# Multiperiod Hierarchical Location Problem of Transit Hub in Urban Agglomeration Area

Authors: Ting-ting Li; Rui Song; Shi-wei He; Ming-kai Bi; Wei-chuan Yin; Ying-qun Zhang
Journal: Mathematical Problems in Engineering
 (2017)

---

## Abstract

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 oftransit 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 real-life 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.


---

## Body

## 1. Introduction

In the process of rapid urbanization in developing countries, regional spatial structure experiences tremendous changes, especially with the development ofurban 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 oftransit hub in urban agglomeration area (THUAA), which contributes to accommodating the hierarchical feature ofpassenger 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 forthe 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. Section2 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 real-life case of Beijing-Tianjin-Hebei Region in China. Finally, main findings and future work directions are summarized in Section 7.


## 2. 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 abouthierarchical 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 [3–5], 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 [18–21]. 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.


## 3. Problem Description

In the process of rapid urbanization, PDUAAs form a hierarchical system (see Figure1), and so are the THUAAs (see Figure 2). In addition, different-level THUAAs serve different-level PDUAAs (see Figure 2).Figure 1
The hierarchical structure of PDUAA.Figure 2
The hierarchical structure of THUAA and the demands served by different-level hubs.The hierarchical structure of PDUAA is presented in Figure1. 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.Figure2 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 different-level demands in UAA and minimizing the total demand weighted travel time. As shown in Figure3, 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.Figure 3
The concept about hierarchical location problem of THUAA.According to the previous descriptions, the hierarchical location problem of THUAA is a HFLP with a multiflow, nested, and noncoherent structure (see Figure4). 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.Figure 4
The hierarchical service network of THUAA with a multiflow, nested and noncoherent structure.


## 4. Modeling

### 4.1. Notation

In order to formulate the model, the following notations are defined.Sets H: the set of demand levels (or set of service levels or set of hub levels), H={1,2,3} I: the set of demand points J: the set of candidate hubs T: the set of periods, T={1,2,3}. More specifically, periods 1, 2, and 3 represent the short-term period, the medium-term period, and the long-term period, respectivelyParameters L i j: the line distance between demand point i and candidate hub j d i j t: the shortest travel time between demand point i and candidate hub j in period t q i s t: level-s demand of demand point i in period t c h: the minimum quantity of level-h demand served by a level-h hub C j s t: the maximum capacity of level-s service provided by candidate hub j in period t R h: the coverage of level-h hubDecision Variables x i j s t: let xijst be one if level-s demand of demand point i is served by hub j in period t, zero otherwise. y j h t: let yjht be one if candidate hub j is opened as a level-h hub in period t, zero otherwise.


### 4.2. Assumptions

(1)
For a demand point, demands at each level are served by only one hub. (2)
The coverage of hubs at each level and the planning periods are specific. (3)
The transportation network and different-level demands of each demand point in each period are known. (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:(1)Minimize∑i∈I∑j∈J∑s∈H∑t∈Tdijtqistxijst(2)Subject  to:∑j∈Jxijst=1∀i∈I,s∈H,t∈T(3)∑h∈Hyjht≤1∀j∈J,t∈T(4)xijst≤∑h∈H∣h≥syjht∀i∈I,j∈J,s∈H,t∈T(5)∑i∈Iqihtxijht≥chyjht∀j∈J,h∈H,t∈T(6)∑i∈Iqistxijst≤∑h∈H∣h≥sCjstyjht∀j∈J,s∈H,t∈T(7)∑h∈H∣h≥syjhtRh≥xijstLij∀i∈I,j∈J,s∈H,t∈T(8)yjht-1≤∑k∈H∣k≥hyjkt∀j∈J,h∈H,t∈2,3(9)xijst∈0,1,yjht∈0,1∀i∈I,j∈J,h∈H,s∈H,t∈T.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.


## 4.1. Notation

In order to formulate the model, the following notations are defined.Sets H: the set of demand levels (or set of service levels or set of hub levels), H={1,2,3} I: the set of demand points J: the set of candidate hubs T: the set of periods, T={1,2,3}. More specifically, periods 1, 2, and 3 represent the short-term period, the medium-term period, and the long-term period, respectivelyParameters L i j: the line distance between demand point i and candidate hub j d i j t: the shortest travel time between demand point i and candidate hub j in period t q i s t: level-s demand of demand point i in period t c h: the minimum quantity of level-h demand served by a level-h hub C j s t: the maximum capacity of level-s service provided by candidate hub j in period t R h: the coverage of level-h hubDecision Variables x i j s t: let xijst be one if level-s demand of demand point i is served by hub j in period t, zero otherwise. y j h t: let yjht be one if candidate hub j is opened as a level-h hub in period t, zero otherwise.


## 4.2. Assumptions

(1)
For a demand point, demands at each level are served by only one hub. (2)
The coverage of hubs at each level and the planning periods are specific. (3)
The transportation network and different-level demands of each demand point in each period are known. (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:(1)Minimize∑i∈I∑j∈J∑s∈H∑t∈Tdijtqistxijst(2)Subject  to:∑j∈Jxijst=1∀i∈I,s∈H,t∈T(3)∑h∈Hyjht≤1∀j∈J,t∈T(4)xijst≤∑h∈H∣h≥syjht∀i∈I,j∈J,s∈H,t∈T(5)∑i∈Iqihtxijht≥chyjht∀j∈J,h∈H,t∈T(6)∑i∈Iqistxijst≤∑h∈H∣h≥sCjstyjht∀j∈J,s∈H,t∈T(7)∑h∈H∣h≥syjhtRh≥xijstLij∀i∈I,j∈J,s∈H,t∈T(8)yjht-1≤∑k∈H∣k≥hyjkt∀j∈J,h∈H,t∈2,3(9)xijst∈0,1,yjht∈0,1∀i∈I,j∈J,h∈H,s∈H,t∈T.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.


## 5. 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 Figure5, the steps of IACSA are as follows.Figure 5
The procedure of IACSA.Step 1 (initialization).
Mfeasible antibodies are generated randomly, composing the initial antibody population P.Step 2 (evaluation).
The affinities of the antibodies inP are calculated according to the affinity function and then are sorted in descending order.Step 3 (clonal proliferation).
Each antibody inP is cloned and then an antibody population C is formed.Step  3.1. The antibody in P 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. Δc=C(m′)-C(m) is defined to compare the affinity of a clone C(m′) with that of its ancestor C(m). 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 Δc≤0, the ancestor is replaced by the clone; otherwise, the clone is accepted with probability exp⁡(-Δc/C(m)).Step  3.3. Steps  3.1 and 3.2 are repeated M times to form an antibody population C.Step 4 (hypermutation).
A new antibody populationC∗ is generated by the hypermutation operation.Step 5 (clonal selection).
Select the antibodies whose affinities rank in the topb from C∗ to replace the antibodies whose affinities rank in the bottom b in P.Step 6 (termination test).
If termination criterion is met, stop and output the antibody with the highest affinity inP; otherwise, go to Step 3.


### 5.2. 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.Figure 6
Encoding of solution.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-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, Figure6 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. Q is a large constant. δ1, δ2, and δ3 are penalty factors. The affinity function F is shown in (10)F=Q∑i∈I∑j∈J∑s∈H∑t∈Tdijtqistxijst+δ1∑j∈J∑h∈H∑t∈Tmax⁡0,chyjht-∑i∈Iqihtxijht+δ2∑j∈J∑s∈H∑t∈Tmax⁡0,∑i∈Iqistxijst-∑h∈H∣h≥sCjstyjht+δ3∑i∈I∑j∈J∑s∈H∑t∈Tmax⁡0,xijstLij-∑h∈H∣h≥syjhtRh-1.(  4) Clonal Factor. The clonal factor of the antibody whose affinity ranks number r is defined as ar=A(1-r/N). Moreover, it is limited that ar≥a0. A and a0 are the maximum and minimum clonal factors, respectively. N is the maximum number of iterations.(  5) Mutation Rate. The mutation rate is defined as ε=εmax(1-n/N). Moreover, it is limited that ε>εmin. εmax and εmin are the maximum and minimum mutation rates, respectively. n is the number of current iterations. N is the maximum number of iterations.(  6) Mutation Operation. A gene in each fragment of an antibody is selected randomly. A random number λ between 0 and 1 is generated. The current gene is replaced if λ<ε; otherwise it remains itself.(  7) Selection Pool Size and Remainder Replacement Size. Selection pool size and remainder replacement size are both defined as b. More specifically, b=B(1-n/N). Moreover, it is limited that b>b0. B and b0 are the maximum and minimum sizes, respectively. n is the current iteration. N is the maximum number of iterations.(  8) Termination Criteria. The algorithm terminates when the number of iterations reaches the predefined maximum number.


