rail station locations (spacing). Waiting time cost depends on headway. In-vehicle time cost is a function of distance between commuters’ residential locations and central business district (CBD).

In this paper, all commuters are assumed to work in the CBD of a monocentric city, and thus homework is a compulsory trip of commuters each day. The long-term planning horizon of rail transit line design is divided into several equal periods. In each period, rail service can be improved. After the implementation of rail service in each period, commuters make residential location choice by trade-off between daily housing rent and generalized travel cost.

The reminder of this paper is organized as follows. In the next section, assumptions and notations are given. Section 3 presents model formation. Some model properties are examined. In Section 4, a numerical example is used to illustrate the insightful findings of the proposed models. Section 5 concludes this paper.

2. Assumptions and Notations

A transportation corridor of 𝐵 km length is proposed, which extends from the CBD towards the boundary of the city, as shown in Figure 1. There is an ordered sequence of stations {1,2,…,𝑁(𝜏)+1}. The symbol 𝐷_𝑠 represents the distance between station 𝑠 and the CBD, 𝑁(𝜏) represents rail station number and 𝐷^𝜏₁ is the rail line length in period 𝜏. The considered designed variables include the combination of rail line length 𝐷^𝜏₁, station location 𝐷_𝑠 or spacing (𝐷_𝑠−1−𝐷_𝑠), train headway ℎ(𝜏), and fare 𝑓_𝑠 [2].

Figure 1 

The rail transit line configuration in a monocentric city.

To facilitate the presentation of the essential ideas, without loss of generality, basic assumptions and notations are made in this paper, as follows.

2.1. Assumptions

(A1) Commuters are assumed to be homogeneous and they have the same preferred arrival time to the workplace located in the CBD and the same preferred daily housing rent for each residential location [13]. This assumption could be extended to multiclass commuters in further studies.

(A2) Commuters are assumed to board trains at the nearest rail station, and the trains stop at every station on the candidate rail transit line. This assumption has also been adopted by many previous works, such as those of Wirasinghe and Ghoneim [14] and Li et al. [2].

(A3) In-vehicle crowding cost in trains and moving costs for commuters from one place to another are not considered, since the proposed model is for long-term planning purpose of rail transit line. Other travel modes are not considered, because the main goal of this paper is to explore the prospect theory based commuters’ residential location choice on the design of a rail transit line. The situation considered here may emerge in a monocentric city with highly compact city centre. Commuters in this monocentric city live in the dispersed surrounding suburban area [15].

(A4) The original population density in the monocentric city is specified as a linear function. The original population density at distance 𝑥 from the CBD in period 𝜏 is defined as 𝑔(𝑥,𝜏)=𝑔₀(𝜏)(1−𝑒𝑥),∀𝑥∈[0,𝐵], where 𝑔₀(𝜏) represents the population density in the CBD of period 𝜏 and 𝑒 represents the density gradient describing how rapidly the density falls as the distance increases. Here, 𝑒>0 represents the fact that more commuters live at CBD area, while 𝑒<0 represents the fact that more commuters live at suburban area. Smaller value of 𝑒 means more decentralized city. Specifically, when 𝑒 equals 0, this linear population density function is reduced to a uniform one. With this assumption, the total number of population 𝐺(𝜏) in period 𝜏 is given by 𝐺(𝜏)=∫^𝐵₀𝑔₀(𝜏)(1−𝑒𝑥)𝑑𝑥 [2].

2.2. Notations

Consider the following: 𝑐(𝑥,𝑠,𝜏): commuters’ actual generalized travel cost for arriving at CBD from location 𝑥 from station 𝑠 by train in the 𝜏th period; 𝑟(𝑥,𝜏): commuters’ actual housing rent at the location 𝑥 in the 𝜏th period, 𝑥∈𝐗, and 𝐗 is the choice set; that is, many types of houses exist at location 𝑥; 𝑐(̃𝑥,𝑠,𝜏): commuters’ perceived generalized travel cost for arriving at CBD from location 𝑥 from station 𝑠 by train in the 𝜏th period; 𝑟(̃𝑥,𝜏): commuters’ perceived housing rent at the location 𝑥 in the 𝜏th period; 𝑐_{WTP,𝑥,𝜏}: commuters’ reference point to decide whether the generalized travel cost is high or low at the location 𝑥 in the 𝜏th period, which is called the commuters’ willingness-to-pay (WTP) on travel cost; 𝑟_{WTP,𝑥,𝜏}: commuters’ reference point to decide whether the housing rent is high or low at the location 𝑥 in the 𝜏th period, which is called the commuters’ willingness-to-pay (WTP) on housing rent; 𝑝(𝑥,𝜏): the probability of obtaining low living cost in terms of the generalized travel cost and housing rent at location 𝑥 in the 𝜏th period; Δ𝑟𝑐(𝑥,𝑠,𝜏): the deviation between perceived living cost and reference points in terms of the generalized travel cost and housing rent at location 𝑥 in the 𝜏th period; 𝜋(𝑝(𝑥,𝜏)): the probability weighting function; 𝑉(Δ𝑟𝑐(𝑥,𝑠,𝜏)): value function of living at location 𝑥 and travelling to the CBD from station 𝑠 by train in the 𝜏th period; PV_{𝑥,𝑠,𝜏}: prospect value of living at location 𝑥 and travelling to the CBD from station 𝑠 by train in the 𝜏th period.

3. Model Formulation

The design of a rail transit line is considered over a planning horizon of [0,𝑇]. This horizon is divided into 𝑀 equal periods. The rail transit line is assumed to be implemented by an operator franchised by government. Social welfare maximization is the decision objective of rail transit line design. Commuters are assumed to make residential location choices as if they are prospect maximizers. This question can be formulated as a mathematically programming model with the objective of social welfare maximization, subjected to the constraints of prospect theory based residential location choice equilibrium condition.

3.1. Prospect Theory Based Residential Location Choice Equilibrium Condition

As stated above, prospect theory can be used to capture the learning process of commuters’ location choice over rail design periods in the planning horizon. As in Avineri [7], Wardrop’s [16] principle of user equilibrium could be extended to prospect theory based equilibrium, “Equilibrium under the condition that no commuter can decrease his/her choice prospect value by unilaterally switching his/her choice behaviour.” Mathematically, this equilibrium condition can be expressed as 𝑞(𝑥,𝑠,𝜏)>0⟹PV_{𝑥,𝑠,𝜏}=minPV_{𝑥,𝑠,𝜏}[],(𝜏)[],,∀𝑥∈0,𝐵𝑠=1,2,…,𝑁,𝜏∈1,𝑀 (1) where PV_{𝑥,𝑠,𝜏} represents prospect value of living at location 𝑥 and travelling to the CBD from station 𝑠 by train in the 𝜏th period and 𝑞(𝑥,𝑠,𝜏) is the peak-hour travel demand density at location 𝑥 to the CBD by train through station 𝑠 in period 𝜏.

