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In contrast to private cars, rail transit systems are a more effective way to deal with the emerging challenges in cities with high population densities, such as congestion, air pollution, and traffic emissions. Rail transit systems, however, are commonly costly, due to substantial investments in construction and maintenance. It is thus necessary to design the candidate rail transit systems carefully to ensure public transport accessibility and sustainability, with consideration of the space-time correlation of population densities. In this paper, the space-time correlations of population densities are incorporated into the design of a candidate rail transit line over years. A closed-formed mathematical programming model is proposed, with an optimisation objective of social welfare budget maximisation. The social welfare budget is defined as the summation of the expected social welfare and social welfare margins. The model decision variables include rail line length, rail station number, and project start time of the candidate rail transit line. The analytical solutions for the proposed rail design model are given explicitly for different scenarios with various constraints.

The preferred travel mode in areas with low population density is a private car, such as the United States. The preferred travel mode in areas with high population density is rail transit, such as Hong Kong. One possible reason is that high congestion and long delay may occur on highways, during morning peak hours on working days in highly populated areas. The generalised travel cost of a private car may be higher than rail transit in such areas.

The travel mode choices were commonly examined, with the assumptions of given and fixed population densities (see, e.g., [

In the transportation corridor of a candidate rail transit line, the population densities in each residential location vary year by year. If the increase of population density in the first year leads to an increase in the second year, positive temporal correlations then exist between population densities in the first year and the second year, and vice versa. Similarly, if the increase of population density in one residential location leads to the increase of population density in another residential location, positive spatial-temporal correlations then exist between population densities in these two residential locations.

The space-time correlation of population densities can be considered, with the nested logit model (e.g., [

In these nested logit and C-logit models, the space-time correlations between alternatives were investigated to calculate the choice probabilities for the residential locations and/or travel choice behaviours of households. In other words, the space-time correlations of population densities cannot explore explicitly with the nested logit and C-logit models. The possible effects of the space-temporal correlation of population densities on the design of a candidate rail transit line cannot be examined explicitly by nested logit or C-logit models.

The space-time correlations of population densities can be taken into account explicitly by the space-time correlation coefficient of population densities [

Zhang et al. [

It was noted that the space-time correlations of population densities were mainly considered for road traffic origin-destination estimation in these previous studies. In this paper, we will incorporate the space-time correlations of population densities by the space-time correlation coefficient of population densities for the design optimisation of a candidate rail transit line.

Based on the space-time correlation coefficient of population densities, a closed-form programming model is introduced to examine the effects of space-time correlation of population densities on the design of a rail transit line in this paper. The optimisation objective of the proposed model is budget social welfare maximisation. The budget social welfare is defined as a summation of expected social welfare and social welfare margin. The model decision variables include rail line length, rail station number, and the project start time.

As shown in Figure

Spatial and temporal correlation of population densities in a candidate rail transit line over years.

Two major extensions to the related literature are made in this paper: (i) the effects of space-time correlation of population densities on the design of a candidate rail transit line over years are investigated by a closed-form mathematical programming model; (ii) the analytical optimal solutions of design variables of the candidate rail transit line over years are obtained with the proposed model.

The remainder of this paper is organised as follows: in the next section, some basic considerations are given. A rail design model is proposed in Section

To facilitate the presentation of the essential ideas, some basic assumptions are made, listed as follows:

A1. The candidate rail transit line is assumed to be linear and start from the CBD and then be built along a linear transportation corridor [

A2. The standard deviation (SD) of the population density is assumed to be an increasing function with respect to its mean value. This function is referred to as the stochastic population density function. In addition, the stochastic population density function is assumed to be a nondecreasing function with respect to its mean value. [

A3. Households’ responses to the quality of the rail service provided are measured by a generalised travel cost that is a weighted combination of in-vehicle time, access time, waiting time, and the fare [

