# Research on the Degree of Coupling between the Urban Public Infrastructure System and the Urban Economic, Social, and Environmental System: A Case Study in Beijing, China

Authors: ZhiMei Tao
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The coordinated development of urban public infrastructure system and urban economic, social, and environmental system is an important goal for the integrated management and sustainable development of urban public infrastructure system. This paper constructs a research model of the degree of coupling coordination between urban public infrastructure system and urban economic, social, and environmental system using the analytic network process (ANP), the analytic hierarchy process (AHP), a combination evaluation method based on game theory, and a coupling coordination degree model. Using Beijing data from 2000 to 2016, the degree of coupling coordination between the Beijing urban public infrastructure system and the urban economic, social, and environmental system is empirically analyzed. This study finds that (1) the supply level of Beijing’s urban public infrastructure system has an obvious impact on the degree of coupling coordination between the two systems. (2) The global financial crisis reduced the supply speed of the urban public infrastructure system in Beijing, and put the dynamic coupling state of the two systems in the low-level symbiosis stage. Beijing needs to improve the supply of urban public infrastructure to support the development of the urban economic, social, and environmental system. (3) Improving the supply level of the urban environmental infrastructure in Beijing, especially improving sewage disposal capacity and increasing the number of special vehicles for urban sanitation and the amount of domestic waste clearance, will positively affect the degree of coupling coordination between two systems. (4) An increase in the GDP of Beijing has a direct positive impact on the degree of coupling coordination. In addition, the increase in the social development level of the employees in three industries in Beijing has a significant impact on the degree of coupling coordination.