Prospect value PV_{𝑥,𝑠,𝜏} could be calculated by the following equations: PV_{𝑥,𝑠,𝜏}=𝜋𝑝(𝑥,𝜏)𝑉(Δ𝑟𝑐(𝑥,𝑠,𝜏)),𝜋𝑝(𝑥,𝜏)=𝑝(𝑥,𝜏)^𝛾𝑝(𝑥,𝜏)^𝛾+1−𝑝(𝑥,𝜏)^1/𝛾,𝑉(Δ𝑟𝑐(𝑥,𝑠,𝜏))=Δ𝑟𝑐(𝑥,𝑠,𝜏)^𝛼′()((,Δ𝑟𝑐𝑥,𝑠,𝜏≥0−𝜆−Δ𝑟𝑐𝑥,𝑠,𝜏))^𝛽′,Δ𝑟𝑐(𝑥,𝑠,𝜏)<0,Δ𝑟𝑐(𝑥,𝑠,𝜏)=𝑐̃(𝑥,𝑠,𝜏)+𝑟̃(𝑥,𝜏)−𝑐_{WTP,𝑥,𝜏}−𝑟_{WTP,𝑥,𝜏},𝑐̃(𝑥,𝑠,𝜏)=𝑐(𝑥,𝑠,𝜏)+𝜉,𝑟̃(𝑥,𝜏)=𝑟(𝑥,𝜏)+𝜓, (2)(3)(4)(5)(6)(7) where 𝜉/𝜓 is a random disturbance term reflecting generalized travel cost/housing rent differences among residential locations.

The generalized travel cost, 𝑐(𝑥,𝑠,𝜏), consists of rail fare and various travel time cost, including access time cost, waiting time cost, and in-vehicle time cost. Specifically, it is defined as 𝑐(𝑥,𝑠,𝜏)=𝑓_𝑠(𝜏)+𝜙_𝑢𝑢_𝑠(𝑥)+𝜙_𝑠𝑤_𝑠(𝜏)+𝜙_𝑡𝑡_𝑠(𝜏), (8) where 𝑓_𝑠(𝜏) represents rail fare from station 𝑠 to the CBD in period 𝜏, 𝑢_𝑠(𝑥) represents commuters’ average access time to station 𝑠 from location 𝑥, 𝑤_𝑠(𝜏) represents commuters’ average waiting time at station 𝑠 in period 𝜏, 𝑡_𝑠(𝜏) represents commuters’ in-vehicle time to the CBD from station 𝑠, and 𝜙_𝑢,𝜙_𝑤,𝜙_𝑡 represent commuters’ value of time for access time, waiting time, and in-vehicle time, respectively.

The commuters’ waiting time at station 𝑠 in period 𝜏, 𝑤_𝑠(𝜏), can be given by 𝑤_𝑠(𝜏)=𝛼ℎ(𝜏),∀𝑠=1,2,…,𝑁(𝜏), (9) where ℎ(𝜏) represents the headway of railway service in period 𝜏 and 𝛼 is a calibration parameter which depends on the distributions of train headway and commuter arrival.

The commuters’ in-vehicle time from station 𝑠 to the CBD in period 𝜏, 𝑡_𝑠(𝜏), can be calculated by 𝑡_𝑠(𝜏)=𝑇_𝑠1(𝜏)+𝑇_𝑠2(𝜏),∀𝑠=1,2,…,𝑁(𝜏), (10) where 𝑇_𝑠1(𝜏)=𝐷_𝑠𝑉_𝑡(𝜏)(𝜏),𝑇,∀𝑠=1,2,…,𝑁_𝑠2=𝛽₀(𝑁(𝜏)+1−𝑠),∀𝑠=1,2,…,𝑁(𝜏), (11) where 𝑉_𝑡(𝜏) represents the average train cruise speed in period 𝜏, 𝐷_𝑠 represents the station distance from station 𝑠 to CBD defined as above, and 𝛽₀ represents the average train dwelling time at a station, which can be calibrated with observed data [17, 18].

To represent the demand-supply relationship of housing rental market, the following housing rent is assumed given by 𝑟(𝑥,𝜏)=𝛼_𝑠𝛽1+^𝑠𝑃(𝑥,𝜏)(𝐻()(,𝑥,𝜏−𝑃𝑥,𝜏)) (12) where 𝐻(𝑥,𝜏) (in terms of housing unit) denotes potential housing supply density at location 𝑥 in the 𝜏th period and 𝛼_𝑠 and 𝛽_𝑠 are positive scalar parameters that represent the fixed and demand-dependent components of the rent function around station 𝑠 [19].

In order to determine travel demand density 𝑞(𝑥,𝑠,𝜏), we first define the potential travel demand density at location 𝑥 in period 𝜏, which is denoted by 𝑃(𝑥,𝜏). Generally speaking, commuters’ destinations are normally distributed along the rail line with more concentration close to the CBD of course. Denote 𝜂 by the proportion of trips with CBD being the destinations in period 𝜏 and denote 𝜌 by the ratio of peak-hour flow to the daily average flow, and then 𝜂𝜌𝑔(𝑥) represents the peak-hour potential travel demand density in terms of (A4). We have 𝑃(𝑥,𝜏)=𝜂𝜌𝑔₀(𝜏)(1−𝑒𝑥)=𝑃₀(𝜏)(1−𝑒𝑥),∀𝑥∈[0,𝐵], (13) where 𝑃₀(𝜏) represents the peak-hour potential travel demand density in the CBD and 𝑃₀(𝜏)=𝜂𝜌𝑔₀(𝜏).

As stated above, travel demand density for rail service, 𝑞(𝑥,𝑠,𝜏), is closely concerned with several design variables, namely, rail line length, rail station or spacing, headway, and fare, in terms of the generalized travel cost. To represent such effect, a negative exponential elastic demand density function is used as follows: 𝑞()()(𝑐̃(,[](𝜏),𝑥,𝑠,𝜏=𝑃𝑥,𝜏exp−𝜃𝑥,𝑠,𝜏))∀𝑥∈0,𝐵,𝑠=1,2,…,𝑁 (14) where 𝜃(𝜃>0) represents the sensitivity parameter for the generalized travel cost and the perceived generalized travel cost 𝑐(̃𝑥,𝑠,𝜏) is given by (6) and (8).