A4. The study period is assumed to be a peak hour, for instance, the morning peak hour, which is usually the most critical period in the day [

A5. Rail station number depends on rail line length and rail station spacings. To obtain the analytical solutions, an even rail station spacing is assumed. In other words, with the assumption of constant rail station spacing, once rail line length is determined, the rail station number is also determined. This assumption is also used in the works of Li et al. [

To take into account the space-time correlation of population densities, it is assumed that there exists a perturbation in the population density. The yearly perturbed population density

To take spatial and temporal correlations between population densities into account, the following spatial and temporal covariance is defined as [

Households are assumed to choose the residential locations to maximise their own utilities subject to budget constraint. A Cobb–Douglas form of the utility function is adopted, shown as follows [

The budget constraints for households are expressed as follows:

Under user equilibrium condition, no households can increase his/her utility by unilaterally changing their location choices. Mathematically, the utility maximisation for households can be expressed as

A similar mathematical formulation has been formulated in Li et al. [

To keep the balance of the supply and demand of housing, it requires that

Substituting equations (

The population conservation equation can be expressed as

The government or the rail operator will build a rail transit line to meet the increasing travel demand of households and eases highway traffic congestion. Social welfare is commonly used to assess the performance of a candidate rail transit line. Due to the yearly uncertainty associated with rail travel demand, the social welfare of the candidate rail transit line is also not a deterministic value. Because of the uncertainty of social welfare, an extra safety margin is assigned to ensure a higher probability of gaining a certain level of social welfare. In view of this, the concept of social welfare budget is proposed as follows:

From equation (

Let

As

The value of

Social welfare of the candidate rail transit line consists of the consumer surplus of households and the profit of the rail operator. Mathematically, expected social welfare

The expected consumer surplus of households

The expected profit of rail operator

In terms of A3, the travel demand function of rail service from residential location

Substituting it into equation (

The expected generalised travel cost consists of fare, access cost from residential locations to rail stations, waiting for cost for rail service at stations, and in-vehicle cost from rail stations to the CBD, shown as follows [

The average headway in year

In terms of equations (

The standard deviation of travel demand for rail service

As stated above, the government or rail operator aims to maximise the social welfare budget of the candidate rail transit line by determining the optimum rail line length, rail station number, and project start time of the candidate rail transit line.

In terms of equations (

For the budget social welfare maximisation problem (

In terms of equation (

For the social welfare budget maximisation problem (

To obtain the optimal solution of the rail line length, the partial derivative of objective function equation (

Similarly, to obtain the optimal solution of the rail station number, the partial derivative of objective function equation (

To obtain the optimal solution of the project start time of the candidate rail transit line, the partial derivative of objective function equation (

To facilitate the presentation of the essential ideas and contributions of this study, two illustrative examples are given below.

The input parameters are summarised in Table

Parameters.

Symbol | Definition | Value |
---|---|---|

Variable cost for rail service | 3 | |

Daily unit fixed maintenance cost of rail line | 10^{6} | |

Daily fixed operation cost of each rail station | ||

Coefficient of variation of population density | 0.3 | |

Fixed component of fare for using the rail service | 4 | |

Variable component of fare per unit distance | 0.1 | |

Fleet size of trains in base year 0 | 5 | |

Average daily household income | 400 | |

Initial value of rail length in year 0 | 20 | |

The planning and operation time horizon | 3 | |

Initial value of rail station number in year 0 | 20 | |

Sensitivity parameter in travel demand function | 0.02 | |

Initial value of total population in year 0 | 100000 | |

Parameters of households’ utility function | 0.2/0.8 | |

Average daily housing rent in the CBD | 300 | |

Average access time | 400 | |

Average train speed in year 0 | 60 | |

Housing supply in year 0 | ||

Interest rate | 0.01 | |

Growth rate of the total population along the transportation corridor | 0.1 | |

Probability of gaining budget social welfare | 95 | |

Constant terminal time | 5/60 | |

Parameter of waiting time function | 0.5 | |

Parameters for travel cost function | 80/100/60 |

Figure

The effects of space-time correlation of population densities on social welfare budget.