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## Body

## 1. Introduction

An infrastructure system consisting of subsystems such as energy, transportation, water resources, postal service and telecommunication, and environmental facilities is an important basis for economic productivity and population welfare [1]. As a supporting system and carrier of urban economic and social activities, urban infrastructure supply faces a series of pressures and challenges under global urbanization. Due to a wide range of economic, social, technological, and other factors, the demand for urban infrastructure is gradually increasing, and many investments have been made to meet the needs of urban public infrastructure. Since 1981, China’s fixed asset investment in urban public infrastructure has continued to increase, and its supply and service levels have continued to improve over the long term [2]. Nevertheless, the supply level of urban infrastructure still fails to meet social demand in China. After summer rainstorms, the drainage in cities is not adequate, and in the summer, “going to the city to see the sea” has become a phenomenon in many cities. Some cities are besieged by garbage; in some places, garbage is blown by the wind and sewage is evaporated, and other problems such as road zippers, traffic congestion, air and water pollution still exist. Improving the supply efficiency of urban infrastructure systems is an important problem for urban managers.The urban public infrastructure system is a multibody, complex, and composite system of energy, transportation, postal services and telecommunications, the environment, water resources, and other subsystems. These subsystems are interdependent and interrelated and form the whole urban public infrastructure system together, providing the basic products and services needed for urban development [3, 4]. Infrastructure supply is affected by the human research view and the specialized division of urban public infrastructure supply, and traditionally, which is provided by each department alone, there is a lack of communication and contact between departments [3, 5, 6]. Over time, people have become increasingly aware that the various subsystems of urban public infrastructure are interrelated and interact, forming a system of urban public infrastructure. Only by including supply management in the system as a whole can we improve the overall supply level of urban public infrastructure in an orderly manner, so that the urban public infrastructure system can meet the needs of urban economic and social development. The United Kingdom proposed studying the relationship between the national infrastructure system and economic and social development and managing the future development plan of the national infrastructure system as a whole [7]. In 2017, China’s “13th Five-Year Plan for Urban Public Infrastructure Construction” propose for the first time to change the way in which previous departments compiled industry plans separately, to consider the overall planning of urban infrastructure in a systematic way and to coordinate the supply of products and services in all sectors to meet the requirements of urban economic and social development.Understanding the interdependence of the various subsystems of urban public infrastructure is a prerequisite for studying the overall supply of urban public infrastructure systems. Different scholars define and classify the interdependence among subsystems of urban public infrastructure. Rinaldi believes that thee physical, information, spatial, and logical interdependence exists among the subsystems of urban public infrastructure [3]. Many scholars have summarized, classified, and simulated the interdependence of urban public infrastructure [8–10]. Jaime presents a System Safety Management System (SSMS) model to interdependency modelling for the case of the Mexico City Metro transport network, which highlights that interdependency in the Metro transportation network occurs vertically and horizontally [11]. The study of the interdependence of urban public infrastructure mainly focuses on the impact of urban public infrastructure interdependence when a sudden event occurs [12]. However, the interdependence of urban public infrastructure exists not only when an emergency arises but also in people’s daily lives, and the impact on the supply of urban public infrastructure products and services is equally important [13]. The research on the interdependence of the various subsystems of urban public infrastructure has been deepening. Tao proposes that the indispensability, completeness, and irreplaceability of the subsystems of urban public infrastructure are the essence and fundamental reasons for the interdependence of the subsystems of urban public infrastructure and an important basis for the formation of the overall supply management system of urban infrastructure. The interdependence of the various subsystems of urban public infrastructure is of great significance to the normal operation of urban public infrastructure and the integrated supply management of the system [4]. Due to the complexity of the interdependence of urban infrastructure systems and their subsystems, studying the role of interdependence of infrastructure systems and the supply management of infrastructure systems is challenging [14].The urban public infrastructure system should promote the city’s economic growth, social welfare, and environmental quality development. An improvement in the supply level of urban public infrastructure systems will increase the labor productivity of the society, expand total social demand, increase the accumulation of fixed capital, increase the total output of the society, and guarantee the economic growth of the city [15, 16]. In addition, an increase in the supply level of urban public infrastructure will promote the convenience of urban production and living activities, attract foreign populations, increase employment opportunities, and improve the social welfare level of cities [17, 18]. The advancement of urbanization has put pressure on the urban environment to a certain extent. Urban public infrastructure can deal with urban garbage and sewage, reduce air pollution, slow the heat island effect, improve urban climate conditions, etc., and have a positive impact on the urban environment [19–21]. On the one hand, an improvement in the urban public infrastructure supply level will increase the level of urban development and the growth of urban social wealth so that the city government will be stronger and have more funds to invest in the construction and operation of urban infrastructure. On the other hand, due to the advancement of urbanization, urban population concentration, economic growth, and environmental quality have put pressure on urban public infrastructure. Urban public infrastructure is interdependent and mutually influential. An urban public infrastructure system based on interdependence is a collection of human activities that actively arrange and integrate the physical facilities and activities of the subsystems of urban public infrastructure and ensure that the subsystems of the urban public infrastructure interact to improve the overall efficiency of the urban public infrastructure system. Through infrastructure management control, the collection of information regarding system infrastructure demand, a consideration of the interdependence between the subsystems of infrastructure, expert opinion and government decision-making, city governments engage in urban infrastructure supply management and make decisions to ensure the supply of urban public infrastructure system can meet to support the development of urban, economic, social, and environmental system. The interaction that occurs between urban public infrastructure systems and urban economic and social environmental systems is shown in Figure 1. The direct goal of coordinated development between urban public infrastructure system and urban economic, social, and environmental system is to realize the integrated management of the supply benefits of urban public infrastructure systems, to improve the economic growth, social welfare, and environmental quality of cities. The long-term goal of the coordinated development of urban public infrastructure systems and urban economic, social, and environmental systems is to achieve the rational allocation of resources to urban public infrastructure systems and realize the long-term sustainable development of cities.Figure 1
Interaction between urban public infrastructure supply and urban economic and social environment.There is no research on the coordinated development of infrastructure and the economic environment in foreign countries. In the UK, the Infrastructure Transformation Research Consortium (ITRC), which consists of researchers from seven universities, including Oxford, Cambridge, Newcastle, and Leeds, proposes that urban infrastructure systems are complex applied systems [22]. Through the use of system-of-system modeling (SOSM), this consortium simulates and studies the relationship between national infrastructure and economic and social development and conducts cross-sectoral performance evaluations of infrastructure systems that face an uncertain future [7]. We compare research on the coordinated development of infrastructure and social development with SOSM research and find the following.The similarities between the two types of studies are as follows:(1)
Both have comprehensive strategic objectives. The ultimate goal of both studies is to achieve sustainable regional or urban development through an orderly allocation of resources.(2)
Both have received considerable attention at the national level. In the UK, the national infrastructure system is highly regarded, while the national finance department proposes using an integrated approach to the development of national infrastructure plans [7]. China’s Ministry of Construction proposes systematically studying the construction and management planning of urban public infrastructure and promoting solutions for urban problems.(3)
Both have a similar research objective: ITRC focuses on the long-term supply management of energy, transportation, water resources, solid waste and wastewater clean-up and recycling, and data and information communication services. The main research objective related to China’s infrastructure and urban coordinated development is the relationship between the supply level of urban infrastructure systems and the level of urban development.The differences between the two studies are as follows:(1)
Their research focus is different. SOSM proposes managing the infrastructure system across departments as a whole for the purpose of transforming, storing, and transmitting infrastructure traffic by managing physical infrastructure entities and the corresponding human system behaviors to allow for the long-term performance evaluation of the various departments involved in infrastructure. Research on the coordinated development of urban infrastructure and urban economic, social, and environmental systems considers the overall supply of urban public infrastructure and the given resources and their ability to support sustainable urban development, mainly by considering the overall management of the interaction between the supply level of infrastructure systems and the level of urban development.(2)
The scope of each type of research is different. The scope of research conducted using SOSM is limited to the UK’s national infrastructure because the UK’s infrastructure supply policy is made at the national level, and data on infrastructure supply, inputs, and outputs throughout the Commonwealth are available. Considering China’s vast geographical scope and regional differences, China’s research on the coordination of infrastructure and urban development mainly considers the synergistic relationship between the infrastructure provided by cities or regions and their jurisdiction and the development level of the cities or regions. Generally, coordinated development studies are not conducted at the national level.(3)
The perspective of each type of research is different. The research framework of SOSM combines top-down infrastructure supply management with bottom-up infrastructure providers and the behavioral performance of service objects based on the quality of service of the infrastructure (such as reliability, cost, security, and environmental impact) and focuses on the long-term management of national infrastructure through the linkage and management of a large number of infrastructure service departments. Coordinated development research generally does not consider subjective demand and differences in infrastructure service objects. This research mainly examines the level of urban public infrastructure supply by considering the number of services provided by urban infrastructure (e.g., electricity supply and garbage removal) and explores the interaction between infrastructure supply levels and the level of urban development.(4)
Each type of research has a different understanding of the interdependence of the infrastructure subsystems. SOSM is based on the premise that the interdependence of infrastructure subsystems is the basis for national infrastructure system integration modeling. It has been proposed that infrastructure involves more than the operation of a single specialized department. It is necessary to present the interdependence of various departments of the infrastructure using a cross-sectoral integration model and evaluate the cross-department performance of each subsystem of infrastructure. Existing research on urban coordinated development of infrastructure and the level of urban development does not recognize the interdependence of infrastructure subsystems and proposes that the urban public infrastructure system should support the level of urban development but not enforce an excessive level of urban development, which would waste social resources.The earliest research on urban infrastructure and urban coordinated development was conducted by Zhang Junyong, who believes that urban infrastructure construction investment and economic development have a mutually reinforcing relationship [23]. At present, Chinese scholars’ research on the coordinated development of infrastructure and cities is divided into two categories: qualitative research and quantitative research. Qualitative research focus on the sharing, co-construction, and coordinated development of environmental infrastructure [24], interact with each other on transportation infrastructure and urbanization [25]. Scholars who conduct quantitative research mainly include the following: Yu et al. [26] construct an index system to evaluate economic and social development and analyze the coordinated development of Qingdao’s urban resources, infrastructure and economic society. Wu et al. [27]construct evaluation indicators for regional economic systems and calculate the coordination degree of the economic system and infrastructure system in various provinces and cities in China. Ying et al. [28] analyze the coordination of Hangzhou infrastructure and climate based on the coupling relationship between climate and green infrastructure.In short, there is still a wide range of research fields that consider the coordinated development of domestic and foreign urban infrastructure supply and the level of urban development. (1) In terms of the evaluation index system, existing research is more focused on the coordinated development of the level of infrastructure supply and the level of urban development. This study incorporates indicators for the level of urban development measuring the environmental pressures in cities and constructs indicators for the urban economic, social, and environmental system by focusing on the three aspects of urban economic development, social welfare, and environmental pressure. This is a new breakthrough in the construction of an index system. (2) In terms of research methods, the methods used to generate the weights of the evaluation indicators are mainly limited to using the coupling coordination degree model. The dynamic coupling and coordination model has not been used. (3) In terms of the research design, this study focuses on the infrastructure system and the interdependence of the infrastructure subsystems. This study applies the network analytic hierarchy process (AHP) and considers the infrastructure of the interdependence of the subsystems to evaluate the supply level of the urban public infrastructure system.This paper constructs a model of the coupling coordination degree of an urban infrastructure system and an urban economic, social, and environmental system using Beijing as an example. In addition, this paper provides a theoretical basis and discusses methods for ensuring the long-term and sustainable supply management of urban public infrastructure systems. The research design used in this paper is shown in Figure2. The second part describes the research model used in this study, which involves AHP, ANP, the entropy method, a combination weight method based game theory, a coupling coordination degree model, and a dynamic coupling coordination degree model. The research model is used to evaluate the coordination of the urban public infrastructure system and the urban economic, social, and environmental system. The last part of this paper describes the empirical analysis of data on Beijing from 2000 to 2016 using the research model.Figure 2
Research design flow chart.