3.2. Social Welfare of Candidate Rail Transit Line

Social welfare of the candidate rail transit line can be calculated by summation of operator’s net profit and consumer surplus of commuters. Mathematically, the social welfare in the planning horizon 𝜏∈[1,𝑀], SW, can be expressed as SW=^𝑀_𝜏=1𝑃_𝜏+CS_𝜏, (15) where 𝑃_𝜏 and CS_𝜏 are the operator’s net profit and consumer surplus of commuters in period 𝜏, respectively.

The operator’s net profit is closely concerned with revenue from fare and related construction and operation cost. Accordingly, 𝑃_𝜏 could be calculated by 𝑃_𝜏=𝑅_𝜏−𝐶_𝜏, (16) where 𝑅_𝜏 is the operator’s revenue in period 𝜏 and 𝐶_𝜏 is the related construction and operation cost in period 𝜏.

The operator’s revenue comes from fare. It could be calculated by summation of the number of commuters boarding at each station multiplied by the corresponding fare; that is, 𝑅_𝜏=^𝑁_𝑠=1𝑓_𝑠(𝜏)𝑄_𝑠,𝜏(1+𝑖)^𝜏−1, (17) where 1/(1+𝑖)^𝜏−1 is the discount factor in period 𝜏, 𝑖 is the interest rate, and 𝑄_𝑠,𝜏 is the travel demand of station 𝑠 in period 𝜏.

In terms of (A2), the travel demand of each station comes from coverage area of this station; that is, 𝑄_𝑠,𝜏=^𝐿𝑠−1_𝐿𝑠𝑞(𝑥,𝑠,𝜏)𝑑𝑥,∀𝑠=1,2,…,𝑁(𝜏), (18) where 𝑞(𝑥,𝑠,𝜏) is peak-hour travel demand density of rail service at location 𝑥 in period 𝜏 given by (14). 𝐿_𝑠 is commuter watershed line, which is located at the middle point of line segment (𝑠,𝑠+1) and the distance of commuter watershed line 𝐿_𝑠 from the CBD is given by 𝐿_𝑠=𝐷_𝑠+𝐷_𝑠+12,∀𝑠=1,2,…,𝑁(𝜏). (19) In particular, 𝐿₀ represents the maximum coverage location of rail service. Beyond this location, no one would patronize the rail service. Thus, 𝐿₀ holds 𝑞𝐿^𝜏₀,1,𝜏=0,𝐿^𝜏₀∈𝐷^𝜏₁,𝐵, (20) where 𝑞(𝐿^𝜏₀,1,𝜏) is travel demand density for station 1 at location 𝐿^𝜏₀ in period 𝜏. On the basis of (8)–(14), the maximum service coverage 𝐿₀ of the rail line can be given by 𝐿₀=1𝑒. (21) According to the (A4), (21) implies that railway service is available for all the residential people in the considered corridor [2]. Substituting (14) and (19) into (21), 𝑄_𝑠,𝜏 can be rewritten as 𝑄_𝑠,𝜏=𝑃₀(𝜏)^𝐿𝑠−1_𝐿𝑠exp−𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝑥+𝜆_𝑠(𝜏)𝑑𝑥−𝑒𝑃₀(𝜏)^𝐿𝑠−1_𝐿𝑠𝑥exp−𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝑥+𝜆_𝑠(𝜏)𝑃𝑑𝑥=−⁰(𝜏)𝑉_𝑡(𝜏)𝜃𝜙_𝑡exp−𝜃𝑐̃𝐿_𝑠−1𝑐̃𝐿,𝑠,𝜏−exp−𝜃_𝑠,𝑠,𝜏+𝑒𝑃₀(𝜏)𝑉²_𝑡(𝜏)𝜃²𝜙²_𝑡⋅𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠−1+1exp−𝜃𝑐̃𝐿_𝑠−1−,𝑠,𝜏𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠+1exp−𝜃𝑐̃𝐿_𝑠,𝑠,𝜏, (22) where 𝜆_𝑠(𝜏)=−𝜃𝑓_𝑠(𝜏)−𝜃𝜙_𝑢𝑢_𝑠(𝑥)−𝜃𝜙_𝑠𝑤_𝑠(𝜏)−𝜃𝜙_𝑡𝛽₀×(𝑁(𝜏)+1−𝑠)+𝜃𝜙_𝑡𝑢(𝑥)𝑉_𝑤𝑉_𝑡(𝜏)(𝜏).−𝜃𝜉,∀𝑠=1,2,…,𝑁 (23)𝑉_𝑤 is denoted as commuters’ walking speed from location 𝑥 to station 𝑠, and 𝑢(𝑥)𝑉_𝑤 is the distance between location 𝑥 and station 𝑠.

The discounted cost 𝐶_𝜏, which consists of three cost components, the train operations cost 𝐶_𝑜,𝜏, rail line cost 𝐶_𝐿,𝜏, and rail station cost 𝐶_𝑠,𝜏, could be expressed as 𝐶_𝜏=𝐶_𝑜,𝜏+𝐶_𝐿,𝜏+𝐶_𝑠,𝜏. (24) The discounted train operating cost 𝐶_𝑜,𝜏 is given by 𝐶_𝑜,𝜏=𝜇_𝑜+𝜇₁𝐹(𝜏)(1+𝑖)^𝜏−1, (25) where 𝜇_𝑜 is the fixed operating cost, 𝜇₁ is the operating cost per train in each period, and 𝐹 is the fleet size (or the number of trains) on that line. 𝐹(𝜏) equals the vehicle round journey time 𝑇_𝑅(𝜏) divided by the headway 𝐻(𝜏). Namely, 𝐹(𝜏)=𝑇_𝑅(𝜏)ℎ(𝜏), (26) where the round journey time 𝑇_𝑅 is composed of the terminal time, line-haul travel time, and train dwelling delays at station [20], which could be expressed as 𝑇_𝑅=𝜍𝑇_𝑜+2𝑇₁₁+𝑇₁₂, (27) where 𝑇_𝑜 is the constant terminal time on the circular line and 𝜍 is the number of terminal times on that line. 𝑇₁₁ and 𝑇₁₂ are, respectively, the total line-haul travel time and total dwelling delay for train’s operations from station 1 to CBD, given by (11).