## 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 Figure5, the steps of IACSA are as follows.Figure 5
The procedure of IACSA.Step 1 (initialization).
Mfeasible antibodies are generated randomly, composing the initial antibody population P.Step 2 (evaluation).
The affinities of the antibodies inP are calculated according to the affinity function and then are sorted in descending order.Step 3 (clonal proliferation).
Each antibody inP is cloned and then an antibody population C is formed.Step  3.1. The antibody in P 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. Δc=C(m′)-C(m) is defined to compare the affinity of a clone C(m′) with that of its ancestor C(m). 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 Δc≤0, the ancestor is replaced by the clone; otherwise, the clone is accepted with probability exp⁡(-Δc/C(m)).Step  3.3. Steps  3.1 and 3.2 are repeated M times to form an antibody population C.Step 4 (hypermutation).
A new antibody populationC∗ is generated by the hypermutation operation.Step 5 (clonal selection).
Select the antibodies whose affinities rank in the topb from C∗ to replace the antibodies whose affinities rank in the bottom b in P.Step 6 (termination test).
If termination criterion is met, stop and output the antibody with the highest affinity inP; otherwise, go to Step 3.


## 5.2. 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.Figure 6
Encoding of solution.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-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, Figure6 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. Q is a large constant. δ1, δ2, and δ3 are penalty factors. The affinity function F is shown in (10)F=Q∑i∈I∑j∈J∑s∈H∑t∈Tdijtqistxijst+δ1∑j∈J∑h∈H∑t∈Tmax⁡0,chyjht-∑i∈Iqihtxijht+δ2∑j∈J∑s∈H∑t∈Tmax⁡0,∑i∈Iqistxijst-∑h∈H∣h≥sCjstyjht+δ3∑i∈I∑j∈J∑s∈H∑t∈Tmax⁡0,xijstLij-∑h∈H∣h≥syjhtRh-1.(  4) Clonal Factor. The clonal factor of the antibody whose affinity ranks number r is defined as ar=A(1-r/N). Moreover, it is limited that ar≥a0. A and a0 are the maximum and minimum clonal factors, respectively. N is the maximum number of iterations.(  5) Mutation Rate. The mutation rate is defined as ε=εmax(1-n/N). Moreover, it is limited that ε>εmin. εmax and εmin are the maximum and minimum mutation rates, respectively. n is the number of current iterations. N is the maximum number of iterations.(  6) Mutation Operation. A gene in each fragment of an antibody is selected randomly. A random number λ between 0 and 1 is generated. The current gene is replaced if λ<ε; otherwise it remains itself.(  7) Selection Pool Size and Remainder Replacement Size. Selection pool size and remainder replacement size are both defined as b. More specifically, b=B(1-n/N). Moreover, it is limited that b>b0. B and b0 are the maximum and minimum sizes, respectively. n is the current iteration. N is the maximum number of iterations.(  8) Termination Criteria. The algorithm terminates when the number of iterations reaches the predefined maximum number.


## 6. 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 Figure7, 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 7
The population and urban residents proportion of cities in Beijing-Tianjin-Hebei Region in 2014.Figure8 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.Figure 8
The population of each administrative district in Beijing in 2014.Figure 9
The population of each administrative district in Tianjin in 2014.In the future, Beijing-Tianjin-Hebei Region will be developed in the light of the spatial pattern presented in Figure10. Moreover, intercity rail will be constructed according to the planning described in Figure 11.Figure 10
The spatial development pattern of Beijing-Tianjin-Hebei Region.Figure 11
Intercity rail planning of Beijing-Tianjin-Hebei Region (years 2014~2030).


### 6.1. Data

The administrative districts in Beijing (see Figure8), 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:Lij: it is measured in arcGIS; see Table 2 in Appendix.Rh: R1 is defined as 70 km. R2 is defined as 150 km. R3 is defined as 300 km.ch: it is estimated according to the passenger transport volume of Beijing-Tianjin-Hebei Region from China’s Urban Statistical Yearbooks in the recent years. c1 is defined as 7000 thousand persons annually. c2 is defined as 3000 thousand persons annually. c3 is defined as 2500 thousand persons annually.dijt: 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.qist: 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.Cjst: 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.