## 2. Research Materials, Research Methods, and Model Construction

### 2.1. Research Materials

#### 2.1.1. Research Object

Beijing is the capital of China. It is the political, economic, and cultural center of the country and has advanced and comprehensive urban public infrastructure. In addition, data on Beijing’s urban public infrastructure are abundant, typical, and representative. Research on the system integration supply management of Beijing’s urban public infrastructure shows that it can be used as a model for other cities.


#### 2.1.2. Evaluation Index System

To measure the relationship between the urban infrastructure system and the urban economic, social, and environmental system, this paper initially develops an index framework based on relevant references [26–30]. Then, according to the specific situation of Beijing and the principles of appropriateness, comparability, and availability, a final evaluation of the index system is conducted. The urban infrastructure system consists of five subsystems, energy, transportation, environment, water resources, postal services and telecommunications, as well as 20 evaluation indicators. The urban economic, social, and environmental system includes three subsystems, the urban economic, social and environmental subsystems, as well as 12 evaluation indicators. The final evaluation indicators are shown in Tables 1 and 2.Table 1
Evaluation indicators of the supply levels of the various subsystems of urban public infrastructure.
Criteria layer
System layer
Indicator layer
Unit
u
IEW
u
ANP
u
cw
∗
Economic benefit
I
1: energy facilities system
I
11: per capita social electricity consumption
Kilowatt-hours/person
0.0307
0.0294
0.0298
I
12: per capita energy consumption
tons of standard coal/person
0.0421
0.0664
0.0593
I
13: natural gas ratio
%
0.0206
0.0157
0.0171
I
14: per capita heating area
m2/person
0.0265
0.0114
0.0158
Social benefit
I
2: road traffic system
I
21: railway and highway facilities density
km/km2
0.0586
0.1575
0.1287
I
22: per capita urban road area
m2/person
0.0277
0.0241
0.0251
I
23: number of public transportation lines
Number
0.0491
0.0442
0.0456
I
24: bus operation passenger volume
10,000 persons
0.0571
0.1011
0.0883
Environment benefit
I
3: environmental protection system
I
31: per capita park green area
m2/person
0.0436
0.0580
0.0538
I
32: sewage treatment capacity
10,000 m3/day
0.0404
0.0302
0.0332
I
33: total number of urban sanitation special vehicle and equipment
unit
0.0335
0.0217
0.0251
I
34: household garbage clearance volume
10,000 tons
0.0519
0.0941
0.0818
I
4: water resources and water supply and drainage system
I
41: per capita annual water supply
m3/person
0.1020
0.0957
0.0975
I
42: density of piped water supply
km/km2
0.0458
0.0590
0.0552
I
43: per capita comprehensive water production capacity
10 thousand m3/day person
0.0400
0.0307
0.0334
I
44: sewage pipe density
km/km2
0.0542
0.0221
0.0314
I
5: postal and telecommunication facilities system
I
51: per capita postal and telecommunications volume
CNY/person
0.0709
0.0864
0.0819
I
52: mobile phone penetration rate
Household/100 people
0.0349
0.0151
0.0209
I
53: mainline penetration rate
%
0.0466
0.0088
0.0198
I
54: bureau number of post and telecommunications
Unit
0.1238
0.0028
0.0380
Note. The relevant data are from the Beijing Statistical Yearbook 2000–2016.Table 2
Urban economic social and environmental system evaluation indicators.
Dimension layer
Indicator layer
Unit
u
IEW
u
AHP
u
cw
∗
S
1: economic aspect
S
11: the actual value of gross domestic product
100 million CNY
0.0728
0.3365
0.2560
S
12: the proportion of the tertiary industry
%
0.0646
0.0586
0.0604
S
13: actual utilization of foreign direct investment
10,000 USD
0.1349
0.0342
0.0649
S
14: fixed assets investment ratio
%
0.0634
0.1103
0.0960
S
2: social aspect
S
21: per capita household disposable income
CNY
0.1194
0.0580
0.0767
S
22: per capita consumption expenditure
CNY
0.1197
0.0941
0.1019
S
23: number of employees in three industries
Person
0.0757
0.0302
0.0441
S
24: urbanization rate
%
0.0725
0.0217
0.0372
S
3: environmental aspects
S
31: annual average of inhalable particles
mg/m3
0.0670
0.0078
0.0259
S
32: total wastewater discharge
10 thousand tons
0.1234
0.0766
0.0909
S
33: general industrial solid waste production
10 thousand tons
0.0375
0.0251
0.0289
S
34: sulfur dioxide emissions
10 thousand tons
0.0492
0.0133
0.0243
Note. The relevant data are from the Beijing Statistical Yearbook 2000–2016.