The discounted rail line cost 𝐶_𝐿,𝜏 is the sum of variable cost 𝛾₁𝐷₁ (e.g., land acquisition cost, line construction cost) which is proportional to the rail line length 𝐷₁ and the fixed cost 𝛾₀ (e.g., line overhead cost, maintenance cost, and labour cost), discounted to present value terms. Namely, 𝐶_𝐿,𝜏=𝛾₁𝐷₁(1+𝜒)^𝜏−1+𝛾₀(1+𝑖)^𝜏−1, (28) where 𝛾₁ is the fixed rail line cost per kilometre in each period. The term 1/(1+𝜒)^𝜏−1 represents the inflation factor. It means that, for the same capacity enhancement, the fixed rail line cost increases 𝜒 each period.

The discounted rail station cost 𝐶_𝑠,𝜏 includes a fixed cost (e.g., station land acquisition cost and design and construction cost) and a variable cost (e.g., station overhead cost, operating cost, and maintenance cost), discounted to present value terms. Mathematically, 𝐶_𝑠,𝜏 can be expressed as 𝐶_𝑠,𝜏=𝜅₀(1+𝜒)^𝜏−1+𝜅₁(𝑁(𝜏)+1)(1+𝑖)^𝜏−1, (29) where 𝜅₀ is the fixed cost and 𝜅₁ is the operating cost per station in each period.

Consumer surplus measures the difference between what consumers would be willing to pay for travel and what they actually pay. In order to obtain consumer surplus, the inverse demand function is calculated as follows: 𝑞(𝑥,𝑠,𝜏)⁻¹𝑞()()1𝑥,𝑠,𝜏=𝑐𝑥,𝑠,𝜏+𝜉=𝜃𝑃()ln𝑥,𝜏𝑞(),𝑥,𝑠,𝜏 (30) with ∀𝑥∈[0,𝐵],𝑠=1,2,…,𝑁(𝜏). The consumer surplus at location 𝑥 in period 𝜏, denoted as CS(𝑥,𝑠,𝜏), can be calculated by CS(𝑥,𝑠,𝜏)=₀^{𝑞(𝑥,𝑠,𝜏)}[𝑞(𝑥,𝑠,𝜏)]⁻¹(𝑤)(𝑐())=𝑞()𝑑𝑤−𝑞𝑥,𝑠,𝜏)(𝑥,𝑠,𝜏+𝜉𝑥,𝑠,𝜏𝜃. (31) The discounted consumer surplus in period 𝜏, CS_𝜏, is then obtained by summing the consumer surplus along the candidate rail transit line, discounted to the present value. Namely, CS_𝜏=∫^𝐿0₀CS(𝑤,𝑠,𝜏)𝑑𝑤(1+𝑖)^𝜏−1. (32)

3.3. Social Welfare Maximization Model

As stated above, the design goal of the rail transit line is social welfare maximization. Mathematically, this problem can be formulated as follows: maxSW𝐃,ℎ(𝜏),𝑓_𝑠(𝜏)=^𝑀_𝜏=1^𝑁(𝜏)_𝑠=1𝑓_𝑠(𝜏)𝑄_𝑠,𝜏(1+𝑖)^𝜏−1−^𝑀_𝜏=1𝜇₀+𝜇₁/ℎ(𝜏)𝜍𝑇₀+2𝐷^𝜏₁/𝑉_𝑡(𝜏)+2𝛽₀𝑁(𝜏)(1+𝑖)^𝜏−1−^𝑀_𝜏=1𝛾₀+𝜅₁(𝑁(𝜏)+1)(1+𝑖)^𝜏−1−(1+𝜒)^𝜏−1×𝛾₁𝐷^𝜏₁+𝜅₀+∫^𝐿𝜏0₀𝑞(𝑤,𝑠,𝜏)/𝜃𝑑𝑤(1+𝑖)^𝜏−1, (33) where 𝐃 represents the vector of station locations; namely, 𝐃=(𝐷_𝑁(𝜏),…,𝐷₂,𝐷^𝜏₁). 𝑄_𝑠,𝜏 can be determined by (22).

The optimal solutions for the rail line length, rail station location, headway, and fare can be obtained by setting the partial derivatives of objective function equation (33) with respect to these decision variables equal to zero and solving them simultaneously. The following proposition gives the optimal solutions. The proof is given in Appendix A.