For a given total population, positive temporal cc means that the increase of population density in the first year leads to the increase of population density in the next year. As a result, households are distributed to limited residential locations and the total population has a centralised distribution.

In summary, as temporal cc increases, the total population has a more centralised distribution, and the social welfare budget of the candidate rail transit line decreases. More centralised population distribution can lead to a lower social welfare budget of the candidate rail transit line. Decentralised population distribution takes a high social welfare budget of the candidate rail transit line.

Similarly, for given temporal cc, as spatial cc increases, the optimal social welfare budget decreases. For instance, for temporal cc 0.8, as spatial cc increases from −1 to 1, the optimal social welfare budget decreases from

Positive spatial cc implies the increase in population density in a residential location and leads to the increase of population density in another residential location. A type of cooperation relationship may exist between these two adjacent residential locations. For instance, the population growth in a new town can lead to an increase in population density in residential locations of the adjacent suburban city.

In summary, as spatial cc increases, the residential locations are more correlated with each other, and the optimal budget social welfare decreases. More correlated residential locations can lead to lower budget social welfare for the candidate rail transit line. Conversely, a competitive relationship between residential locations leads to the availability of a high budget social welfare for the candidate rail transit line.

It is also noted that the effects of temporal cc on the optimal social welfare budget are more significant than spatial. For instance, as temporal cc increases from −1 to 1, the optimal social welfare budget decreases from level of

Compared with traditional studies assuming a spatial and temporal cc of 0, the optimal social welfare is overestimated in parts of (a) and (b) in Figure

Table

Numerical results.

Temporal cc | Spatial cc | |||||||
---|---|---|---|---|---|---|---|---|

−1 | −1 | 30.98 | 17.49 | 9.59 | 28 | 16 | 9 | 11.19 |

−1 | −0.8 | 29.95 | 16.91 | 9.27 | 27 | 15 | 8 | 11.20 |

−1 | −0.6 | 28.93 | 16.34 | 8.95 | 26 | 15 | 8 | 11.21 |

−1 | −0.4 | 17.91 | 15.76 | 8.64 | 25 | 14 | 8 | 11.22 |

−1 | −0.2 | 26.88 | 15.18 | 8.32 | 24 | 14 | 8 | 11.22 |

−1 | 0 | 15.86 | 14.60 | 8.00 | 24 | 13 | 7 | 11.23 |

−1 | 0.2 | 14.84 | 14.02 | 7.68 | 23 | 13 | 7 | 11.24 |

−1 | 0.4 | 13.81 | 13.45 | 7.37 | 22 | 12 | 7 | 11.25 |

−1 | 0.6 | 22.79 | 12.87 | 7.05 | 21 | 12 | 6 | 11.26 |

−1 | 0.8 | 22.77 | 12.29 | 6.73 | 20 | 11 | 6 | 11.27 |

−1 | 1 | 20.74 | 11.71 | 6.42 | 19 | 11 | 6 | 11.29 |

1 | −1 | 22.09 | 14.36 | 9.21 | 20 | 13 | 8 | 8.60 |

1 | −0.8 | 21.36 | 13.88 | 8.90 | 19 | 13 | 8 | 8.60 |

1 | −0.6 | 20.63 | 13.41 | 8.60 | 19 | 12 | 8 | 8.61 |

1 | −0.4 | 19.90 | 12.93 | 8.29 | 18 | 12 | 8 | 8.61 |

1 | −0.2 | 19.17 | 12.46 | 7.99 | 17 | 11 | 7 | 8.62 |

1 | 0 | 18.44 | 11.99 | 7.68 | 17 | 11 | 7 | 8.62 |

1 | 0.2 | 17.71 | 11.51 | 7.38 | 16 | 10 | 7 | 8.63 |

1 | 0.4 | 16.98 | 11.04 | 7.08 | 15 | 10 | 6 | 8.64 |

1 | 0.6 | 16.25 | 10.56 | 6.77 | 15 | 10 | 6 | 8.65 |

1 | 0.8 | 15.52 | 10.09 | 6.47 | 14 | 9 | 6 | 8.66 |

1 | 1 | 14.79 | 9.61 | 6.16 | 13 | 9 | 6 | 8.67 |

Note: “cc” represents covariance coefficient.