### 2.2. Research Method

#### 2.2.1. Preliminary Processing of Data

The relevant data on the supply level of various subsystems of Beijing’s infrastructure and on the development level of Beijing’s urban economic, social, and environmental system (2000–2016) were acquired from the Beijing Statistical Yearbook (Beijing Bureau of Statistics 2000–2016) through simple processing. All per capita indicators (includingI11, I12, I14, I22, I41, I43, and I51) represent the corresponding statistical data divided by the number of permanent residents in that year. All density indicators (including I21, I42, and I44) represent the corresponding statistical data divided by the area of Beijing. The indicator I21 is the sum of railway mileage and highway mileage divided by the area of Beijing. The data are standardized using formulas (1) and (2) (as shown in Tables 3 and 4).Positive indicator:(1)Yij=Xij−minXjmaxXj−minXj.Negative indicator:(2)Yij=maxXj−XijmaxXj−minXj,where Xij represents the observed value of the jth index in the ith year; maxXj represents the maximum observation; minXj represents the minimum observation; and Yij is a standardized value.Table 3
The standardized data of urban public infrastructure system.
year
Per capita social electricity consumptionI11
Per capita energy consumptionI12
Natural gas ratioI13
Per capita heating areaI14
Railway and highway facilities densityI21
Per capita urban road areaI22
Number of public transportation linesI23
Bus operation passenger trafficI24
Per capita park green areaI31
Sewage treatment capacityI32
Total number of urban vehicle sanitation special equipmentI33
Domestic garbage removal volumeI34
Per capita annual water supplyI41
Tap water supply pipe densityI42
Per capita tap water comprehensive production capacityI43
Sewage pipe densityI44
Per capita postal and telecommunications volumeI51
Mobile phone penetration rateI52
Fixed line mainline penetration rateI53
Bureau number of post and telecommunicationsI54
2000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.3735
0.0000
0.0000
0.0000
0.0000
0.0000
1.0000
0.0000
0.6906
0.0000
0.0000
0.0000
0.0375
0.0000
2001
0.0364
0.0355
0.4163
0.1323
0.0329
0.4355
0.0000
0.1052
0.0637
0.0310
0.1859
0.0238
0.8757
0.0573
0.6780
0.0515
0.0002
0.1214
0.2048
0.0047
2002
0.2486
0.1929
0.4871
0.2382
0.0878
1.0000
0.1088
0.2088
0.1398
0.1078
0.2048
0.0447
0.5718
0.1010
1.0000
0.1335
0.0405
0.2386
0.3106
0.0177
2003
0.3754
0.3775
0.4421
0.4598
0.0961
0.0347
0.1912
0.0487
0.2748
0.1789
0.2572
0.1140
0.5935
0.1783
0.9396
0.1741
0.0989
0.3087
0.5085
0.0250
2004
0.5346
1.0000
0.4167
0.5401
0.1169
0.3915
0.1941
0.2276
0.2780
0.2616
0.3969
0.1911
0.4806
0.2535
0.6726
0.1751
0.1406
0.3923
0.8464
0.0213
2005
0.6168
0.6046
0.5918
0.6281
0.1248
0.6958
0.2088
0.2715
0.3634
0.4043
0.3861
0.2756
0.4204
0.2375
0.2708
0.1108
0.2142
0.4234
1.0000
0.0273
2006
0.7935
0.8251
0.6777
0.6882
0.8050
0.5245
0.2029
0.1504
0.3634
0.4188
0.4271
0.4224
0.3393
0.4586
0.3365
0.2561
0.3042
0.4430
0.8362
0.0278
2007
0.8480
0.9660
0.8351
0.7056
0.8344
0.5360
0.2765
0.1991
0.4565
0.4540
0.4337
0.5292
0.2861
0.5905
0.3396
0.4149
0.4704
0.4265
0.7713
0.1077
2008
0.9348
0.5646
0.9242
0.7949
0.7852
0.8167
0.3647
0.4542
0.6118
0.4155
0.5139
0.6257
0.2100
0.6958
0.2889
0.4317
0.5685
0.4015
0.6143
0.1062
2009
0.6154
0.4736
0.9181
0.7843
0.8338
0.7522
0.4294
0.6161
0.7516
0.4705
0.5081
0.6248
0.1509
0.7678
0.2873
0.4378
0.6478
0.4430
0.5461
0.1202
2010
0.6977
0.5010
0.9482
0.7856
0.8759
0.6693
0.5059
0.6919
0.8292
0.4891
0.5768
0.5847
0.0588
0.9124
0.2762
0.4351
0.7866
0.5070
0.4471
0.1949
2011
0.6670
0.3221
0.9069
0.8526
0.9162
0.5281
0.6147
0.7720
0.8758
0.4982
0.6396
0.5871
0.0501
1.0000
0.3496
0.4825
0.1624
0.6229
0.4027
0.2003
2012
0.7493
0.3294
0.9535
0.8643
0.9388
0.4847
0.6000
0.8674
0.9068
0.5377
0.8046
0.6113
0.0110
0.6863
0.0000
0.6432
0.2056
0.7785
0.3652
0.3074
2013
0.7987
0.3475
0.9351
0.8849
0.9602
0.5308
0.8088
0.9729
0.9379
0.5471
0.8536
0.6518
0.0000
0.7361
0.1110
0.7472
0.2914
0.8176
0.3072
0.4034
2014
0.8186
0.3363
0.9293
0.9136
0.9817
0.5898
1.0000
1.0000
0.9689
0.6132
0.9078
0.7595
0.0175
0.7895
0.3443
0.7759
0.3696
1.0000
0.2253
0.4405
2015
0.8369
0.2923
0.9769
0.9405
0.9859
0.5739
0.9971
0.8107
0.9845
0.6432
0.9661
0.8574
0.0312
0.8351
0.3379
0.8787
0.5771
0.9835
0.1433
0.6948
2016
1.0000
0.4077
1.0000
1.0000
1.0000
0.6353
1.0000
0.8023
1.0000
1.0000
1.0000
1.0000
0.0519
0.8695
0.3264
1.0000
1.0000
0.9304
0.0000
0.8269Table 4
The standardized data of urban economic social and environmental system.
Year
Gross regional product actual valueS11
The proportion of the tertiary industriesS12
Actual use of foreign investment amountS13
Fixed assets investment ratioS14
Per capita household disposable incomeS21
Per capita consumption expenditureS22
Number of employees in three industriesS23
Urbanization rateS24
Annual average inhalable particlesS31
Total wastewater dischargeS32
General industrial solid waste production volumeS33
Sulfur dioxide emissionsS34
2000
0.0000
0.2210
0.0000
0.7978
0.0000
0.0000
0.0000
0.0000
0.9459
0.0000
0.7017
1.0000
2001
0.0674
0.2918
0.0035
0.8216
0.0262
0.0144
0.0160
0.0574
0.9865
0.0044
0.6976
0.8959
2002
0.1294
0.3938
0.0195
0.8978
0.0451
0.0602
0.0997
0.1127
1.0000
0.0532
0.5834
0.8572
2003
0.1952
0.1303
0.0343
1.0000
0.0753
0.0884
0.1398
0.1680
0.6622
0.0555
0.7664
0.8161
2004
0.2817
0.0000
0.1175
0.8840
0.1127
0.1245
0.3908
0.2219
0.7703
0.1116
0.9274
0.0000
2005
0.3315
0.3173
0.1567
0.7514
0.1556
0.1596
0.4306
0.6778
0.6757
0.1498
0.8256
0.8511
2006
0.3833
0.4023
0.2482
0.8470
0.2052
0.2127
0.5000
0.7574
0.9324
0.2016
1.0000
0.7837
2007
0.4862
0.3598
0.2934
0.7235
0.2480
0.2297
0.5383
0.7757
0.7568
0.2383
0.8886
0.6771
2008
0.5498
1.0000
0.2478
0.1641
0.3063
0.2677
0.6019
0.8205
0.4054
0.3091
0.7260
0.5501
2009
0.5350
0.3088
0.2934
0.6932
0.3492
0.3158
0.6308
0.8322
0.3919
0.6672
0.8442
0.5303
2010
0.6347
0.0992
0.3835
0.5891
0.3990
0.3844
0.6863
0.9384
0.3919
0.6100
0.8805
0.5136
2011
0.7641
0.5099
0.4086
0.3365
0.4806
0.4533
0.7497
0.9691
0.2973
0.7277
0.6832
0.4370
2012
0.8096
0.4136
0.4698
0.3146
0.5566
0.5226
0.8123
0.9652
0.2297
0.6602
0.6536
0.4190
2013
0.8707
0.3739
0.5575
0.2521
0.6387
0.5974
0.8683
0.9767
0.2162
0.7161
0.5711
0.3884
2014
0.8795
0.4561
0.6461
0.2460
0.7152
0.6557
0.8945
0.9878
0.3243
0.7959
0.5390
0.3519
2015
0.9002
0.7932
0.9971
0.1736
0.9059
0.9458
0.9434
1.0000
0.1351
0.8091
0.1111
0.3175
2016
1.0000
0.5694
1.0000
0.0000
1.0000
1.0000
1.0000
0.9991
0.0000
1.0000
0.0000
0.1482