Proposition 1. With the given population density in a particular period, the optimal rail line length, rail station location, headway, and fare solutions with the objectives of social welfare maximization satisfy the systems of equations 𝜕𝑆𝑊(⋅)𝜕𝐷_𝑠=^𝑀_𝜏=1^𝑠+1_{𝑖=𝑠−1}𝑓_𝑠(𝜏)(1+𝑖)^𝜏−1𝜕𝑄_𝑖,𝜏𝜕𝐷_𝑠−Δ_𝑠×^𝑀_𝜏=11(1+𝑖)^𝜏−12𝜇₁ℎ(𝜏)𝑉_𝑡(𝜏)+1+𝜒^𝜏−1𝛾₁ℎ(𝜏)==0,𝜇₁𝜍𝑇₀+2𝐷^𝜏₁/𝑉_𝑡(𝜏)+2𝛽₀𝑁(𝜏)𝛼𝜙_𝑠Δ₃−𝑓_𝑠(𝜏)∑^𝑁(𝜏)_𝑖=1Δ^𝑠₁−Δ^𝑠₂,𝑓_𝑠(𝜏)=∑^𝑀_𝜏=1∑^𝑁(𝜏)_𝑖=1𝑄_𝑖,𝜏−∑^𝑀_𝜏=1Δ₃∑^𝑀_𝜏=1∑^𝑁(𝜏)_𝑖=1Δ^𝑠₂−Δ^𝑠₁, (34) where Δ_𝑠=1 if 𝑠=1, and 0 otherwise. Δ^𝑠₁, Δ^𝑠₂, and Δ₃ are given by Δ^𝑠₁=𝑃₀(𝜏)𝑉_𝑡(𝜏)𝜙_𝑡exp−𝜃𝑐̃𝐿_𝑠−1𝑐̃𝐿,𝑠,𝜏−exp−𝜃_𝑠,Δ,𝑠,𝜏_𝑠2=𝑒𝑃₀(𝜏)𝑉²_𝑡(𝜏)𝜃𝜙²_𝑡𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠−1+1exp−𝜃𝑐̃𝐿_𝑠−1−,𝑠,𝜏𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠+1exp−𝜃𝑐̃𝐿_𝑠,Δ,𝑠,𝜏₃=𝑃₀(𝜏)exp−𝜃𝑐̃𝐿^𝜏₀(𝜏)×,𝑁,𝜏𝜃𝜙²_𝑡+𝑒𝜃𝜙_𝑡𝜙_𝑠𝑉_𝑡(𝜏)+𝑒𝜃𝜙_𝑠𝑉²_𝑡(𝜏)𝜃𝜙²_𝑡−exp(−𝜃𝑐̃(0,0,𝜏))𝜃𝜙²_𝑡+𝑒𝜙_𝑠𝑉²_𝑡(𝜏)𝜃𝜙²_𝑡, (35) and 𝜕𝑄_𝑖,𝜏/𝜕𝐷_𝑠 are given by 𝜕𝑄_{𝑠−1,𝜏}𝜕𝐷_𝑠𝑃=−⁰(𝜏)2exp−𝜃𝑐̃𝐿_𝑠−1,𝑠,𝜏+𝑒𝑃₀(𝜏)𝑉_𝑡(𝜏)2𝜃𝜙_𝑡×𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠−1+12exp−𝜃𝑐̃𝐿_𝑠−1,(𝜏),𝑠,𝜏∀𝑠=2,…,𝑁𝜕𝑄_1,𝜏𝜕𝐷^𝜏₁𝑃=−⁰(𝜏)2exp−𝜃𝑐̃𝐿₁+,𝑠,𝜏𝑒𝑃₀(𝜏)𝑉_𝑡(𝜏)𝑉_𝑡(𝜏)−𝜃𝜙_𝑡2𝜃²𝜙²_𝑡exp−𝜃𝑐̃𝐿₁,𝑠,𝜏𝜕𝑄_𝑠,𝜏𝜕𝐷_𝑠=𝑃₀(𝜏)2exp−𝜃𝑐̃𝐿_𝑠−1𝑐̃𝐿,𝑠,𝜏−exp−𝜃_𝑠,𝑠,𝜏−𝑒𝑃₀(𝜏)𝑉_𝑡(𝜏)2𝜃𝜙_𝑡×𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠−1+12⋅exp−𝜃𝑐̃𝐿_𝑠−1−,𝑠,𝜏𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠+12exp−𝜃𝑐̃𝐿_𝑠,(𝜏),,𝑠,𝜏∀𝑠=2,…,𝑁𝜕𝑄_𝑠+1,𝜏𝜕𝐷_𝑠=𝑃₀(𝜏)2exp−𝜃𝑐̃𝐿_𝑠+1,𝑠,𝜏−𝑒𝑃₀(𝜏)𝑉_𝑡(𝜏)2𝜃𝜙_𝑡⋅𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠+12exp−𝜃𝑐̃𝐿_𝑠,(𝜏),𝑠,𝜏∀𝑠=1,…,𝑁−1. (36)

Proposition 1 presents the partial derivatives of travel demand 𝑄_𝑖,𝜏 with respect to railway line length 𝐷^𝜏₁ and railway station location 𝐷_𝑠. There is another alternative approach to determine these partial derivatives, implementing equilibrium sensitivity analysis of travel demand with respect to railway line length and railway station location. Details on sensitivity analysis approach could be seen in Friesz et al. [21] and Yan and Lam [22].

By contrast, the closed-form solutions, given by Proposition 1, can be used to examine the interrelationship between the optimal solutions of rail design variables directly. For instance, it could be seen that the optimal headway ℎ(𝜏) will increase if the railway operating cost per train 𝜇₁ increased. Li et al. [2] proposed a similar closed-form analysis with the objective of profit maximization based on utility theory and in a static situation.

To highlight the difference between the optimal solutions of the rail design variables based on prospect theory and traditional utility theory, the following proposition is given. The proof is given in Appendix B.

Proposition 2. Overestimation exists for the optimal solutions of rail line length, headway, and fare based on traditional utility theory, compared with prospect theory.

The most widely used solution algorithm for solving concave problem is the Frank-Wolfe searching algorithm. For solving the prospect theory based residential location equilibrium problem (1), this algorithm reduces to a sequence of shortest path computations and one-dimensional minimizations [23]. For the optimization of rail design variables, the heuristic algorithm proposed by Li et al. [2] is used here, which is directly based on the first-order optimality conditions of the social welfare with respect to the above rail design variables, as shown in Proposition 1.

4. Numerical Example

To facilitate the presentation of the essential ideas and contributions of this paper, an illustrative example is employed. Specifically, the difference between the optimal solutions of rail design variables based on traditional utility theory and prospect theory are compared.

The alignment of the rail transit line concerned is shown in Figure 1. The corridor length is fixed as 40 km. The time horizon is 3 years and 𝑀 is 3. Without loss of generality, even station spacing is set as 1.0 km. Other parameters are given in the following Table 2.

Table 2 

Parameters.

From Table 3, it could be seen that the optimal solutions of rail line length, fare, and headway based on prospect theory were less than those based on utility theory. This result was in accord with Proposition 2. However, the social welfare based on prospect theory was larger than that based on utility theory; namely, 26669520>24278400. These results can be attributed to the long-term learning behaviour of commuters’ on residential location choice. This long-term learning behaviour reduced the investment of the rail transit line, but increased the social welfare.

Table 3 

Optimal solutions of rail design variables.

5. Conclusions

This paper proposed closed-form models to explore the impacts of prospect theory based residential location choice on the design of a rail transit line in a monocentric city. Prospect theory was used to model the long-term learning behaviour of commuters’ on residential location choice over a planning horizon. Trade-off exists between daily housing rent and generalized travel cost for commuters.

The analytical optimal solutions of rail design variables with social welfare maximization have been given. It is concluded that overestimation exists based on traditional utility theory, compared with prospect theory.

This study provides a new avenue for the design of a rail transit line. Further research is needed in the following directions.(1)In this paper, a monocentric city is assumed, with only one CBD and several other residential locations. Thus, the commuters’ mobility between different CBD(s) in larger cities cannot be explored. The city boundary is not explicitly considered. The proposed model can be extended into polycentric CBD model in a further study [24–26].(2)All commuters were assumed to be homogenous in this study. However, previous studies have shown that income levels dominated the residential location choices [27, 28]. Therefore, the proposed model can be extended to incorporate the income levels for determining the residential location choices and population density.(3)Only rail travel mode is considered in this paper. To investigate the effects of commuters’ travel mode choice behaviour on the design of a rail transit line; more travel modes should be taken into account, for instance, autobus or park-and-ride modes [29–31].

