Multiperiod Hierarchical Location Problem of Transit Hub in Urban Agglomeration Area

With the rapid urbanization in developing countries, urban agglomeration area (UAA) forms. Also, transportation demand in UAA grows rapidly and presents hierarchical feature. Therefore, it is imperative to develop models for transit hubs to guide the development of UAA and better meet the time-varying and hierarchical transportation demand. In this paper, the multiperiod hierarchical location problem of transit hub in urban agglomeration area (THUAA) is studied. A hierarchical service network of THUAA with a multiflow, nested, and noncoherent structure is described. Then a multiperiod hierarchical mathematical programming model is proposed, aiming at minimizing the total demand weighted travel time. Moreover, an improved adaptive clonal selection algorithm is presented to solve the model. Both the model and algorithm are verified by the application to a reallife problem of Beijing-Tianjin-Hebei Region in China. The results of different scenarios in the case show that urban population migration has a great impact on the THUAA location scheme. Sustained and appropriate urban population migration helps to reduce travel time for urban residents.


Introduction
In the process of rapid urbanization in developing countries, regional spatial structure experiences tremendous changes, especially with the development of urban agglomeration area (UAA).On the one hand, because of the increasing close connection between cities in UAA, intercity passenger demand grows rapidly and presents hierarchical features from the aspects of travel distance and travel space.Also, more convenient transport service is required by passengers.On the other hand, the development of UAA is uncoordinated and meets challenges.More specifically, some megacities develop fast but face conflicts between their growing population and decreasing urban carrying capacity, while some small and medium-sized cities have low proportion of urban residents and their developments lag behind.Therefore, it is required to intensify the transportation network, promote the distribution of key industries and public resources, and shift away some of the megacities' functions and population, so as to help small and medium-sized cities to develop industries and attract residents to UAA.
In addition, it is demonstrated that transit hubs have the potential to drive the development of their surrounding areas.Thus, it is necessary to study the hierarchical location problem of transit hub in urban agglomeration area (THUAA), which contributes to accommodating the hierarchical feature of passenger demand in urban agglomeration area (PDUAA) and coordinating the development of UAA.Moreover, the location problem of THUAA is a long-term planning.Transportation network and passenger demand in the process of rapid urbanization are constantly changing.The hubs are hard to rebuild due to its huge resource consumption and political impacts.Considering all these, it is imperative to make overall plan for the multiperiod hierarchical location problem (MHLP) of THUAA.
MHLP of THUAA studied in this paper is a strategic planning.The planning horizon is divided into several time periods.It decides which hierarchy a city or an administrative district should be in each specific period.The goal is to maximize the accessibility of population to get transportation services in a UAA.Therefore, it does not involve the detailed location problem of stations.
The main contributions of this paper are (i) a hierarchical service network of THUAA with a multiflow, nested and noncoherent structure; (ii) a mathematical programming model and a solution algorithm for MHLP of THUAA; and (iii) application to a real-life problem in China to meet changes in the process of rapid urbanization.
The remainder of this paper is organized as follows.Section 2 reviews the related hierarchical and multiperiod facility location problem literatures.Section 3 defines the hierarchical service network of THUAA.Section 4 presents the optimization model.Section 5 proposes a solution method to solve the model.In Section 6 the model is applied to a reallife case of Beijing-Tianjin-Hebei Region in China.Finally, main findings and future work directions are summarized in Section 7.