### 6.2. Computational Results

The IACSA is coded in C#, running on a laptop with 2.50 GHz Intel Core I5-2450M processor and 4 GB of RAM under the system Microsoft Windows 7. Also, the optimization software CPLEX version 12.2 is utilized to solve the problem. The parameters of the IACSA are defined as follows:M = 30, A=10, a0=5, ɛmax=0.9, ɛmin=0.1, B=15, b0=5, N=100, Q=106, δ1=106, δ2=106, and δ3=106.The problem is run 10 times by IACSA. The iterative process of IACSA is presented in Figure12.Figure 12
The iterative process of IACSA.The objective function value and computational time of IACSA are compared with the results from CPLEX, as shown in Table1.  ZCPLEX is the optimal objective function value of the model, and ZIACSA is the objective function value of IACSA. Gap is defined as follows:(11)Gap=ZIACSA-ZCPLEXZCPLEX×100%.Table 1
The results comparison between IACSA and CPLEX.
The number of areas in UAA Objective function value Computational time (s) Z C P L E X Z I A C S A Gap (%) CPLEX IACSA 38 83523 83635 0.1 952.6 12.3 300 3652384 3681603 0.6 42485.7 1845.1Table 2
Distances between areas in Beijing-Tianjin-Hebei Region (km).
Area 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 1 0 6 10 18 19 20 36 24 36 40 29 29 58 74 76 81 138 140 122 168 114 93 92 164 162 145 101 58 300 160 293 178 451 402 193 226 216 272 2 6 0 18 13 15 18 32 26 43 42 23 25 61 81 80 82 140 142 125 170 117 95 94 166 164 147 108 61 294 154 295 181 445 396 194 230 223 266 3 10 18 0 29 31 24 49 19 30 44 38 43 53 68 72 84 133 135 117 163 109 88 87 159 157 140 95 58 312 172 288 173 462 414 196 222 211 281 4 18 13 29 0 11 18 23 37 60 54 21 24 74 93 92 94 132 140 123 156 122 101 99 171 170 146 120 64 285 145 300 186 436 387 206 242 221 264 5 19 15 31 11 0 16 24 38 61 50 30 15 75 94 94 90 140 148 132 164 128 107 105 178 176 154 122 72 286 146 307 192 437 388 202 244 230 273 6 20 18 24 18 16 0 37 45 48 37 37 23 62 86 81 78 159 161 143 189 136 114 113 185 183 166 113 79 299 159 314 199 450 401 190 231 237 280 7 36 32 49 23 24 37 0 60 78 71 24 36 92 111 110 111 160 157 145 190 137 115 119 186 184 167 138 81 265 125 320 206 416 367 223 260 225 265 8 24 26 19 37 38 45 60 0 26 61 44 51 50 61 87 101 126 127 110 155 102 80 72 151 150 132 88 57 323 183 274 159 474 425 213 237 203 287 9 36 43 30 60 61 48 78 26 0 47 68 68 24 41 42 87 153 155 138 183 130 108 100 179 177 160 88 84 338 198 302 187 488 440 199 192 231 315 10 40 42 44 54 50 37 71 61 47 0 62 44 55 98 73 43 175 177 159 200 151 130 128 201 199 181 125 98 326 186 330 215 477 428 155 223 253 305 11 29 23 38 21 30 37 24 44 68 62 0 43 83 101 101 103 121 129 121 145 113 92 97 163 161 144 129 53 281 141 298 183 431 383 215 251 202 244 12 29 25 43 24 15 23 36 51 68 44 43 0 84 102 102 83 154 161 145 178 144 122 121 193 191 167 130 86 288 148 322 208 439 390 195 252 242 284 13 58 61 53 74 75 62 92 50 24 55 83 84 0 59 21 94 183 184 167 212 159 138 120 208 207 189 97 108 353 213 322 207 504 455 206 180 261 325 14 74 81 68 93 94 86 111 61 41 98 101 102 59 0 50 138 139 146 170 163 144 140 70 165 137 192 34 116 374 234 263 149 524 476 250 160 263 347 15 76 80 72 92 94 81 110 87 42 73 101 102 21 50 0 113 202 204 186 232 179 157 109 228 226 209 88 127 373 233 310 196 524 475 225 153 280 344 16 81 82 84 94 90 78 111 101 87 43 103 83 94 138 113 0 218 220 203 248 195 173 172 244 242 225 168 141 366 226 373 259 517 468 135 267 296 348 17 138 140 133 132 140 159 160 126 153 175 121 154 183 139 202 218 0 11 22 25 16 38 77 42 68 40 116 83 321 181 266 122 457 423 320 298 114 251 18 140 142 135 140 148 161 157 127 155 177 129 161 184 146 204 220 11 0 37 18 31 53 85 36 71 51 123 91 333 193 266 122 467 435 333 298 124 262 19 122 125 117 123 132 143 145 110 138 159 121 145 167 170 186 203 22 37 0 48 20 44 106 72 83 35 144 64 293 153 281 137 443 395 312 313 114 241 20 168 170 163 156 164 189 190 155 183 200 145 178 212 163 232 248 25 18 48 0 45 65 99 30 80 59 138 110 338 198 261 126 463 421 350 302 119 257 21 114 117 109 122 128 136 137 102 130 151 113 144 159 144 179 195 16 31 20 45 0 22 81 57 82 46 119 66 304 164 277 133 455 406 304 308 128 253 22 93 95 88 101 107 114 115 80 108 130 92 122 138 140 157 173 38 53 44 65 22 0 54 80 89 68 126 35 315 175 284 140 466 417 283 307 139 263 23 92 94 87 99 105 113 119 72 100 128 97 121 120 70 109 172 77 85 106 99 81 54 0 103 84 128 47 70 375 235 210 96 526 477 281 238 199 324 24 164 166 159 171 178 185 186 151 179 201 163 193 208 165 228 244 42 36 72 30 57 80 103 0 41 83 134 116 362 222 233 108 482 440 354 284 138 276 25 162 164 157 170 176 183 184 150 177 199 161 191 207 137 226 242 68 71 83 80 82 89 84 41 0 130 103 113 378 238 199 55 528 480 351 231 182 319 26 145 147 140 146 154 166 167 132 160 181 144 167 189 192 209 225 40 51 35 59 46 68 128 83 130 0 168 87 297 157 326 182 425 383 336 359 81 219 27 101 108 95 120 122 113 138 88 88 125 129 130 97 34 88 168 116 123 144 138 119 126 47 134 103 168 0 151 415 275 230 116 566 517 277 147 239 363 28 58 61 58 64 72 79 81 57 84 98 53 86 108 116 127 141 83 91 64 110 66 35 70 116 113 87 151 0 253 132 250 129 379 335 212 194 137 218 29 