Appendices

A. Proof of Proposition 1

To obtain the optimal solutions of rail line length and rail station locations, the partial derivatives of the objective function SW(⋅) with respect to 𝐷_𝑠 are set to zero; namely, 𝜕SW(⋅)𝜕𝐷_𝑠=^𝑀_𝜏=1𝑓_𝑠(𝜏)(1+𝑖)^𝜏−1𝜕∑^𝑁_𝑖=1𝑄_𝑖,𝜏𝜕𝐷_𝑠−Δ_𝑠×^𝑀_𝜏=11(1+𝑖)^𝜏−12𝜇₁ℎ(𝜏)𝑉_𝑡(𝜏)+(1+𝜒)^𝜏−1𝛾₁+1(1+𝑖)^𝜏−1𝜕∫^𝐿𝜏0₀𝑞(𝑤,𝑠,𝜏)/𝜃𝑑𝑤𝜕𝐷_𝑠(𝜏),=0,∀𝑠=1,2,…,𝑁 (A.1) where Δ_𝑠=1 if 𝑠=1, and 0 otherwise. In terms of (18)–(23), 𝑄_𝑠,𝜏 is a function of 𝐷_𝑠,𝜆_𝑠,𝜆_𝑠−1, which are functions of 𝐷_𝑠−1,𝐷_𝑠 and 𝐷_𝑠+1; namely, 𝑄_𝑠,𝜏=𝑄_𝑠,𝜏𝐷_𝑠−1,𝐷_𝑠,𝐷_𝑠+1,∀𝑠=1,2,…,𝑁(𝜏). (A.2) Thus, the following equation holds: 𝜕𝑄_𝑖,𝜏𝜕𝐷_𝑠=0,∀𝑠≠𝑠−1,𝑠,𝑠+1. (A.3)