Literature Review
In the context of rapid urbanization, the system of THUAA is multilevel so as to provide hierarchical service for various passengers.However, the hierarchical feature of THUAA system has been neglected in most researches.Furthermore, the multiperiod nature of the hierarchical location problem of THUAA attracted few attention.According to the previous definitions, the location problem of THUAA can be viewed as a facility location problem.Therefore, the hierarchical and multiperiod facility location models can be applied to MHLP of THUAA, which are reviewed as follows.
There has been a rich body of literatures about hierarchical facility location problem (HFLP), as seen in the recent reviews [1,2].The HFLP models are now mainly employed in the following areas: health care system [3][4][5], education system [6], disaster management [7,8], solid waste management [9], and telecommunications network [10,11], and so forth.However, the HFLP models have not been applied in the location problem of THUAA.Exact solution methods, such as branch-and-bound [12], are utilized to solve HFLPs.With the increasing size and complexity of real-life HFLPs, nonexact solution methods have also been proposed (e.g., approximation algorithm [13,14], tabu search [15]).
As a kind of dynamic facility location models (see the review [16]), multiperiod facility location model divides the planning horizon into several time periods, which helps to better deal with the time-varying parameters [17].Till now, a large number of researches have been done on multiperiod facility location models [18][19][20][21].However, there are few literatures involving multiperiod HFLP models.Hinojosa et al. [22] presented a mix integer programming model to address the location-allocation problem of the production plants and intermediate warehouses in different time periods.The model was solved by Lagrangian relaxation combined with a heuristic procedure.Wang et al. [23] addressed the location problem of regional distribution centers and stores in a distribution system across several time periods.A genetic algorithm based heuristic was utilized to solve the problem.Antunes et al. [24] proposed a multiperiod hierarchical location model for urban hierarchy planning, aiming at minimizing the total demand weighted travel time for the residents to get service from different-level facilities during several time periods.Pehlivan et al. [3] studied the adjustment of hierarchical perinatal network, that is, when and where to locate different-level maternity facilities.The objective of the model is to minimize the total cost.Also, the service quality of each facility is ensured above a given level.The preceding analyses show that multiperiod HFLP modeling needs more researches and the solution methods for solving large-size problems in realistic situations should be paid more attention.Therefore, the application of multiperiod HFLP models to MHLP of THUAA is innovative and has practical significance.

Problem Description
In the process of rapid urbanization, PDUAAs form a hierarchical system (see Figure 1), and so are the THUAAs (see Figure 2).In addition, different-level THUAAs serve different-level PDUAAs (see Figure 2).
The hierarchical structure of PDUAA is presented in Figure 1.According to the travel distance and the travel space, PDUAAs are divided into 3 levels of demands, in which the intercity short-distance and middle-distance travel are called level-1 demand, the external middle-distance travel is called level-2 demand, and the external long-distance travel is called level-3 demand.
Figure 2 presents the hierarchical structure of THUAA and the demands served by different-level hubs.THUAA  system consists of 3 levels of hubs.Different-level hubs provide different-level services.More specifically, level-3 hub provides all services (i.e., it can serve all demands).Level-2 hub provides level-2 service and level-1 service (i.e., it can serve level-2 demand and level-1 demand).Level-1 hub provides level-1 service (i.e., it can only serve level-1 demand).
The essence of MHLP of THUAA can be explained as follows: (i) in a given UAA, the transportation network and passenger demand are known, and the planning horizon is divided into several time periods; (ii) a hub can only serve the demands in its coverage area; and (iii) the target is to choose appropriate administrative districts or cities for different-level hubs in each period, thus satisfying differentlevel demands in UAA and minimizing the total demand weighted travel time.As shown in Figure 3, the yellow point, the red point, the blue point, and the gray point represent level-3 hub, level-2 hub, level-1 hub, and demand point, respectively.The coverage area of a hub is marked with a dashed line circle.
According to the previous descriptions, the hierarchical location problem of THUAA is a HFLP with a multiflow, nested, and noncoherent structure (see Figure 4).It is a multiflow system because any lower-level demand can be assigned to any higher-level hub.For instance, level-1 demand of C1 is assigned to a level-2 hub B2.Because higher-level hubs provide more services than lower-level hubs, it is nested.Since the structure is noncoherent, the demand points served by the same hub for their lower-level demands may be served by different hubs for their higher-level demands.For instance, level-1 demands of B0 and C0 are both assigned to B1, while level-3 demands of B0 and C0 are assigned to A3 and B3, respectively.: the set of periods,  = {1, 2, 3}.More specifically, periods 1, 2, and 3 represent the short-term period, the medium-term period, and the long-term period, respectively

Parameters
Subject to: The objective function (1) minimizes the total demand weighted travel time in the planning horizon.Constraints (2) ensure that all demands in each period are satisfied.Constraints (3) represent that different-level hubs cannot be located at the same candidate hub in each period.Constraints (4) impose that demands at each level can only be served by a hub of equal or higher level in each period.Constraints (5) enforce that the highest-level demand served by a hub in each period exceeds the minimum quantity.Constraints (6) state that different-level demands served by a hub in each period do not exceed the maximum capacity.Constraints (7) limit that a hub can only serve the demand points in its coverage area.Constraints (8) guarantee that a hub cannot be downgraded once it is opened.Constraints (9) are the binary constraints for the variables.