#### 2.2.2. The ANP Is Used to Evaluate the Interdependence of the Subsystems of Urban Public Infrastructure

The ANP is a multiobjective decision-making method developed by Professor SaatyTL and is based on the AHP, which is especially applicable to complex system decision-making problems with internal interdependence and feedback from indicator elements [31]. The ANP method is used to evaluate the decision-making of the interdependence subsystem in the network to obtain weights of subsystem in the paper.


#### 2.2.3. The Weight of the Evaluation Index of Each Subsystem Is Obtained by Using a Combination Weighting Method Based on Game Theory

Game theory is an important part of operations research and is used for studying competitive elements. Drawing on game theory, relevant experts have developed a game theory combination weighting method [32, 33] that uses comprehensive subjective and objective weighting to overcome the limitations of subjective weighting and objective weighting. The paper uses a combined weight method based on game theory to calculate the weight of the evaluation indicators of the urban public infrastructure system and the urban economic social and environmental system.The calculation process of combination weighting method based on game theory is as follows:(1)
Obtainn weights according to the n-type weighting method, and then construct a basic weight vector set U=u1,u2,…,un These n vectors are arbitrarily linearly combined to form a possible set of weights:(3)U=∑k=1nWkukTwk≥0,where u is a possible weight vector of a possible set of weight vectors and Wk is the weight coefficient.We use game theory to find the most satisfactoryu∗ in the possible vector set. The most satisfactory weight vector can be transformed by optimizing the linear combination weight coefficient Wk. The goal of optimization is to minimize the dispersion of u from each uk. That is,(4)min∑j=1nwj×ujT−ujT2,i=1,2,…,n.Based on the differential properties of the matrix, the first derivative condition of the optimization of equation (4) is(5)∑j=1nWj×uj×ujT=ui×ujT,i=1,2,…,n.Equation (5) corresponds to a linear system of equations:(6)u1⋅u1Tu1⋅u2T⋯u1⋅unTu2⋅u1Tu2⋅u2T⋯u2⋅unT⋯⋯⋯⋯un⋅u1Tun⋅u2T⋯un⋅unTw1w2⋯wn=u1⋅u1Tu1⋅u2T⋯un⋅unT.After obtainingw1,w2,…,wn using formula (6), we normalize it:(7)Wk∗=wk∑k=1nwk.Finally, the combined weight is(8)u∗=∑k=1nwkukT.


### 2.3. Research Model Construction

#### 2.3.1. Model Used to Evaluate the Supply Level of the Urban Public Infrastructure System

The model used to evaluate the supply level of the urban public infrastructure system (see Figure3) is divided into three levels. The first level is the rule layer, which includes three aspects: economic growth, social welfare, and environmental pressure on urban public infrastructure systems. The second layer of the model is the network layer. Considering the purpose of the research, five subsystems of the public infrastructure system are included in the model, namely, water resources and water supply and drainage system (I1), transportation system (I2), postal power system (I3), environmental facilities system (I4), and energy facilities system (I5), which are consistent with Table 1. The third layer of the model is the system indicator layer, which consists of the evaluation indicators for the supply level of each subsystem.Figure 3
Model used to evaluate the management of the supply level of the urban public infrastructure system.The main process used to evaluate the supply level of the urban public infrastructure system is described below:Step 1.
According to the urban development orientation, the established stock of the urban public infrastructure and the interdependence of the public infrastructure determine the structural relationship of the urban public infrastructure system and act as the basis for constructing the structure of the urban public infrastructure system ANP management activities. This process needs to be conducted by experts.Step 2.
The ANP super matrix and the weighted super matrix are calculated. The evaluation criteria used for the ANP act as the basis for evaluating the relationships in the system and reflect the overall goal of the evaluation. Let the evaluation criteria in the ANP structure beCi,i=1,2,…,m. The system’s network layer has subsystems Ij,j=1,2,…,n. For evaluating the supply level of urban public infrastructure systems and interdependence, Ci is used as a criterion to judge the interdependence of urban public infrastructure subsystems. On this basis, the judgment matrix is constructed to form the feature vector W1j,W2j,…,Wij. When the feature vector passes the consistency test, it is expressed as a matrix form, and a local weight vector matrix Wij is generated. m supermatrices W are formed under the influence of the control criterion index Ci; however, W is not a normalized matrix. To ensure the calculation results are objective and comparable, the supermatrix columns are normalized. The corresponding weighting factor Yij,i,j=1,2,3,…,n is set, and the superweighted matrix is Wij¯=yijwij.
To accurately reflect the interaction between the subsystems, the stability of the supermatrix must be processed. Stability processing is performed on the superweighting matrixWij¯, generating an ANP limit matrix W∞. The processing method is as follows:(9)W∞=limk=⟶∞1m∑i=1mW¯k.
Equation (9) is convergent and unique, and the value of the corresponding column in the original matrix is the weight of the stability of the urban public infrastructure subsystem.Step 3.
After the subsystem weights are obtained, the weights of the internal indicators of each subsystem are obtained using AHP.Step 4.
The entropy method is used to obtain the evaluation index weight of each subsystem [34, 35].Step 5.
Using a combination weighting method based on game theory, the weights of the evaluation indexes of each subsystem are obtained.Step 6.
The supply level of each subsystem and the overall system are obtained.
The supply level of each subsystem and the system supply level are obtained by using index weights are calculated using a combination weighting method based on game theory.(10)Ssubsystem,i,r=∑j=1nujr∗Yij,where Ssubsystem,i,r represents the supply level of each subsystem; 1, 2, ..., 5 represent 5 different subsystems; i represents the year; and ujr∗ represents the weight of the combination index (which is based on game theory) for each subsystem.
The supply level of the urban public infrastructure system is obtained by using the following formula:(11)Ssystem,i=∑j=1nuj∗Yij,where ujr∗ represents the supply level of the urban public infrastructure system and u∗ represents the weight of the combination index (which is based on game theory) of the urban public infrastructure system.