300 294 312 285 286 299 265 323 338 326 281 288 353 374 373 366 321 333 293 338 304 315 375 362 378 297 415 253 0 130 484 367 160 109 309 440 212 110 30 160 154 172 145 146 159 125 183 198 186 141 148 213 234 233 226 181 193 153 198 164 175 235 222 238 157 275 132 130 0 367 250 265 220 224 315 136 130 31 293 295 288 300 307 314 320 274 302 330 298 322 322 263 310 373 266 266 281 261 277 284 210 233 199 326 230 250 484 367 0 130 574 543 409 183 294 412 32 178 181 173 186 192 199 206 159 187 215 183 208 207 149 196 259 122 122 137 126 133 140 96 108 55 182 116 129 367 250 130 0 470 433 311 152 188 306 33 451 445 462 436 437 450 416 474 488 477 431 439 504 524 524 517 457 467 443 463 455 466 526 482 528 425 566 379 160 265 574 470 0 53 468 576 286 165 34 402 396 414 387 388 401 367 425 440 428 383 390 455 476 475 468 423 435 395 421 406 417 477 440 480 383 517 335 109 220 543 433 53 0 418 527 252 130 35 193 194 196 206 202 190 223 213 199 155 215 195 206 250 225 135 320 333 312 350 304 283 281 354 351 336 277 212 309 224 409 311 468 418 0 275 326 345 36 226 230 222 242 244 231 260 237 192 223 251 252 180 160 153 267 298 298 313 302 308 307 238 284 231 359 147 194 440 315 183 152 576 527 275 0 315 413 37 216 223 211 221 230 237 225 203 231 253 202 242 261 263 280 296 114 124 114 119 128 139 199 138 182 81 239 137 212 136 294 188 286 252 326 315 0 120 38 272 266 281 264 273 280 265 287 315 305 244 284 325 347 344 348 251 262 241 257 253 263 324 276 319 219 363 218 110 130 412 306 165 130 345 413 120 0Table 3
Travel time between areas in Beijing-Tianjin-Hebei Region in 2020 (h).
Area 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 1 0.0 0.1 0.1 0.1 0.2 0.2 0.3 0.2 0.3 0.3 0.2 0.2 0.5 0.6 0.6 0.7 1.2 1.2 1.0 1.4 1.0 0.8 0.8 1.4 1.4 1.2 0.8 0.2 1.2 0.6 4.2 1.5 1.8 1.6 0.8 1.9 0.9 3.9 2 0.1 0.0 0.1 0.1 0.1 0.2 0.3 0.2 0.4 0.3 0.2 0.2 0.5 0.7 0.7 0.7 1.2 1.2 1.0 1.4 1.0 0.8 0.8 1.4 1.4 1.2 0.9 0.2 1.2 0.6 4.2 1.5 1.8 1.6 0.8 1.9 0.9 3.8 3 0.1 0.1 0.0 0.2 0.3 0.2 0.4 0.2 0.2 0.4 0.3 0.4 0.4 0.6 0.6 0.7 1.1 1.1 1.0 1.4 0.9 0.7 0.7 1.3 1.3 1.2 0.8 0.2 1.2 0.7 4.1 1.4 1.8 1.7 0.8 1.8 0.8 4.0 4 0.1 0.1 0.2 0.0 0.1 0.2 0.2 0.3 0.5 0.4 0.2 0.2 0.6 0.8 0.8 0.8 1.1 1.2 1.0 1.3 1.0 0.8 0.8 1.4 1.4 1.2 1.0 0.3 1.1 0.6 4.3 1.5 1.7 1.5 0.8 2.0 0.9 3.8 5 0.2 0.1 0.3 0.1 0.0 0.1 0.2 0.3 0.5 0.4 0.2 0.1 0.6 0.8 0.8 0.8 1.2 1.2 1.1 1.4 1.1 0.9 0.9 1.5 1.5 1.3 1.0 0.3 1.1 0.6 4.4 1.6 1.7 1.6 0.8 2.0 0.9 3.9 6 0.2 0.2 0.2 0.2 0.1 0.0 0.3 0.4 0.4 0.3 0.3 0.2 0.5 0.7 0.7 0.6 1.3 1.3 1.2 1.6 1.1 1.0 0.9 1.5 1.5 1.4 0.9 0.3 1.2 0.6 4.5 1.7 1.8 1.6 0.8 1.9 0.9 4.0 7 0.3 0.3 0.4 0.2 0.2 0.3 0.0 0.5 0.6 0.6 0.2 0.3 0.8 0.9 0.9 0.9 1.3 1.3 1.2 1.6 1.1 1.0 1.0 1.6 1.5 1.4 1.2 0.3 1.1 0.5 4.6 1.7 1.7 1.5 0.9 2.2 0.9 3.8 8 0.2 0.2 0.2 0.3 0.3 0.4 0.5 0.0 0.2 0.5 0.4 0.4 0.4 0.5 0.7 0.8 1.0 1.1 0.9 1.3 0.9 0.7 0.6 1.3 1.2 1.1 0.7 0.2 1.3 0.7 3.9 1.3 1.9 1.7 0.9 2.0 0.8 4.1 9 0.3 0.4 0.2 0.5 0.5 0.4 0.6 0.2 0.0 0.4 0.6 0.6 0.2 0.3 0.4 0.7 1.3 1.3 1.1 1.5 1.1 0.9 0.8 1.5 1.5 1.3 0.7 0.3 1.4 0.8 4.3 1.6 2.0 1.8 0.8 1.6 0.9 4.5 10 0.3 0.3 0.4 0.4 0.4 0.3 0.6 0.5 0.4 0.0 0.5 0.4 0.5 0.8 0.6 0.4 1.5 1.5 1.3 1.7 1.3 1.1 1.1 1.7 1.7 1.5 1.0 0.4 1.3 0.7 4.7 1.8 1.9 1.7 0.6 1.9 1.0 4.4 11 0.2 0.2 0.3 0.2 0.2 0.3 0.2 0.4 0.6 0.5 0.0 0.4 0.7 0.8 0.8 0.9 1.0 1.1 1.0 1.2 0.9 0.8 0.8 1.4 1.3 1.2 1.1 0.2 1.1 0.6 4.3 1.5 1.7 1.5 0.9 2.1 0.8 3.5 12 0.2 0.2 0.4 0.2 0.1 0.2 0.3 0.4 0.6 0.4 0.4 0.0 0.7 0.9 0.8 0.7 1.3 1.3 1.2 1.5 1.2 1.0 1.0 1.6 1.6 1.4 1.1 0.3 1.2 0.6 4.6 1.7 1.8 1.6 0.8 2.1 1.0 4.1 13 0.5 0.5 0.4 0.6 0.6 0.5 0.8 0.4 0.2 0.5 0.7 0.7 0.0 0.5 0.2 0.8 1.5 1.5 1.4 1.8 1.3 1.1 1.0 1.7 1.7 1.6 0.8 0.4 1.4 0.9 4.6 1.7 2.0 1.8 0.8 1.5 1.0 4.6 14 0.6 0.7 0.6 0.8 0.8 0.7 0.9 0.5 0.3 0.8 0.8 0.9 0.5 0.0 0.4 1.2 1.2 1.2 1.4 1.4 1.2 1.2 0.6 1.4 1.1 1.6 0.3 0.5 1.5 0.9 3.8 1.2 2.1 1.9 1.0 1.3 1.1 5.0 15 0.6 0.7 0.6 0.8 0.8 0.7 0.9 0.7 0.4 0.6 0.8 0.8 0.2 0.4 0.0 0.9 1.7 1.7 1.6 1.9 1.5 1.3 0.9 1.9 1.9 1.7 0.7 0.5 1.5 0.9 4.4 1.6 2.1 1.9 0.9 1.3 1.1 4.9 16 0.7 0.7 0.7 0.8 0.8 0.6 0.9 0.8 0.7 0.4 0.9 0.7 0.8 1.2 0.9 0.0 1.8 1.8 1.7 2.1 1.6 1.4 1.4 2.0 2.0 1.9 1.4 0.6 1.5 0.9 5.3 2.2 2.1 1.9 0.5 2.2 1.2 5.0 17 1.2 1.2 1.1 1.1 1.2 1.3 1.3 1.0 1.3 1.5 1.0 1.3 1.5 1.2 1.7 1.8 0.0 0.1 0.2 0.2 0.1 0.3 0.6 0.4 0.6 0.3 1.0 0.3 2.7 0.7 3.8 0.5 3.8 2.1 1.3 4.3 0.5 3.6 18 1.2 1.2 1.1 1.2 1.2 1.3 1.3 1.1 1.3 1.5 1.1 1.3 1.5 1.2 1.7 1.8 0.1 0.0 0.3 0.1 0.3 0.4 0.7 0.3 0.6 0.4 1.0 0.4 2.8 0.8 3.8 0.5 3.9 2.2 1.3 4.3 0.5 3.7 19 1.0 1.0 1.0 1.0 1.1 1.2 1.2 0.9 1.1 1.3 1.0 1.2 1.4 1.4 1.6 1.7 0.2 0.3 0.0 0.4 0.2 0.4 0.9 0.6 0.7 0.3 1.2 0.3 2.4 0.6 4.0 0.5 3.7 2.0 1.2 4.5 0.5 3.4 20 1.4 1.4 1.4 1.3 1.4 1.6 1.6 1.3 1.5 1.7 1.2 1.5 1.8 1.4 1.9 2.1 0.2 0.1 0.4 0.0 0.4 0.5 0.8 0.3 0.7 0.5 1.1 0.4 2.8 0.8 3.7 0.5 3.9 2.1 1.4 4.3 0.5 3.7 21 1.0 1.0 0.9 1.0 1.1 1.1 1.1 0.9 1.1 1.3 0.9 1.2 1.3 1.2 1.5 1.6 0.1 0.3 0.2 0.4 0.0 0.2 0.7 0.5 0.7 0.4 1.0 0.3 2.5 0.7 4.0 0.5 3.8 2.0 1.2 4.4 0.5 3.6 22 0.8 0.8 0.7 0.8 0.9 1.0 1.0 0.7 0.9 1.1 0.8 1.0 1.1 1.2 1.3 1.4 0.3 0.4 0.4 0.5 0.2 0.0 0.4 0.7 0.7 0.6 1.0 0.1 2.6 0.7 4.1 0.6 3.9 2.1 1.1 4.4 0.6 3.8 23 0.8 0.8 0.7 0.8 0.9 0.9 1.0 0.6 0.8 1.1 0.8 