Solution Method: Improved Adaptive Clonal Selection Algorithm (IACSA)
Adaptive clonal selection algorithm (ACSA) [25,26] is widely used because of its fast convergence.Simulated annealing (SA) [27] is a parallel and global optimization algorithm, not depending on the initial population.Considering the feature of the encoding and decoding for the MHLP model, improved adaptive clonal selection algorithm (IACSA) is put forward, in which the idea of SA is introduced into ACSA, thus ensuring global optimization and fast convergence.
5.1.The Procedure of IACSA.In immunology, clone means asexual propagation.A clone refers to one or more offspring derived from a single ancestor, whose genetic composition is identical to that of the ancestor.In IACSA, (i) an antibody represents a solution; (ii) affinity measures the fitness of an antibody; and (iii) antibody population is a group of solutions.As is shown in Figure 5, the steps of IACSA are as follows.
Step 1 (initialization). feasible antibodies are generated randomly, composing the initial antibody population .
Step 2 (evaluation).The affinities of the antibodies in  are calculated according to the affinity function and then are sorted in descending order.
Step 3 (clonal proliferation).Each antibody in  is cloned and then an antibody population  is formed.
Step 3.1.The antibody in  is cloned by neighborhood search.The clonal factor of an antibody (i.e., the number of clones created for an antibody) is proportional to its affinity.Namely, an antibody with a higher affinity will have a higher clonal factor.Step 3.2.Δ = (  ) − () is defined to compare the affinity of a clone (  ) with that of its ancestor ().Specifically, the idea of SA is introduced to use probability proportion to accept the worse clones, thus expanding the search space and preventing premature.According to the Metropolis acceptance criteria (i.e., an acceptance theory based on probability): if Δ ≤ 0, the ancestor is replaced by the clone; otherwise, the clone is accepted with probability exp(−Δ/()).
Step 3.3.Steps 3.1 and 3.2 are repeated  times to form an antibody population .
Step 4 (hypermutation).A new antibody population  * is generated by the hypermutation operation.
Step 5 (clonal selection).Select the antibodies whose affinities rank in the top  from  * to replace the antibodies whose affinities rank in the bottom  in .
Step 6 (termination test).If termination criterion is met, stop and output the antibody with the highest affinity in ; otherwise, go to Step 3.

The Detail of IACSA.
The detailed design of IACSA is as follows.
(1) Encoding and Decoding.Integer encoding is used.Each antibody is composed of 9 fragments.Each fragment of an antibody represents the serial numbers of hubs which serve different-level demands of each demand point in each period.For example, in Figure 6, level-1 and level-2 demands of demand point 1 are both served by hub 1 in period 1, while level-3 demand of demand point 1 is served by hub 2 in period 1.
Moreover, each antibody is divided into 3 parts by period.The hub level in each period can be judged by each part of the antibody.Firstly, a hub which serves level-3 demand is judged as a level-3 hub.Secondly, a hub serves level-2 demand but no Level-2 demand Fragment 2 Level-3 demand Fragment 3 Antibody fragments Period 2 Level-1 demand Fragment 4 Level-2 demand Fragment 5 Level-3 demand Fragment 6 Antibody fragments Period 3 Level-1 demand Fragment 7 Level-2 demand Fragment 8 Level-3 demand Fragment 9 level-3 demand is judged as a level-2 hub.Finally, a hub serves only level-1 demand is judged as a level-1 hub.For instance, Figure 6 shows that (i) in period 1, hubs 2 and 4 are both level-3 hubs, while hubs 0 and 1 are both level-2 hubs and hub 3 is a level-1 hub; (ii) in period 2, hubs 0, 2, and 4 are all level-3 hubs, while hub 1 and hub 3 are level-2 hub and level-1 hub, respectively; and (iii) in period 3, hubs 0, 2, and 4 are all level-3 hubs, while hubs 1 and 3 are both level-2 hubs.
(2) Initialization.Because different-level demands are served by different-level hubs, each gene of the antibody is generated according to the corresponding candidate hub set.More specifically, level-3 demand can be served by level-3 candidate hubs.Level-2 demand can be served by level-2 and level-3 candidate hubs.Level-1 demand can be served by all candidate hubs.Firstly, a candidate hub is selected randomly from the corresponding candidate hub set for each gene in period 1.Then the antibody fragments of period 1 are copied to periods 2 and 3, respectively.Thus, an initial antibody is formed.
(3) Affinity Function.According to the above encoding and initialization method, constraints (2), ( 3), (4), and ( 9) are all satisfied.Constraints ( 8) are judged whether it is satisfied when a new antibody is generated.Therefore, constraints (5), (6), and ( 7) are dealt with the penalty function method, adding to the objective function. is a large constant. 1 ,  2 , and  3 are penalty factors.The affinity function  is shown in (4) Clonal Factor.The clonal factor of the antibody whose affinity ranks number  is defined as   = (1 − /).Moreover, it is limited that   ≥  0 . and  0 are the maximum and minimum clonal factors, respectively. is the maximum number of iterations.(5) Mutation Rate.The mutation rate is defined as  =  max (1− /).Moreover, it is limited that  >  min . max and  min are the maximum and minimum mutation rates, respectively. is the number of current iterations. is the maximum number of iterations.
(6) Mutation Operation.A gene in each fragment of an antibody is selected randomly.A random number  between 0 and 1 is generated.The current gene is replaced if  < ; otherwise it remains itself.