#### 2.3.2. Coupling Coordination Analysis

(1) Coupling Coordination Model (CCM). Coupling is a term used in physics to indicate the degree to which two or more systems interact and affect each other to reach a degree of synergy. Coupling is used to measure the degree of agreement in terms of the level of system development. Liao was the first to develop a model to evaluate the degree of coupling coordination within a system or between different systems and further differentiated the levels of coupling coordination degree [36]. This paper constructs a CCM of the urban public infrastructure system and the urban economic social and environmental system to analyze the degree of coordinated development between the urban public infrastructure system and the level of urban development. The calculation process of the CCM is as follows:(1)
The evaluation value of the urban public infrastructure system and the urban economic, social, and environmental system is calculated, which are represented byfx and gy, respectively. These are shown in Table 5.(2)
The coupling degree of the urban public infrastructure system and the urban economic social and environmental system is calculated:(12)C=fx∗gyfx+gy/221/2,T=afx+bgy,Dx,y=C⋅T,where D represents the degree of the coupling between the two systems, D∈0,1, C represents the degree of coordination between the two systems, C∈0,1, T represents the comprehensive coordination index of the two systems, and a and b represent the contribution of the two systems. Let fx.Table 5
Evaluation of the supply level of urban infrastructure system and the demand level of the urban economic, social, and environmental system.
Years
Urban infrastructure system supply level (fx)
Urban economic, social, and environmental system evaluation value (gy)
2000
0.1383
0.1590
2001
0.1675
0.1882
2002
0.2247
0.2310
2003
0.2413
0.2474
2004
0.3200
0.2683
2005
0.3204
0.3344
2006
0.4425
0.3979
2007
0.5045
0.4131
2008
0.5320
0.4138
2009
0.5528
0.4670
2010
0.5862
0.4884
2011
0.5519
0.5416
2012
0.5568
0.5556
2013
0.6076
0.5870
2014
0.6606
0.6210
2015
0.6860
0.6924
2016
0.6372
0.7097(2) Dynamic Coupling Coordination Model (DCCM). The DCCM is used to determine the coupling degree of the composite system [37]. Li and Ding was the first to propose a DCCM and used it to evaluate the coordination of resource environment systems and social economic systems and then calculated the coordination degree of the composite system [38].To clarify the relationship between the urban public infrastructure system and the urban economic, social, and environmental system, this paper assumes that both of these systems and their relationship form a composite system. Based on theory of the evolution of subsystems in a general system, we construct a DCCM to analyze the evolution state and coupling state of the composite systems. The changes between the urban public infrastructure system and the urban economic, social, and environmental system are nonlinear, so the evolution equation can be expressed as(13)dxtdt=fx1,x2,…,xn;i=1,2,…,n,where f is a nonlinear function of xi. The motion stability of a nonlinear system depends on the characteristic root property of the first approximation system. The high-order term can guarantee the stability of motion and be used to obtain an approximate linear system.(14)dxtdt=∑i=1naixi;i=1,2,…,n.Using the above method, we establish the general function of the urban public infrastructure system and the urban economic, social, and environmental system.(15)fx=∑i=1naixi∗;i=1,2,…,n,gy=∑i=1nbiyi∗;i=1,2,…,n.We assume that the urban public infrastructure system and the urban economic, social, and environmental system and their relationship are in the same system, which has two elementsFx and Fy. According to Beta Langfi’s general system theory, the evolution equation of a composite system is expressed as(16)Fx=dfxdt=T1fx+T2gy,VFx=dFxdt,(17)Gy=dgxdt=U1fx+U2gy,VGy=dGydt.F
x and Gy are the evolutionary states of the urban public infrastructure system and the urban economic, social, and environmental system under their own interaction and external conditions. VFx and VGy are the respective evolutionary speeds of the urban public infrastructure system and the urban economic, social, and environmental system. In a composite system, because the entire system contains only two elements Fx and Gy, any changes in the subsystem that occur when Fx and Gy interact will cause the entire system to change. The evolution rate V of a composite system can be seen as a function of VFx and VGy, so VGy. Therefore, VFx and VGy are used as control variables; the coupling relationship between the composite system and the two subsystems can be obtained by studying the variation in V. Because the evolution of the entire system satisfies the S-type development mechanism, we assume that the dynamic coupling and coordination relationship between the urban public infrastructure system and the urban economic, social, and environmental system is cyclical. Since a change in V is caused by VFx and VGy, we analyze V on the two-dimensional coordinate plane (VFx, VGy). The change in V is an ellipse in the coordinate system (the change in the urban economic, social, and environmental system is not as rapid as that of the urban public infrastructure system, and the amplitude is relatively small), as shown in Figure 4.Figure 4
Dynamic coupling degree between urban public infrastructure and the urban economic, social, and environmental system (created by author based on Li Chongming’s study [38]).In this case, the variableV is an ellipse. The variable β represents the angle between VFx and VGy. According to the value of β, we can determine the value of the dynamic coupling coordination degree of the composite system. Based on relevant research [38], this paper proposes the development stage and state of the dynamic coupling coordination degree of the composite system, as shown in Table 6.(18)β=arctanVFxVGy.Table 6
Dynamic coupling and coordination of the composite system.
Range ofβ
Stage
System development stage
System status
−
90
∘
≺
β
≤
0
∘
I
Low-level symbiosis
At this stage of development, the development speed of urban public infrastructure systems is very low, and the impact of urban public infrastructure systems on urban economic, social, and environmental systems is almost 0.
0
∘
≺
β
≺
45
∘
II
Primary coordinated development stage
V
F
x
<
V
G
y The development speed of urban public infrastructure system is lower than that of urban economic, social, and environmental system. Urban public infrastructure system can not promote the development of urban economic, social, and environmental system, showing that urban infrastructure system can not carry the development of urban economic, social, and environmental system.
β
=
45
∘
Coordinated development stage
V
F
x
=
V
G
y The speed of urban public infrastructure system is equal to the urban economic, social, and environmental system, and the urban public infrastructure system and the urban economic, social, and environmental system are coordinated.
45
∘
≺
β
≤
90
∘
Coordinated development stage
V
F
x
>
V
G
y With the development of the urban public infrastructure system, the two systems began to interact with each other. The restrictions of the urban economic, social, and environmental system on the urban public infrastructure system began to appear, but it was not obvious.
90
∘
≺
β
≤
180
∘
III
Extreme development stage
With the accelerated development of urban economic, social, and environmental systems, urban public infrastructure systems have increased the demand for urban development, and the contradiction between infrastructure systems and urban systems has emerged, and began to limit the improvement of urban development level.
−
180
∘
≺
β
≤
−
90
∘
IV
High-level coordinated development stage
The composite system gradually transforms into a common development stage, and finally reaches the high-level coordinated development of the urban public infrastructure system and the urban economic, social, and environmental system
The authors edited the table by themselves according to Li Chongming’s literature [38].