1.0 1.0 0.6 0.9 1.4 0.6 0.7 0.9 0.8 0.7 0.4 0.0 0.9 0.7 1.1 0.4 0.3 3.1 0.9 3.0 0.4 4.4 2.4 1.1 3.4 0.8 4.6 24 1.4 1.4 1.3 1.4 1.5 1.5 1.6 1.3 1.5 1.7 1.4 1.6 1.7 1.4 1.9 2.0 0.4 0.3 0.6 0.3 0.5 0.7 0.9 0.0 0.3 0.7 1.1 0.5 3.0 0.9 3.3 0.4 4.0 2.2 1.4 4.1 0.6 3.9 25 1.4 1.4 1.3 1.4 1.5 1.5 1.5 1.2 1.5 1.7 1.3 1.6 1.7 1.1 1.9 2.0 0.6 0.6 0.7 0.7 0.7 0.7 0.7 0.3 0.0 1.1 0.9 0.5 3.1 1.0 2.8 0.2 4.4 2.4 1.4 3.3 0.7 4.6 26 1.2 1.2 1.2 1.2 1.3 1.4 1.4 1.1 1.3 1.5 1.2 1.4 1.6 1.6 1.7 1.9 0.3 0.4 0.3 0.5 0.4 0.6 1.1 0.7 1.1 0.0 1.4 0.3 2.5 0.6 4.7 0.7 3.5 1.9 1.3 5.1 0.3 3.1 27 0.8 0.9 0.8 1.0 1.0 0.9 1.2 0.7 0.7 1.0 1.1 1.1 0.8 0.3 0.7 1.4 1.0 1.0 1.2 1.1 1.0 1.0 0.4 1.1 0.9 1.4 0.0 0.6 3.5 1.1 3.3 0.5 4.7 2.6 1.1 2.1 1.0 5.2 28 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.2 0.3 0.4 0.2 0.3 0.4 0.5 0.5 0.6 0.3 0.4 0.3 0.4 0.3 0.1 0.3 0.5 0.5 0.3 0.6 0.0 2.1 0.5 3.6 1.1 1.5 4.8 3.0 2.8 2.0 3.1 29 1.2 1.2 1.2 1.1 1.1 1.2 1.1 1.3 1.4 1.3 1.1 1.2 1.4 1.5 1.5 1.5 2.7 2.8 2.4 2.8 2.5 2.6 3.1 3.0 3.1 2.5 3.5 2.1 0.0 0.5 4.0 1.5 0.6 0.4 1.2 1.8 3.0 0.4 30 0.6 0.6 0.7 0.6 0.6 0.6 0.5 0.7 0.8 0.7 0.6 0.6 0.9 0.9 0.9 0.9 0.7 0.8 0.6 0.8 0.7 0.7 0.9 0.9 1.0 0.6 1.1 0.5 0.5 0.0 1.8 1.0 1.1 0.9 0.9 1.3 1.9 1.9 31 4.2 4.2 4.1 4.3 4.4 4.5 4.6 3.9 4.3 4.7 4.3 4.6 4.6 3.8 4.4 5.3 3.8 3.8 4.0 3.7 4.0 4.1 3.0 3.3 2.8 4.7 3.3 3.6 4.0 1.8 0.0 0.5 8.2 7.8 5.8 2.6 1.2 5.9 32 1.5 1.5 1.4 1.5 1.6 1.7 1.7 1.3 1.6 1.8 1.5 1.7 1.7 1.2 1.6 2.2 0.5 0.5 0.5 0.5 0.5 0.6 0.4 0.4 0.2 0.7 0.5 1.1 1.5 1.0 0.5 0.0 3.9 3.6 2.6 2.2 2.7 4.4 33 1.8 1.8 1.8 1.7 1.7 1.8 1.7 1.9 2.0 1.9 1.7 1.8 2.0 2.1 2.1 2.1 3.8 3.9 3.7 3.9 3.8 3.9 4.4 4.0 4.4 3.5 4.7 1.5 0.6 1.1 8.2 3.9 0.0 0.2 1.9 2.3 4.1 0.7 34 1.6 1.6 1.7 1.5 1.6 1.6 1.5 1.7 1.8 1.7 1.5 1.6 1.8 1.9 1.9 1.9 2.1 2.2 2.0 2.1 2.0 2.1 2.4 2.2 2.4 1.9 2.6 4.8 0.4 0.9 7.8 3.6 0.2 0.0 2.1 2.6 3.6 1.9 35 0.8 0.8 0.8 0.8 0.8 0.8 0.9 0.9 0.8 0.6 0.9 0.8 0.8 1.0 0.9 0.5 1.3 1.3 1.2 1.4 1.2 1.1 1.1 1.4 1.4 1.3 1.1 3.0 1.2 0.9 5.8 2.6 1.9 2.1 0.0 2.3 2.7 4.9 36 1.9 1.9 1.8 2.0 2.0 1.9 2.2 2.0 1.6 1.9 2.1 2.1 1.5 1.3 1.3 2.2 4.3 4.3 4.5 4.3 4.4 4.4 3.4 4.1 3.3 5.1 2.1 2.8 1.8 1.3 2.6 2.2 2.3 2.6 2.3 0.0 4.5 5.9 37 0.9 0.9 0.8 0.9 0.9 0.9 0.9 0.8 0.9 1.0 0.8 1.0 1.0 1.1 1.1 1.2 0.5 0.5 0.5 0.5 0.5 0.6 0.8 0.6 0.7 0.3 1.0 2.0 3.0 1.9 1.2 2.7 4.1 3.6 2.7 4.5 0.0 1.7 38 3.9 3.8 4.0 3.8 3.9 4.0 3.8 4.1 4.5 4.4 3.5 4.1 4.6 5.0 4.9 5.0 3.6 3.7 3.4 3.7 3.6 3.8 4.6 3.9 4.6 3.1 5.2 3.1 0.4 1.9 5.9 4.4 2.4 1.9 4.9 5.9 1.7 0.0Table 4
Different-level demands and maximum capacities of areas in Beijing-Tianjin-Hebei Region in 2020 (ten thousand persons annually).
Area q i 11 q i 21 q i 31 C j 11 C j 21 C j 31 1 40301 2529 1651 61131 5227 5119 2 57598 3615 2360 87369 6142 5119 3 172395 9306 7065 261502 9648 8462 4 101128 6347 4144 153398 6534 13285 5 28755 1805 1178 43617 0 0 6 159168 7631 6523 241438 8574 0 7 32837 2061 1346 49809 6403 0 8 40057 2514 1642 60762 6860 5119 9 25249 1585 1035 38299 6599 9645 10 71147 4465 2916 107920 6272 0 11 50313 3158 2062 76318 6991 6399 12 12141 810 498 18417 0 0 13 11543 725 473 17510 0 0 14 10670 670 437 16185 3920 0 15 12187 765 499 18487 4247 0 16 7312 459 300 11092 0 0 17 90939 5430 3415 101352 5440 5701 18 12920 686 485 14399 2176 2851 19 14584 871 548 16254 3264 0 20 12774 763 480 14236 0 0 21 14568 870 547 16236 2984 0 22 20439 1085 767 22779 3228 0 23 16224 861 609 18082 3264 0 24 52152 2768 1958 58124 3445 5131 25 8552 454 321 9531 0 0 26 13815 733 519 15397 0 0 27 16345 868 614 18216 0 0 28 20616 3259 1481 23344 8299 0 29 53405 8443 3837 60471 19151 14505 30 40085 6337 2880 45388 12129 2872 31 13925 2201 1000 15767 2160 0 32 37114 5868 2666 42024 13086 8703 33 44258 6997 3180 50113 11810 2843 34 29118 4603 2092 32970 0 0 35 20911 3306 1502 23678 3541 0 36 14930 2360 1073 16905 3126 0 37 31108 4918 2235 35224 5832 0 38 17556 2775 1261 19878 2869 0According to Table1, 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 an hour. Gap between IACSA and CPLEX is still less than 1%. Therefore, it can be concluded that IACSA is producing reasonable results in large-size problems.Figure13 presents the location of THUAA in Beijing-Tianjin-Hebei Region. In the short-term period, the location planning of THUAA is consistent with the spatial development pattern, the transportation network, and the population distribution. In the medium-term and long-term periods, along with the development of intercity rail, the increase of urban population and the urban population migration from city center to suburban, more and more level-2 and level-3 hubs are opened.Figure 13
The location of THUAA in Beijing-Tianjin-Hebei Region.
(a)
The short-term location planning of THUAA in Beijing-Tianjin-Hebei Region (b)
The medium-term location planning of THUAA in Beijing-Tianjin-Hebei Region (c)
The long-term location planning of THUAA in Beijing-Tianjin-Hebei RegionIn 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. Figure14 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.Figure 14
The demand weighted travel time in different scenarios.Figure 15
The number of hubs at each level in different scenarios.