(7) Selection Pool Size and Remainder Replacement Size.
Selection pool size and remainder replacement size are both defined as .More specifically,  = (1 − /).Moreover, it is limited that  >  0 . and  0 are the maximum and minimum sizes, respectively. is the current iteration. is the maximum number of iterations.

Case Study
This section applies the model to a real-life problem of Beijing-Tianjin-Hebei Region in China.Beijing-Tianjin-Hebei Region is an important UAA in China.However, the development of Beijing-Tianjin-Hebei Region is not coordinated enough.In Figure 7, the population and urban residents proportion of cities in Beijing-Tianjin-Hebei Region in 2014 are presented, which show that the population and urban residents proportion of Beijing and Tianjin are much higher than those of other cities. Figure 8 presents the population of each administrative district in Beijing in 2014.It can be seen that the population in the central area is much higher than that of suburban districts.More specifically, Beijing is facing great pressure of population, which needs functional dispersal and decentralization population in the future.Figure 9 presents the population of each administrative district in Tianjin in 2014.The center of Tianjin has high population density, while the population of Binhai and other suburban areas in Tianjin are small.
In the future, Beijing-Tianjin-Hebei Region will be developed in the light of the spatial pattern presented in Figure 10.Moreover, intercity rail will be constructed according to the planning described in Figure 11.8), the administrative districts in Tianjin (see Figure 9), and the other 11 cities in Beijing-Tianjin-Hebei Region (see Figure 7) are considered as both demand points and candidate hubs.Periods 1∼3 represent the short-term period (year  2015∼2020), the medium-term period (year 2020∼2025), and the long-term period (year 2025∼2030), respectively.The data source and the value of the parameters are shown as follows:

Data. The administrative districts in Beijing (see Figure
: it is measured in arcGIS; see Table 2 in Appendix.
ℎ :  1 is defined as 70 km. 2 is defined as 150 km. 3 is defined as 300 km.
ℎ : it is estimated according to the passenger transport volume of Beijing-Tianjin-Hebei Region from China's Urban Statistical Yearbooks in the recent years. 1 is defined as 7000 thousand persons annually. 2 is defined as 3000 thousand persons annually. 3 is defined as 2500 thousand persons annually.  : considering the intercity rail planning presented in Figure 11, it is calculated by arcGIS.For example, the travel time between areas in period 1 is presented in Table 3 in Appendix.  : considering the population migration in the future and the increase of urban residents, it is reckoned by the population and trip frequency.For  example, demands at each level in period 1 are presented in Table 4 in Appendix.
: it is estimated according to the intercity rail planning presented in Figure 11 and the passenger transport volume of Beijing-Tianjin-Hebei Region from China's Urban Statistical Yearbooks (2015).For example, capacities at each level in period 1 are presented in Table 4 in Appendix.
The problem is run 10 times by IACSA.The iterative process of IACSA is presented in Figure 12.
The objective function value and computational time of IACSA are compared with the results from CPLEX, as shown in Table 1. CPLEX is the optimal objective function value of the model, and  IACSA is the objective function value of IACSA.Gap is defined as follows: According to Table 1, Gap between IACSA and CPLEX is less than 1% for the case.Also, the average computational time of IACSA is shorter than that of CPLEX.For large-size problem, random numbers are generated (see Table 1).In large-size problem, the computational time of CPLEX is more than 10 h.However, IACSA can get the near optimum in half  city center to suburban, more and more level-2 and level-3 hubs are opened.In addition, 3 scenarios are designed to investigate the impact of urban population migration on the location of THUAA.Scenario 0, the basic scenario with the parameters defined in the previous sections, includes 3 periods of urban population migration.In scenario 1, urban population migration happens in periods 1 and 2. In scenario 2, urban population migration happens only in period 1. Figure 14 shows that the more the urban population migration (from scenario 2 to scenario 0) is, the less the total demand weighted travel time is.In Figure 15, we can find that, in all scenarios as time goes, the number of level-1 hub reduces, while the number of level-2 hub increases and the number of level-3 hub remains the same or increases.Moreover, in each period, with more and more urban population migration (from scenario 2 to scenario 0), the number of level-1 hub remains the same or becomes lower, while both the numbers of level-2 hub and level-3 hub remain the same or become higher.In all, urban population migration helps to reduce travel time for urban residents, which needs a sustained and appropriate implementation.

Conclusion and Further Research
In this paper, the changes of passenger demand and the development of transportation network in the process of rapid urbanization are considered and MHLP of THUAA is studied.According to the hierarchical feature of PDUAA and the hierarchical nature of THUAA, a hierarchical service network of THUAA with a multiflow, nested and noncoherent structure is described.To better meet the time-varying demand in the process of rapid urbanization, the multiperiod nature of the hierarchical location problem of THUAA is taken into account.Thus, a mathematical programming model and a solution algorithm for multiperiod location of THUAA hierarchical system are proposed.The proposed model and algorithm are verified by the application to a reallife problem of Beijing-Tianjin-Hebei Region in China.The results of different scenarios in the case show that urban population migration has a great impact on the hub location scheme.Also, sustained and appropriate urban population migration helps to reduce travel time for urban residents.Due to the uncertainty of the parameters (e.g., demands at each level, the coverages of different-level hubs, and the maximum capacity of each candidate hub), MHLP of THUAA under uncertainty needs further research in the future.

Figure 2 :
Figure 2: The hierarchical structure of THUAA and the demands served by different-level hubs.

4. 1 .
Notation.In order to formulate the model, the following notations are defined.Sets: the set of demand levels (or set of service levels or set of hub levels),  = {1, 2, 3} : the set of demand points : the set of candidate hubs

Figure 3 :Figure 4 :
Figure 3: The concept about hierarchical location problem of THU-AA.

Figure 9 :
Figure 9: The population of each administrative district in Tianjin in 2014.

Figure 14 :
Figure 14: The demand weighted travel time in different scenarios.

Figure 15 :
Figure 15: The number of hubs at each level in different scenarios.
: let   be one if level- demand of demand point  is served by hub  in period , zero otherwise. ℎ : let  ℎ be one if candidate hub  is opened as a level-ℎ hub in period , zero otherwise.
: the line distance between demand point  and candidate hub    : the shortest travel time between demand point  and candidate hub  in period    : level- demand of demand point  in period   ℎ : the minimum quantity of level-ℎ demand served by a level-ℎ hub   : the maximum capacity of level- service provided by candidate hub  in period   ℎ : the coverage of level-ℎ hub Decision Variables (4) The minimum quantity of the highest-level demand served by different-level hubs is a known constant.(5) The maximum capacities of different-level services provided by each candidate hub in each period are limited and known.4.3.Mathematical Formulation.The multiperiod hierarchical location model of THUAA is formulated as follows: Minimize ∑ ∈ ∑ ∈ ∑ ∈ ∑ ∈

Table 1 :
The results comparison between IACSA and CPLEX.

Table 3 :
Travel time between areas in Beijing-Tianjin-Hebei Region in 2020 (h).

Table 4 :
Different-level demands and maximum capacities of areas in Beijing-Tianjin-Hebei Region in 2020 (ten thousand persons annually).