## 2.1. Research Materials

### 2.1.1. Research Object

Beijing is the capital of China. It is the political, economic, and cultural center of the country and has advanced and comprehensive urban public infrastructure. In addition, data on Beijing’s urban public infrastructure are abundant, typical, and representative. Research on the system integration supply management of Beijing’s urban public infrastructure shows that it can be used as a model for other cities.


### 2.1.2. Evaluation Index System

To measure the relationship between the urban infrastructure system and the urban economic, social, and environmental system, this paper initially develops an index framework based on relevant references [26–30]. Then, according to the specific situation of Beijing and the principles of appropriateness, comparability, and availability, a final evaluation of the index system is conducted. The urban infrastructure system consists of five subsystems, energy, transportation, environment, water resources, postal services and telecommunications, as well as 20 evaluation indicators. The urban economic, social, and environmental system includes three subsystems, the urban economic, social and environmental subsystems, as well as 12 evaluation indicators. The final evaluation indicators are shown in Tables 1 and 2.Table 1
Evaluation indicators of the supply levels of the various subsystems of urban public infrastructure.
Criteria layer
System layer
Indicator layer
Unit
u
IEW
u
ANP
u
cw
∗
Economic benefit
I
1: energy facilities system
I
11: per capita social electricity consumption
Kilowatt-hours/person
0.0307
0.0294
0.0298
I
12: per capita energy consumption
tons of standard coal/person
0.0421
0.0664
0.0593
I
13: natural gas ratio
%
0.0206
0.0157
0.0171
I
14: per capita heating area
m2/person
0.0265
0.0114
0.0158
Social benefit
I
2: road traffic system
I
21: railway and highway facilities density
km/km2
0.0586
0.1575
0.1287
I
22: per capita urban road area
m2/person
0.0277
0.0241
0.0251
I
23: number of public transportation lines
Number
0.0491
0.0442
0.0456
I
24: bus operation passenger volume
10,000 persons
0.0571
0.1011
0.0883
Environment benefit
I
3: environmental protection system
I
31: per capita park green area
m2/person
0.0436
0.0580
0.0538
I
32: sewage treatment capacity
10,000 m3/day
0.0404
0.0302
0.0332
I
33: total number of urban sanitation special vehicle and equipment
unit
0.0335
0.0217
0.0251
I
34: household garbage clearance volume
10,000 tons
0.0519
0.0941
0.0818
I
4: water resources and water supply and drainage system
I
41: per capita annual water supply
m3/person
0.1020
0.0957
0.0975
I
42: density of piped water supply
km/km2
0.0458
0.0590
0.0552
I
43: per capita comprehensive water production capacity
10 thousand m3/day person
0.0400
0.0307
0.0334
I
44: sewage pipe density
km/km2
0.0542
0.0221
0.0314
I
5: postal and telecommunication facilities system
I
51: per capita postal and telecommunications volume
CNY/person
0.0709
0.0864
0.0819
I
52: mobile phone penetration rate
Household/100 people
0.0349
0.0151
0.0209
I
53: mainline penetration rate
%
0.0466
0.0088
0.0198
I
54: bureau number of post and telecommunications
Unit
0.1238
0.0028
0.0380
Note. The relevant data are from the Beijing Statistical Yearbook 2000–2016.Table 2
Urban economic social and environmental system evaluation indicators.
Dimension layer
Indicator layer
Unit
u
IEW
u
AHP
u
cw
∗
S
1: economic aspect
S
11: the actual value of gross domestic product
100 million CNY
0.0728
0.3365
0.2560
S
12: the proportion of the tertiary industry
%
0.0646
0.0586
0.0604
S
13: actual utilization of foreign direct investment
10,000 USD
0.1349
0.0342
0.0649
S
14: fixed assets investment ratio
%
0.0634
0.1103
0.0960
S
2: social aspect
S
21: per capita household disposable income
CNY
0.1194
0.0580
0.0767
S
22: per capita consumption expenditure
CNY
0.1197
0.0941
0.1019
S
23: number of employees in three industries
Person
0.0757
0.0302
0.0441
S
24: urbanization rate
%
0.0725
0.0217
0.0372
S
3: environmental aspects
S
31: annual average of inhalable particles
mg/m3
0.0670
0.0078
0.0259
S
32: total wastewater discharge
10 thousand tons
0.1234
0.0766
0.0909
S
33: general industrial solid waste production
10 thousand tons
0.0375
0.0251
0.0289
S
34: sulfur dioxide emissions
10 thousand tons
0.0492
0.0133
0.0243
Note. The relevant data are from the Beijing Statistical Yearbook 2000–2016.


## 2.1.1. Research Object

Beijing is the capital of China. It is the political, economic, and cultural center of the country and has advanced and comprehensive urban public infrastructure. In addition, data on Beijing’s urban public infrastructure are abundant, typical, and representative. Research on the system integration supply management of Beijing’s urban public infrastructure shows that it can be used as a model for other cities.