## 6.1. Data

The administrative districts in Beijing (see Figure8), 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:Lij: it is measured in arcGIS; see Table 2 in Appendix.Rh: R1 is defined as 70 km. R2 is defined as 150 km. R3 is defined as 300 km.ch: it is estimated according to the passenger transport volume of Beijing-Tianjin-Hebei Region from China’s Urban Statistical Yearbooks in the recent years. c1 is defined as 7000 thousand persons annually. c2 is defined as 3000 thousand persons annually. c3 is defined as 2500 thousand persons annually.dijt: 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.qist: 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.Cjst: 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.


## 6.2. Computational Results

The IACSA is coded in C#, running on a laptop with 2.50 GHz Intel Core I5-2450M processor and 4 GB of RAM under the system Microsoft Windows 7. Also, the optimization software CPLEX version 12.2 is utilized to solve the problem. The parameters of the IACSA are defined as follows:M = 30, A=10, a0=5, ɛmax=0.9, ɛmin=0.1, B=15, b0=5, N=100, Q=106, δ1=106, δ2=106, and δ3=106.The problem is run 10 times by IACSA. The iterative process of IACSA is presented in Figure12.Figure 12
The iterative process of IACSA.The objective function value and computational time of IACSA are compared with the results from CPLEX, as shown in Table1.  ZCPLEX is the optimal objective function value of the model, and ZIACSA is the objective function value of IACSA. Gap is defined as follows:(11)Gap=ZIACSA-ZCPLEXZCPLEX×100%.Table 1
The results comparison between IACSA and CPLEX.
The number of areas in UAA Objective function value Computational time (s) Z C P L E X Z I A C S A Gap (%) CPLEX IACSA 38 83523 83635 0.1 952.6 12.3 300 3652384 3681603 0.6 42485.7 1845.1Table 2
Distances between areas in Beijing-Tianjin-Hebei Region (km).
Area 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 1 0 6 10 18 19 20 36 24 36 40 29 29 58 74 76 81 138 140 122 168 114 93 92 164 162 145 101 58 300 160 293 178 451 402 193 226 216 272 2 6 0 18 13 15 18 32 26 43 42 23 25 61 81 80 82 140 142 125 170 117 95 94 166 164 147 108 61 294 154 295 181 445 396 194 230 223 266 3 10 18 0 29 31 24 49 19 30 44 38 43 53 68 72 84 133 135 117 163 109 88 87 159 157 140 95 58 312 172 288 173 462 414 196 222 211 281 4 18 13 29 0 11 18 23 37 60 54 21 24 74 93 92 94 132 140 123 156 122 101 99 171 170 146 120 64 285 145 300 186 436 387 206 242 221 264 5 19 15 31 11 0 16 24 38 61 50 30 15 75 94 94 90 140 148 132 164 128 107 105 178 176 154 122 72 286 146 307 192 437 388 202 244 230 273 6 20 18 24 18 16 0 37 45 48 37 37 23 62 86 81 78 159 161 143 189 136 114 113 185 183 166 113 79 299 159 314 199 450 401 190 231 237 280 7 36 32 49 23 24 37 0 60 78 71 24 36 92 111 110 111 160 157 145 190 137 115 119 186 184 167 138 81 265 125 320 206 416 367 223 260 225 265 8 24 26 19 37 38 45 60 0 26 61 44 51 50 61 87 101 126 127 110 155 102 80 72 151 150 132 88 57 323 183 274 159 474 425 213 237 203 287 9 36 43 30 60 61 48 78 26 0 47 68 68 24 41 42 87 153 155 138 183 130 108 100 179 177 160 88 84 338 198 302 187 488 440 199 192 231 315 10 40 42 44 54 50 37 71 61 47 0 62 44 55 98 73 43 175 177 159 200 151 130 128 201 199 181 125 98 326 186 330 215 477 428 155 223 253 305 11 29 23 38 21 30 37 24 44 68 62 0 43 83 101 101 103 121 129 121 145 113 92 97 163 161 144 129 53 281 141 298 183 431 383 215 251 202 244 12 29 25 43 24 15 23 36 51 68 44 43 0 84 102 102 83 154 161 145 178 144 122 121 193 191 167 130 86 288 148 322 208 439 390 195 252 242 284 13 58 61 53 74 75 62 92 50 24 55 83 84 0 59 21 94 183 184 167 212 159 138 120 208 207 189 97 108 353 213 322 207 504 455 206 180 261 325 14 74 81 68 93 94 86 111 61 41 98 101 102 59 0 50 138 139 146 170 163 144 140 70 165 137 192 34 116 374 234 263 149 524 476 250 160 263 347 15 76 80 72 92 94 81 110 87 42 73 101 102 21 50 0 113 202 204 186 232 179 157 109 228 226 209 88 127 373 233 310 196 524 475 225 153 280 344 16 81 82 84 94 90 78 111 101 87 43 103 83 94 138 113 0 218 220 203 248 195 173 172 244 242 225 168 141 366 226 373 259 517 468 135 267 296 348 17 138 140 133 132 140 159 160 126 153 175 121 154 183 139 202 218 0 11 22 25 16 38 77 42 68 40 116 83 321 181 266 122 457 423 320 298 114 251 18 140 142 135 140 148 161 157 127 155 177 129 161 184 146 204 220 11 0 37 18 31 53 85 36 71 51 123 91 333 193 266 122 467 435 333 298 124 262 19 122 125 117 123 132 143 145 110 138 159 121 145 167 170 186 203 22 37 0 48 20 44 106 72 83 35 144 64 293 153 281 137 443 395 312 313 114 241 20 168 170 163 156 164 189 190 155 183 200 145 178 212 163 232 248 25 18 48 0 45 65 99 30 80 59 138 110 338 198 261 126 463 421 350 302 119 257 21 114 117 109 122 128 136 137 102 130 151 113 144 159 144 179 195 16 31 20 45 0 22 81 57 82 46 119 66 304 164 277 133 455 406 304 308 128 253 22 93 95 88 101 107 114 115 80 108 130 92 122 138 140 157 173 38 53 44 65 22 0 54 80 89 68 126 35 315 175 284 140 466 417 283 307 139 263 23 92 94 87 99 105 113 119 72 100 128 97 121 120 70 109 172 77 85 106 99 81 54 0 103 84 128 47 70 375 235 210 96 526 477 281 238 199 324 24 164 166 159 171 178 185 186 151 179 201 163 193 208 165 228 244 42 36 72 30 57 80 103 0 41 83 134 116 362 222 233 108 482 440 354 284 138 276 25 162 164 157 170 176 183 184 150 177 199 161 191 207 137 226 242 68 71 83 80 82 89 84 41 0 130 103 113 378 238 199 55 528 480 351 