The derivative of ∫^𝐿𝜏0₀(𝑞(𝑤,𝑠,𝜏)/𝜃)𝑑𝑤 with respect to 𝐷_𝑠 is calculated as follows: 𝜕∫^𝐿𝜏0₀𝑞(𝑤,𝑠,𝜏)/𝜃𝑑𝑤𝜕𝐷_𝑠=𝜕∫^𝐷𝑠₀𝑞(𝑤,𝑠,𝜏)/𝜃𝑑𝑤𝜕𝐷_𝑠+𝜕∫^𝐿𝜏0_𝐷𝑠𝑞(𝑤,𝑠,𝜏)/𝜃𝑑𝑤𝜕𝐷_𝑠. (A.4) Since 𝜕∫^𝐷𝑠₀𝑞(𝑤,𝑠,𝜏)/𝜃𝑑𝑤𝜕𝐷_𝑠=𝑞𝐷_𝑠,𝑠,𝜏𝜃,∀𝑠=1,2,…,𝑁(𝜏),𝜕∫^𝐿𝜏0_𝐷𝜏1𝑞(𝑤,𝑠,𝜏)/𝜃𝑑𝑤𝜕𝐷^𝜏₁=1𝜃𝑞𝐿₀,1,𝜏𝜕𝐿^𝜏₀𝜕𝐷^𝜏₁−𝑞𝐷^𝜏₁𝑞𝐷,1,𝜏=−^𝜏1,1,𝜏𝜃,𝜕∫^𝐿𝜏0_𝐷𝑠𝑞(𝑤,𝑠,𝜏)/𝜃𝑑𝑤𝜕𝐷_𝑠𝑞𝐷=−^𝑠,𝑠,𝜏𝜃,∀𝑠=2,3,…,𝑁(𝜏), (A.5) we have 𝜕∫^𝐿𝜏0₀𝑞(𝑤,𝑠,𝜏)/𝜃𝑑𝑤𝜕𝐷_𝑠=0,∀𝑠=1,2,…,𝑁(𝜏). (A.6) Substituting (A.3) and (A.6) into (A.1), one immediately obtains 𝑓_𝑠(𝜏)(1+𝑖)^𝜏−1∑^𝑠+1_{𝑖=𝑠−1}𝜕𝑄_𝑖,𝜏𝜕𝐷_𝑠−Δ_𝑠1(1+𝑖)^𝜏−12𝜇₁ℎ(𝜏)𝑉_𝑡(𝜏)+(1+𝜒)^𝜏−1𝛾₁(𝜏).=0,∀𝑠=1,…,𝑁 (A.7) The partial derivative of the objective function SW(⋅) with respect to headway ℎ(𝜏) is 𝜕SW(⋅)(𝜏)=𝜕ℎ_𝑀𝜏=1𝑓_𝑠(𝜏)(1+𝑖)^𝜏−1𝜕∑^𝑁_𝑖=1𝑄_𝑖,𝜏(𝜏)+𝜕ℎ_𝑀𝜏=1𝜇₁/ℎ²(𝜏)𝜍𝑇₀+2𝐷^𝜏₁/𝑉_𝑡(𝜏)+2𝛽₀𝑁(𝜏)(1+𝑖)^𝜏−1+1(1+𝑖)^𝜏−1𝜕∫^𝐿𝜏0₀𝑞(𝑤,𝑠,𝜏)/𝜃𝑑𝑤(𝜏)(𝜏).𝜕ℎ=0,∀𝑠=1,2,…,𝑁 (A.8) From (13), 𝑄_𝑠,𝜏 is a function of 𝑐(̃𝐿_𝑠−1,𝑠,𝜏) and 𝑐(̃𝐿_𝑠,𝑠,𝜏) and thus a function of ℎ(𝜏). In terms of (5) and (15), 𝐿_𝑠(𝑠=0,…,𝑁(𝜏)) is independent of headway ℎ(𝜏). With the given population density at the CBD in period 𝜏, 𝑃₀(𝜏), we have 𝜕𝑄_𝑠,𝜏(𝜏)=𝜕ℎ𝛼𝑃₀(𝜏)𝜙_𝑠𝑉_𝑡(𝜏)𝜙_𝑡×exp−𝜃𝑐̃𝐿_𝑠−1,𝑠,𝜏−exp−𝜃𝑐̃𝐿_𝑠−,𝑠,𝜏𝛼𝑒𝑃₀(𝜏)𝜙_𝑠𝑉²_𝑡(𝜏)𝜃𝜙²_𝑡⋅𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠−1+1exp−𝜃𝑐̃𝐿_𝑠−1−,𝑠,𝜏𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠+1exp−𝜃𝑐̃𝐿_𝑠,𝑠,𝜏. (A.9) The derivative of ∫^𝐿𝜏0₀(𝑞(𝑤,𝑠,𝜏)/𝜃)𝑑𝑤 with respect to ℎ(𝜏) is 𝜕∫^𝐿𝜏0₀𝑞(𝑤,𝑠,𝜏)/𝜃𝑑𝑤(𝜏)=𝜕ℎ_𝐿𝜏00−𝛼𝜙_𝑠𝑃₀(𝜏)(1−𝑒𝑤)exp(−𝜃𝑐̃(𝑤,𝑠,𝜏))𝑑𝑤=−𝛼𝜙_𝑠𝑃₀(𝜏)(𝑐̃(×exp−𝜃𝑤,𝑠,𝜏))1+𝑒𝜙_𝑠𝑉²_𝑡(𝜏)𝜃𝜙²_𝑡𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝑤+1^𝐿𝜏0₀=−𝛼𝜙_𝑠𝑃₀(𝜏)exp−𝜃𝑐̃𝐿^𝜏₀(𝜏)×𝜙,𝑁,𝜏_2𝑡+𝑒𝜙_𝑡𝜙_𝑠𝑉_𝑡(𝜏)+𝑒𝜙_𝑠𝑉²_𝑡(𝜏)𝜃𝜙²_𝑡−exp(−𝜃𝑐̃(0,0,𝜏))𝜃𝜙²_𝑡+𝑒𝜙_𝑠𝑉²_𝑡(𝜏)𝜃𝜙²_𝑡. (A.10) Combining (A.10) and (A.13), one immediately obtains ℎ(𝜏)=𝜇₁𝜍𝑇₀+2𝐷^𝜏₁/𝑉_𝑡(𝜏)+2𝛽₀𝑁(𝜏)𝛼𝜙_𝑠Δ₃−𝑓∑^𝑁(𝜏)_𝑖=1Δ^𝑠₁−Δ^𝑠₂, (A.11) where Δ^𝑠₁=𝑃₀(𝜏)𝑉_𝑡(𝜏)𝜙_𝑡×exp−𝜃𝑐̃𝐿_𝑠−1,𝑠,𝜏−exp−𝜃𝑐̃𝐿_𝑠,Δ,𝑠,𝜏_𝑠2=𝑒𝑃₀(𝜏)𝑉²_𝑡(𝜏)𝜃𝜙²_𝑡×𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠−1+1exp−𝜃𝑐̃𝐿_𝑠−1−,𝑠,𝜏𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠+1exp−𝜃𝑐̃𝐿_𝑠,Δ,𝑠,𝜏₃=𝑃₀(𝜏)exp−𝜃𝑐̃𝐿^𝜏₀(𝜏)×𝜙,𝑁,𝜏_2𝑡+𝑒𝜙_𝑡𝜙_𝑠𝑉_𝑡(𝜏)+𝑒𝜙_𝑠𝑉²_𝑡(𝜏)𝜙²_𝑡−exp(−𝜃𝑐̃(0,0,𝜏))𝜃𝜙²_𝑡+𝑒𝜙_𝑠𝑉²_𝑡(𝜏)𝜃𝜙²_𝑡. (A.12) The partial derivative of the objective function SW(⋅) with respect to flat fare 𝑓_𝑠(𝜏) is 𝜕SW(⋅)𝜕𝑓_𝑠(𝜏)=^𝑀_𝜏=11(1+𝑖)^𝜏−1×^𝑁(𝜏)_𝑖=1𝑄_𝑖,𝜏+𝑓_𝑠(𝜏)𝜕∑^𝑁(𝜏)_𝑖=1𝑄_𝑖,𝜏𝜕𝑓_𝑠(𝜏)+𝜕∫^𝐿𝜏0₀𝑞(𝑤,𝑠,𝜏)/𝜃𝑑𝑤𝜕𝑓_𝑠(𝜏)=0, (A.13) where 𝜕𝑄_𝑠,𝜏𝜕𝑓_𝑠(𝜏)=𝑃₀(𝜏)𝑉_𝑡(𝜏)𝜙_𝑡exp−𝜃𝑐̃𝐿_𝑠−1,𝑠,𝜏−exp−𝜃𝑐̃𝐿_𝑠−,𝑠,𝜏𝑒𝑃₀(𝜏)𝑉²_𝑡(𝜏)𝜃𝜙²_𝑡⋅𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠−1+1exp−𝜃𝑐̃𝐿_𝑠−1−,𝑠,𝜏𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠+1exp−𝜃𝑐̃𝐿_𝑠,𝑠,𝜏,𝜕∫^𝐿𝜏0₀𝑞(𝑤,𝑠,𝜏)/𝜃𝑑𝑤𝜕𝑓_𝑠(𝜏)=^𝐿𝜏0₀−𝑃₀(𝜏)(1−𝑒𝑤)exp(−𝜃𝑐̃(𝑤,𝑠,𝜏))𝑑𝑤=−𝑃₀(𝜏)(𝑐̃(×exp−𝜃𝑤,𝑠,𝜏))1+𝑒𝜙_𝑠𝑉²_𝑡(𝜏)𝜃𝜙²_𝑡𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝑤+1^𝐿𝜏0₀=−𝑃₀(𝜏)exp−𝜃𝑐̃𝐿^𝜏₀(𝜏)×𝜙,𝑁,𝜏_2𝑡+𝑒𝜙_𝑡𝜙_𝑠𝑉_𝑡(𝜏)+𝑒𝜙_𝑠𝑉²_𝑡(𝜏)𝜙²_𝑡−exp(−𝜃𝑐̃(0,0,𝜏))𝜃𝜙²_𝑡+𝑒𝜙_𝑠𝑉²_𝑡(𝜏)𝜃𝜙²_𝑡. (A.14) Thus, 𝑓_𝑠(𝜏)=∑^𝑀_𝜏=1∑^𝑁(𝜏)_𝑖=1𝑄_𝑖,𝜏−∑^𝑀_𝜏=1Δ₃∑^𝑀_𝜏=1∑^𝑁(𝜏)_𝑖=1Δ^𝑠₂−Δ^𝑠₁, (A.15) where Δ^𝑠₁, Δ^𝑠₂, and Δ₃ are the same as in (A.11). In view of the above system of equations, which consist of (A.7), (A.11), and (A.15), the optimal rail line length, rail station location (or spacing), headway, and fare can be calculated.