It can also be seen that the optimal rail line length

From Table

Figure

The housing unit prices map around MTR stations for Hong Kong in the first half-year of 2015.

Table

The housing rent list along the Western Island Line.

Station | Representing housing estates | Housing price (HK$/Sqft) | Daily housing rent (HK$) |
---|---|---|---|

(1) Central (CBD) | Winner Building | 31864 | 676.34 |

(2) Sheung Wan | Hollywood Terrace | 17214 | 366.52 |

(3) Sai Ying Pun | Island Grest | 21645 | 460.88 |

(4) HKU | The Belcher’s | 20298 | 432.88 |

(5) Kennedy Town | Smithfield Terrace | 15283 | 325.43 |

Note. The housing rent price ratio is around 3% in Hong Kong at year 2015. The housing rent price ratio is a measure of the relative affordability of renting and buying in a given housing market. It is calculated as the ratio of home prices to annual rental rates. This data comes from Chiefgroup of Hong Kong, linked by www.chiefgroup.com.hk_upload/pdf1_20140226170953_BJ_20131230.pdf. The average flat size is 36.5 sqft, according to Housing Authority Annual Report 2014–2015 (www.housing.wa.gov.au/housingDocuments). Daily housing rent = (36.5

Figure

The effects of space-time correlation of population densities on social welfare budget for the Western Island Line.

This paper proposes a closed-form model to investigate the effects of space-time correlation of population densities on the design of a candidate rail transit line over years. The traditional studies with an assumption of independence of irrelevant alternatives (IIA) population densities, namely, space-time correlation of population densities of 0, are special cases of the proposed model in this paper.

The proposed model offers several insights. For example, the decentralised population distribution takes the high social welfare budget of the candidate rail transit line. Competition between residential locations takes the high social welfare budget of the candidate rail transit line. The effects of the temporal correlation coefficient (cc) on the optimal social welfare budget are more significant than the spatial correlation coefficient. The optimal rail line length

The proposed model also offers some managerial implications. For instance, from Proposition

This paper provides a new avenue for the modelling and analysis of space-time correlation of population densities on the design of a candidate rail transit line over years.

In this paper, the population are assumed to be homogeneous with trips commuting only from residences to CBD. The proposed model can be extended to incorporate the effects of households’ risk preference on early and late arrival to CBD on the design of a candidate rail transit line over years. [

The decision to extend a rail line involves consideration of technological, social, and economic factors. The prime reason could be social or in other words a desire to make life more convenient as regards manoeuvrability for a specific set of people, namely, those living in the vicinity of the line and new stations to be constructed. However, only pressing economic factor is considered in this paper. More detailed social factors can be taken into account in further studies, for instance, appreciation of land value along the rail line. [

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The work described in this paper was jointly supported by the National Natural Science Foundation of China (Grant no. 71473060), the Science and Technology Development Center, Ministry of Education of China (Grant no. 2018A01025), Humanities and Social Sciences Fund of the Ministry of Education (Grant no. 20YJCZH225), Shanghai “Science and Technology Innovation Action Plan” Soft Science Key Project (Grant no. 20692190900), and Shenzhen Philosophy and Social Sciences Planning Project of China (Grant no. SZ2019C004). The authors would like to thank Prof. W.H.K. Lam for his comments and suggestions and Mrs. Elaine Anson for her proofreading of this manuscript.