231 182 319 26 145 147 140 146 154 166 167 132 160 181 144 167 189 192 209 225 40 51 35 59 46 68 128 83 130 0 168 87 297 157 326 182 425 383 336 359 81 219 27 101 108 95 120 122 113 138 88 88 125 129 130 97 34 88 168 116 123 144 138 119 126 47 134 103 168 0 151 415 275 230 116 566 517 277 147 239 363 28 58 61 58 64 72 79 81 57 84 98 53 86 108 116 127 141 83 91 64 110 66 35 70 116 113 87 151 0 253 132 250 129 379 335 212 194 137 218 29 300 294 312 285 286 299 265 323 338 326 281 288 353 374 373 366 321 333 293 338 304 315 375 362 378 297 415 253 0 130 484 367 160 109 309 440 212 110 30 160 154 172 145 146 159 125 183 198 186 141 148 213 234 233 226 181 193 153 198 164 175 235 222 238 157 275 132 130 0 367 250 265 220 224 315 136 130 31 293 295 288 300 307 314 320 274 302 330 298 322 322 263 310 373 266 266 281 261 277 284 210 233 199 326 230 250 484 367 0 130 574 543 409 183 294 412 32 178 181 173 186 192 199 206 159 187 215 183 208 207 149 196 259 122 122 137 126 133 140 96 108 55 182 116 129 367 250 130 0 470 433 311 152 188 306 33 451 445 462 436 437 450 416 474 488 477 431 439 504 524 524 517 457 467 443 463 455 466 526 482 528 425 566 379 160 265 574 470 0 53 468 576 286 165 34 402 396 414 387 388 401 367 425 440 428 383 390 455 476 475 468 423 435 395 421 406 417 477 440 480 383 517 335 109 220 543 433 53 0 418 527 252 130 35 193 194 196 206 202 190 223 213 199 155 215 195 206 250 225 135 320 333 312 350 304 283 281 354 351 336 277 212 309 224 409 311 468 418 0 275 326 345 36 226 230 222 242 244 231 260 237 192 223 251 252 180 160 153 267 298 298 313 302 308 307 238 284 231 359 147 194 440 315 183 152 576 527 275 0 315 413 37 216 223 211 221 230 237 225 203 231 253 202 242 261 263 280 296 114 124 114 119 128 139 199 138 182 81 239 137 212 136 294 188 286 252 326 315 0 120 38 272 266 281 264 273 280 265 287 315 305 244 284 325 347 344 348 251 262 241 257 253 263 324 276 319 219 363 218 110 130 412 306 165 130 345 413 120 0Table 3
Travel time between areas in Beijing-Tianjin-Hebei Region in 2020 (h).
Area 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 1 0.0 0.1 0.1 0.1 0.2 0.2 0.3 0.2 0.3 0.3 0.2 0.2 0.5 0.6 0.6 0.7 1.2 1.2 1.0 1.4 1.0 0.8 0.8 1.4 1.4 1.2 0.8 0.2 1.2 0.6 4.2 1.5 1.8 1.6 0.8 1.9 0.9 3.9 2 0.1 0.0 0.1 0.1 0.1 0.2 0.3 0.2 0.4 0.3 0.2 0.2 0.5 0.7 0.7 0.7 1.2 1.2 1.0 1.4 1.0 0.8 0.8 1.4 1.4 1.2 0.9 0.2 1.2 0.6 4.2 1.5 1.8 1.6 0.8 1.9 0.9 3.8 3 0.1 0.1 0.0 0.2 0.3 0.2 0.4 0.2 0.2 0.4 0.3 0.4 0.4 0.6 0.6 0.7 1.1 1.1 1.0 1.4 0.9 0.7 0.7 1.3 1.3 1.2 0.8 0.2 1.2 0.7 4.1 1.4 1.8 1.7 0.8 1.8 0.8 4.0 4 0.1 0.1 0.2 0.0 0.1 0.2 0.2 0.3 0.5 0.4 0.2 0.2 0.6 0.8 0.8 0.8 1.1 1.2 1.0 1.3 1.0 0.8 0.8 1.4 1.4 1.2 1.0 0.3 1.1 0.6 4.3 1.5 1.7 1.5 0.8 2.0 0.9 3.8 5 0.2 0.1 0.3 0.1 0.0 0.1 0.2 0.3 0.5 0.4 0.2 0.1 0.6 0.8 0.8 0.8 1.2 1.2 1.1 1.4 1.1 0.9 0.9 1.5 1.5 1.3 1.0 0.3 1.1 0.6 4.4 1.6 1.7 1.6 0.8 2.0 0.9 3.9 6 0.2 0.2 0.2 0.2 0.1 0.0 0.3 0.4 0.4 0.3 0.3 0.2 0.5 0.7 0.7 0.6 1.3 1.3 1.2 1.6 1.1 1.0 0.9 1.5 1.5 1.4 0.9 0.3 1.2 0.6 4.5 1.7 1.8 1.6 0.8 1.9 0.9 4.0 7 0.3 0.3 0.4 0.2 0.2 0.3 0.0 0.5 0.6 0.6 0.2 0.3 0.8 0.9 0.9 0.9 1.3 1.3 1.2 1.6 1.1 1.0 1.0 1.6 1.5 1.4 1.2 0.3 1.1 0.5 4.6 1.7 1.7 1.5 0.9 2.2 0.9 3.8 8 0.2 0.2 0.2 0.3 0.3 0.4 0.5 0.0 0.2 0.5 0.4 0.4 0.4 0.5 0.7 0.8 1.0 1.1 0.9 1.3 0.9 0.7 0.6 1.3 1.2 1.1 0.7 0.2 1.3 0.7 3.9 1.3 1.9 1.7 0.9 2.0 0.8 4.1 9 0.3 0.4 0.2 0.5 0.5 0.4 0.6 0.2 0.0 0.4 0.6 0.6 0.2 0.3 0.4 0.7 1.3 1.3 1.1 1.5 1.1 0.9 0.8 1.5 1.5 1.3 0.7 0.3 1.4 0.8 4.3 1.6 2.0 1.8 0.8 1.6 0.9 4.5 10 0.3 0.3 0.4 0.4 0.4 0.3 0.6 0.5 0.4 0.0 0.5 0.4 0.5 0.8 0.6 0.4 1.5 1.5 1.3 1.7 1.3 1.1 1.1 1.7 1.7 1.5 1.0 0.4 1.3 0.7 4.7 1.8 1.9 1.7 0.6 1.9 1.0 4.4 11 0.2 0.2 0.3 0.2 0.2 0.3 0.2 0.4 0.6 0.5 0.0 0.4 0.7 0.8 0.8 0.9 1.0 1.1 1.0 1.2 0.9 0.8 0.8 1.4 1.3 1.2 1.1 0.2 1.1 0.6 4.3 1.5 1.7 1.5 0.9 2.1 0.8 3.5 12 0.2 0.2 0.4 0.2 0.1 0.2 0.3 0.4 0.6 0.4 0.4 0.0 0.7 0.9 0.8 0.7 1.3 1.3 1.2 1.5 1.2 1.0 1.0 1.6 1.6 1.4 1.1 0.3 1.2 0.6 4.6 1.7 1.8 1.6 0.8 2.1 1.0 4.1 13 0.5 0.5 0.4 0.6 0.6 0.5 0.8 0.4 0.2 0.5 0.7 0.7 0.0 0.5 0.2 0.8 1.5 1.5 1.4 1.8 1.3 1.1 1.0 1.7 1.7 1.6 0.8 0.4 1.4 0.9 4.6 1.7 2.0 1.8 0.8 1.5 1.0 4.6 14 0.6 0.7 0.6 0.8 0.8 0.7 0.9 0.5 0.3 0.8 0.8 0.9 0.5 0.0 0.4 1.2 1.2 1.2 1.4 1.4 1.2 1.2 0.6 1.4 1.1 1.6 0.3 0.5 1.5 0.9 3.8 1.2 2.1 1.9 1.0 1.3 1.1 5.0 15 0.6 0.7 0.6 0.8 0.8 0.7 0.9 0.7 0.4 0.6 0.8 0.8 0.2 0.4 0.0 0.9 1.7 1.7 1.6 1.9 1.5 1.3 0.9 1.9 1.9 1.7 0.7 0.5 1.5 0.9 4.4 1.6 2.1 1.9 0.9 1.3 1.1 4.9 16 0.7 0.7 0.7 0.8 0.8 0.6 0.9 0.8 0.7 0.4 0.9 0.7 0.8 1.2 0.9 0.0 1.8 1.8 1.7 2.1 1.6 1.4 1.4 2.0 2.0 1.9 1.4 0.6 1.5 0.9 5.3 2.2 2.1 1.9 0.5 2.2 1.2 5.0 17 1.2 1.2 1.1 1.1 1.2 1.3 1.3 1.0 1.3 1.5 1.0 1.3 1.5 1.2 1.7 1.8 0.0 0.1 0.2 0.2 0.1 0.3 0.6 0.4 0.6 0.3 1.0 0.3 2.7 0.7 3.8 0.5 3.8 2.1 1.3 4.3 0.5 3.6 18 1.2 1.2 1.1 1.2 1.2 1.3 1.3 1.1 1.3 1.5 1.1 1.3 1.5 1.2 1.7 1.8 0.1 0.0 0.3 0.1 0.3 0.4 0.7 0.3 0.6 0.4 1.0 0.4 2.8 0.8 3.8 0.5 3.9 2.2 1.3 4.3 0.5 3.7 19 1.0 1.0 1.0 1.0 1.1 1.2 1.2 0.9 1.1 1.3 1.0 1.2 1.4 1.4 1.6 1.7 0.2 0.3 0.0 0.4 0.2 0.4 0.9 0.6 0.7 0.3 1.2 0.3 2.4 0.6 4.0 0.5 3.7 2.0 1.2 4.5 0.5 3.4 20 1.4 1.4 1.4 1.3 1.4 1.6 1.6 1.3 1.5 1.7 1.2 1.5 1.8 1.4 1.9 2.1 0.2 0.1 0.4 0.0 0.4 0.5 0.8 0.3 0.7 0.5 1.1 0.4 2.8 0.8 3.7 0.5 3.9 2.1 1.4 4.3 0.5 3.7 21 1.0 1.0 0.9 1.0 1.1 1.1 1.1 0.9 1.1 1.3 0.9 1.2 1.3 1.2 1.5 1.6 0.1 0.3 0.2 0.4 0.0 0.2 0.7 0.5 0.7 0.4 1.0 0.3 2.5 0.7 4.0 0.5 3.8 2.0 1.2 4.4 0.5 3.6 22 0.8 0.8 0.7 0.8 0.9 1.0 1.0 0.7 0.9 1.1 0.8 1.0 1.1 1.2 1.3 1.4 