B. Proof of Proposition 2

In terms of Proposition 1, the rail length 𝐷^𝜏₁ and rail station location (or spacing) 𝐷_𝑠 can be determined by 𝑓_𝑠(𝜏)𝑃₀(𝜏)2exp−𝜃𝑐̃𝐿₂,𝑠,𝜏−exp−𝜃𝑐̃𝐿₁,𝑠,𝜏+𝑓_𝑠(𝜏)𝑒𝑃₀(𝜏)𝑉_𝑡(𝜏)𝑉_𝑡(𝜏)−𝜃𝜙_𝑡2𝜃²𝜙²_𝑡−𝑒𝑃₀(𝜏)𝑉_𝑡(𝜏)2𝜃𝜙_𝑡⋅𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿₁+12exp−𝜃𝑐̃𝐿₁−1,𝑠,𝜏(1+𝑖)^𝜏−12𝜇₁ℎ(𝜏)𝑉_𝑡(𝜏)+(1+𝜒)^𝜏−1𝛾₁=0,exp−𝜃𝑐̃𝐿_𝑠+1,𝑠,𝜏−exp−𝜃𝑐̃𝐿_𝑠(𝜏),,𝑠,𝜏=0,∀𝑠=2,…,𝑁 (B.1)(B.2) combined with (19).

The condition of traditional stochastic user equilibrium based on utility theory can be expressed as 𝑐(̃𝑥,𝑠,𝜏)=min{𝑐(̃𝑥,𝑠,𝜏)}, for locations with 𝑞(𝑥,𝑠,𝜏)>0. Submitting this condition into (19) and (B.1), we could have the optimal solution of rail line length for traditional stochastic user equilibrium based on utility theory, (𝐷^𝜏₁)^∗, shown as follows: (𝐷^𝜏₁)^∗𝑉=−^𝑡(𝜏)𝜃𝜙_𝑡1×ln(1+𝑖)^𝜏−12𝜇₁ℎ(𝜏)𝑉_𝑡(𝜏)+1+𝜒^𝜏−1𝛾₁×𝑓𝑒𝑃₀(𝜏)𝑉_𝑡(𝜏)𝑉_𝑡(𝜏)−𝜃𝜙_𝑡2𝜃²𝜙²_𝑡−𝑒𝑃₀(𝜏)𝑉_𝑡(𝜏)2𝜃𝜙_𝑡×𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿₁+12⁻¹−𝑉_𝑡(𝜏)𝜙_𝑡𝑓_𝑠(𝜏)+𝜙_𝑢𝑢_𝑠𝐿₁+𝜙_𝑠𝛼ℎ(𝜏)+𝜙_𝑡𝛽₀𝑁(𝜏),1+𝜉0<(1+𝑖)^𝜏−12𝜇₁ℎ(𝜏)𝑉_𝑡(𝜏)+1+𝜒^𝜏−1𝛾₁×𝑓_𝑠(𝜏)𝑒𝑃₀(𝜏)𝑉_𝑡(𝜏)𝑉_𝑡(𝜏)−𝜃𝜙_𝑡2𝜃²𝜙²_𝑡−𝑒𝑃₀(𝜏)𝑉_𝑡(𝜏)2𝜃𝜙_𝑡×𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿₁+12⁻¹<1. (B.3) In contrast with the optimal solution of rail line length for the proposed prospect theory based residential location choice equilibrium, (𝐷^𝜏₁)⁺, we could have (𝐷^𝜏₁)^∗<(𝐷^𝜏₁)⁺, (B.4) if and only if (𝑓_𝑠(𝜏)𝑃₀(𝜏)/2)[exp(−𝜃𝑐(̃𝐿₂,𝑠,𝜏))−exp(−𝜃𝑐(̃𝐿₁,𝑠,𝜏))]>0.

Under the proposed prospect theory based residential location choice equilibrium, travel cost will be higher for commuters living far away from the CBD; thus, 𝑐̃𝐿₂,𝑠,𝜏−𝑐̃𝐿₁,𝑠,𝜏<0 (B.5) exists. Since 𝜃>0 and 𝑓_𝑠(𝜏)𝑃₀(𝜏)/2>0, we have 𝑓_𝑠(𝜏)𝑃₀(𝜏)2exp−𝜃𝑐̃𝐿₂,𝑠,𝜏−exp−𝜃𝑐̃𝐿₁,𝑠,𝜏>0. (B.6) Therefore, compared to the proposed prospect theory based residential location choice equilibrium, overestimation exists for the optimal solution of rail line length with the traditional stochastic user equilibrium based on utility theory.

Similarly, under the traditional stochastic user equilibrium based on utility theory, we have Δ^𝑠₁=𝑃₀(𝜏)𝑉_𝑡(𝜏)𝜙_𝑡×exp−𝜃𝑐̃𝐿_𝑠−1,𝑠,𝜏−exp−𝜃𝑐̃𝐿_𝑠Δ,𝑠,𝜏=0,_𝑠2=𝑒𝑃₀(𝜏)𝑉²_𝑡(𝜏)𝜃𝜙²_𝑡𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠−1+1exp−𝜃𝑐̃𝐿_𝑠−1−,𝑠,𝜏𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠+1exp−𝜃𝑐̃𝐿_𝑠=,𝑠,𝜏𝑒𝑃₀(𝜏)𝑉²_𝑡(𝜏)𝜃𝜙²_𝑡𝜃𝜙_𝑡𝑉_𝑡(𝜏)𝐿_𝑠−1−𝐿_𝑠×exp(−𝜃min{𝑐̃(𝑥,𝑠,𝜏)}). (B.7) In terms of Proposition 1, we have (ℎ(𝜏))^∗<(ℎ(𝜏))⁺ and (𝑓_𝑠(𝜏))^∗<(𝑓_𝑠(𝜏))⁺. In conclusion, overestimation exists for the optimal solutions of headway and fare, comparing the traditional stochastic user equilibrium based on utility theory with the proposed prospect theory based residential location choice equilibrium.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work described in this paper was jointly supported by a Grant from the Research Grant Council of the Hong Kong Special Administrative Region (Project no. PolyU 5215/09E) and a Postgraduate Studentship from the Research Committee of the Hong Kong Polytechnic University. The authors would like to thank the anonymous referees for their valuable comments.