0.3 0.4 0.4 0.5 0.2 0.0 0.4 0.7 0.7 0.6 1.0 0.1 2.6 0.7 4.1 0.6 3.9 2.1 1.1 4.4 0.6 3.8 23 0.8 0.8 0.7 0.8 0.9 0.9 1.0 0.6 0.8 1.1 0.8 1.0 1.0 0.6 0.9 1.4 0.6 0.7 0.9 0.8 0.7 0.4 0.0 0.9 0.7 1.1 0.4 0.3 3.1 0.9 3.0 0.4 4.4 2.4 1.1 3.4 0.8 4.6 24 1.4 1.4 1.3 1.4 1.5 1.5 1.6 1.3 1.5 1.7 1.4 1.6 1.7 1.4 1.9 2.0 0.4 0.3 0.6 0.3 0.5 0.7 0.9 0.0 0.3 0.7 1.1 0.5 3.0 0.9 3.3 0.4 4.0 2.2 1.4 4.1 0.6 3.9 25 1.4 1.4 1.3 1.4 1.5 1.5 1.5 1.2 1.5 1.7 1.3 1.6 1.7 1.1 1.9 2.0 0.6 0.6 0.7 0.7 0.7 0.7 0.7 0.3 0.0 1.1 0.9 0.5 3.1 1.0 2.8 0.2 4.4 2.4 1.4 3.3 0.7 4.6 26 1.2 1.2 1.2 1.2 1.3 1.4 1.4 1.1 1.3 1.5 1.2 1.4 1.6 1.6 1.7 1.9 0.3 0.4 0.3 0.5 0.4 0.6 1.1 0.7 1.1 0.0 1.4 0.3 2.5 0.6 4.7 0.7 3.5 1.9 1.3 5.1 0.3 3.1 27 0.8 0.9 0.8 1.0 1.0 0.9 1.2 0.7 0.7 1.0 1.1 1.1 0.8 0.3 0.7 1.4 1.0 1.0 1.2 1.1 1.0 1.0 0.4 1.1 0.9 1.4 0.0 0.6 3.5 1.1 3.3 0.5 4.7 2.6 1.1 2.1 1.0 5.2 28 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.2 0.3 0.4 0.2 0.3 0.4 0.5 0.5 0.6 0.3 0.4 0.3 0.4 0.3 0.1 0.3 0.5 0.5 0.3 0.6 0.0 2.1 0.5 3.6 1.1 1.5 4.8 3.0 2.8 2.0 3.1 29 1.2 1.2 1.2 1.1 1.1 1.2 1.1 1.3 1.4 1.3 1.1 1.2 1.4 1.5 1.5 1.5 2.7 2.8 2.4 2.8 2.5 2.6 3.1 3.0 3.1 2.5 3.5 2.1 0.0 0.5 4.0 1.5 0.6 0.4 1.2 1.8 3.0 0.4 30 0.6 0.6 0.7 0.6 0.6 0.6 0.5 0.7 0.8 0.7 0.6 0.6 0.9 0.9 0.9 0.9 0.7 0.8 0.6 0.8 0.7 0.7 0.9 0.9 1.0 0.6 1.1 0.5 0.5 0.0 1.8 1.0 1.1 0.9 0.9 1.3 1.9 1.9 31 4.2 4.2 4.1 4.3 4.4 4.5 4.6 3.9 4.3 4.7 4.3 4.6 4.6 3.8 4.4 5.3 3.8 3.8 4.0 3.7 4.0 4.1 3.0 3.3 2.8 4.7 3.3 3.6 4.0 1.8 0.0 0.5 8.2 7.8 5.8 2.6 1.2 5.9 32 1.5 1.5 1.4 1.5 1.6 1.7 1.7 1.3 1.6 1.8 1.5 1.7 1.7 1.2 1.6 2.2 0.5 0.5 0.5 0.5 0.5 0.6 0.4 0.4 0.2 0.7 0.5 1.1 1.5 1.0 0.5 0.0 3.9 3.6 2.6 2.2 2.7 4.4 33 1.8 1.8 1.8 1.7 1.7 1.8 1.7 1.9 2.0 1.9 1.7 1.8 2.0 2.1 2.1 2.1 3.8 3.9 3.7 3.9 3.8 3.9 4.4 4.0 4.4 3.5 4.7 1.5 0.6 1.1 8.2 3.9 0.0 0.2 1.9 2.3 4.1 0.7 34 1.6 1.6 1.7 1.5 1.6 1.6 1.5 1.7 1.8 1.7 1.5 1.6 1.8 1.9 1.9 1.9 2.1 2.2 2.0 2.1 2.0 2.1 2.4 2.2 2.4 1.9 2.6 4.8 0.4 0.9 7.8 3.6 0.2 0.0 2.1 2.6 3.6 1.9 35 0.8 0.8 0.8 0.8 0.8 0.8 0.9 0.9 0.8 0.6 0.9 0.8 0.8 1.0 0.9 0.5 1.3 1.3 1.2 1.4 1.2 1.1 1.1 1.4 1.4 1.3 1.1 3.0 1.2 0.9 5.8 2.6 1.9 2.1 0.0 2.3 2.7 4.9 36 1.9 1.9 1.8 2.0 2.0 1.9 2.2 2.0 1.6 1.9 2.1 2.1 1.5 1.3 1.3 2.2 4.3 4.3 4.5 4.3 4.4 4.4 3.4 4.1 3.3 5.1 2.1 2.8 1.8 1.3 2.6 2.2 2.3 2.6 2.3 0.0 4.5 5.9 37 0.9 0.9 0.8 0.9 0.9 0.9 0.9 0.8 0.9 1.0 0.8 1.0 1.0 1.1 1.1 1.2 0.5 0.5 0.5 0.5 0.5 0.6 0.8 0.6 0.7 0.3 1.0 2.0 3.0 1.9 1.2 2.7 4.1 3.6 2.7 4.5 0.0 1.7 38 3.9 3.8 4.0 3.8 3.9 4.0 3.8 4.1 4.5 4.4 3.5 4.1 4.6 5.0 4.9 5.0 3.6 3.7 3.4 3.7 3.6 3.8 4.6 3.9 4.6 3.1 5.2 3.1 0.4 1.9 5.9 4.4 2.4 1.9 4.9 5.9 1.7 0.0Table 4
Different-level demands and maximum capacities of areas in Beijing-Tianjin-Hebei Region in 2020 (ten thousand persons annually).
Area q i 11 q i 21 q i 31 C j 11 C j 21 C j 31 1 40301 2529 1651 61131 5227 5119 2 57598 3615 2360 87369 6142 5119 3 172395 9306 7065 261502 9648 8462 4 101128 6347 4144 153398 6534 13285 5 28755 1805 1178 43617 0 0 6 159168 7631 6523 241438 8574 0 7 32837 2061 1346 49809 6403 0 8 40057 2514 1642 60762 6860 5119 9 25249 1585 1035 38299 6599 9645 10 71147 4465 2916 107920 6272 0 11 50313 3158 2062 76318 6991 6399 12 12141 810 498 18417 0 0 13 11543 725 473 17510 0 0 14 10670 670 437 16185 3920 0 15 12187 765 499 18487 4247 0 16 7312 459 300 11092 0 0 17 90939 5430 3415 101352 5440 5701 18 12920 686 485 14399 2176 2851 19 14584 871 548 16254 3264 0 20 12774 763 480 14236 0 0 21 14568 870 547 16236 2984 0 22 20439 1085 767 22779 3228 0 23 16224 861 609 18082 3264 0 24 52152 2768 1958 58124 3445 5131 25 8552 454 321 9531 0 0 26 13815 733 519 15397 0 0 27 16345 868 614 18216 0 0 28 20616 3259 1481 23344 8299 0 29 53405 8443 3837 60471 19151 14505 30 40085 6337 2880 45388 12129 2872 31 13925 2201 1000 15767 2160 0 32 37114 5868 2666 42024 13086 8703 33 44258 6997 3180 50113 11810 2843 34 29118 4603 2092 32970 0 0 35 20911 3306 1502 23678 3541 0 36 14930 2360 1073 16905 3126 0 37 31108 4918 2235 35224 5832 0 38 17556 2775 1261 19878 2869 0According to Table1, 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 an hour. Gap between IACSA and CPLEX is still less than 1%. Therefore, it can be concluded that IACSA is producing reasonable results in large-size problems.Figure13 presents the location of THUAA in Beijing-Tianjin-Hebei Region. In the short-term period, the location planning of THUAA is consistent with the spatial development pattern, the transportation network, and the population distribution. In the medium-term and long-term periods, along with the development of intercity rail, the increase of urban population and the urban population migration from city center to suburban, more and more level-2 and level-3 hubs are opened.Figure 13
The location of THUAA in Beijing-Tianjin-Hebei Region.
(a)
The short-term location planning of THUAA in Beijing-Tianjin-Hebei Region (b)
The medium-term location planning of THUAA in Beijing-Tianjin-Hebei Region (c)
The long-term location planning of THUAA in Beijing-Tianjin-Hebei RegionIn 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. Figure14 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.Figure 14
The demand weighted travel time in different scenarios.Figure 15
The number of hubs at each level in different scenarios.


## 7. 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 real-life 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.



---

*Source: 7189060-2017-01-12.xml*