## 2.1.2. Evaluation Index System

To measure the relationship between the urban infrastructure system and the urban economic, social, and environmental system, this paper initially develops an index framework based on relevant references [26–30]. Then, according to the specific situation of Beijing and the principles of appropriateness, comparability, and availability, a final evaluation of the index system is conducted. The urban infrastructure system consists of five subsystems, energy, transportation, environment, water resources, postal services and telecommunications, as well as 20 evaluation indicators. The urban economic, social, and environmental system includes three subsystems, the urban economic, social and environmental subsystems, as well as 12 evaluation indicators. The final evaluation indicators are shown in Tables 1 and 2.Table 1
Evaluation indicators of the supply levels of the various subsystems of urban public infrastructure.
Criteria layer
System layer
Indicator layer
Unit
u
IEW
u
ANP
u
cw
∗
Economic benefit
I
1: energy facilities system
I
11: per capita social electricity consumption
Kilowatt-hours/person
0.0307
0.0294
0.0298
I
12: per capita energy consumption
tons of standard coal/person
0.0421
0.0664
0.0593
I
13: natural gas ratio
%
0.0206
0.0157
0.0171
I
14: per capita heating area
m2/person
0.0265
0.0114
0.0158
Social benefit
I
2: road traffic system
I
21: railway and highway facilities density
km/km2
0.0586
0.1575
0.1287
I
22: per capita urban road area
m2/person
0.0277
0.0241
0.0251
I
23: number of public transportation lines
Number
0.0491
0.0442
0.0456
I
24: bus operation passenger volume
10,000 persons
0.0571
0.1011
0.0883
Environment benefit
I
3: environmental protection system
I
31: per capita park green area
m2/person
0.0436
0.0580
0.0538
I
32: sewage treatment capacity
10,000 m3/day
0.0404
0.0302
0.0332
I
33: total number of urban sanitation special vehicle and equipment
unit
0.0335
0.0217
0.0251
I
34: household garbage clearance volume
10,000 tons
0.0519
0.0941
0.0818
I
4: water resources and water supply and drainage system
I
41: per capita annual water supply
m3/person
0.1020
0.0957
0.0975
I
42: density of piped water supply
km/km2
0.0458
0.0590
0.0552
I
43: per capita comprehensive water production capacity
10 thousand m3/day person
0.0400
0.0307
0.0334
I
44: sewage pipe density
km/km2
0.0542
0.0221
0.0314
I
5: postal and telecommunication facilities system
I
51: per capita postal and telecommunications volume
CNY/person
0.0709
0.0864
0.0819
I
52: mobile phone penetration rate
Household/100 people
0.0349
0.0151
0.0209
I
53: mainline penetration rate
%
0.0466
0.0088
0.0198
I
54: bureau number of post and telecommunications
Unit
0.1238
0.0028
0.0380
Note. The relevant data are from the Beijing Statistical Yearbook 2000–2016.Table 2
Urban economic social and environmental system evaluation indicators.
Dimension layer
Indicator layer
Unit
u
IEW
u
AHP
u
cw
∗
S
1: economic aspect
S
11: the actual value of gross domestic product
100 million CNY
0.0728
0.3365
0.2560
S
12: the proportion of the tertiary industry
%
0.0646
0.0586
0.0604
S
13: actual utilization of foreign direct investment
10,000 USD
0.1349
0.0342
0.0649
S
14: fixed assets investment ratio
%
0.0634
0.1103
0.0960
S
2: social aspect
S
21: per capita household disposable income
CNY
0.1194
0.0580
0.0767
S
22: per capita consumption expenditure
CNY
0.1197
0.0941
0.1019
S
23: number of employees in three industries
Person
0.0757
0.0302
0.0441
S
24: urbanization rate
%
0.0725
0.0217
0.0372
S
3: environmental aspects
S
31: annual average of inhalable particles
mg/m3
0.0670
0.0078
0.0259
S
32: total wastewater discharge
10 thousand tons
0.1234
0.0766
0.0909
S
33: general industrial solid waste production
10 thousand tons
0.0375
0.0251
0.0289
S
34: sulfur dioxide emissions
10 thousand tons
0.0492
0.0133
0.0243
Note. The relevant data are from the Beijing Statistical Yearbook 2000–2016.


## 2.2. Research Method

### 2.2.1. Preliminary Processing of Data

The relevant data on the supply level of various subsystems of Beijing’s infrastructure and on the development level of Beijing’s urban economic, social, and environmental system (2000–2016) were acquired from the Beijing Statistical Yearbook (Beijing Bureau of Statistics 2000–2016) through simple processing. All per capita indicators (includingI11, I12, I14, I22, I41, I43, and I51) represent the corresponding statistical data divided by the number of permanent residents in that year. All density indicators (including I21, I42, and I44) represent the corresponding statistical data divided by the area of Beijing. The indicator I21 is the sum of railway mileage and highway mileage divided by the area of Beijing. The data are standardized using formulas (1) and (2) (as shown in Tables 3 and 4).Positive indicator:(1)Yij=Xij−minXjmaxXj−minXj.Negative indicator:(2)Yij=maxXj−XijmaxXj−minXj,where Xij represents the observed value of the jth index in the ith year; maxXj represents the maximum observation; minXj represents the minimum observation; and Yij is a standardized value.Table 3
The standardized data of urban public infrastructure system.
year
Per capita social electricity consumptionI11
Per capita energy consumptionI12
Natural gas ratioI13
Per capita heating areaI14
Railway and highway facilities densityI21
Per capita urban road areaI22
Number of public transportation linesI23
Bus operation passenger trafficI24
Per capita park green areaI31
Sewage treatment capacityI32
Total number of urban vehicle sanitation special equipmentI33
Domestic garbage removal volumeI34
Per capita annual water supplyI41
Tap water supply pipe densityI42
Per capita tap water comprehensive production capacityI43
Sewage pipe densityI44
Per capita postal and telecommunications volumeI51
Mobile phone penetration rateI52
Fixed line mainline penetration rateI53
Bureau number of post and telecommunicationsI54
2000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.3735
0.0000
0.0000
0.0000
0.0000
0.0000
1.0000
0.0000
0.6906
0.0000
0.0000
0.0000
0.0375
0.0000
2001
0.0364
0.0355
0.4163
0.1323
0.0329
0.4355
0.0000
0.1052
0.0637
0.0310
0.1859
0.0238
0.8757
0.0573
0.6780
0.0515
0.0002
0.1214
0.2048
0.0047
2002
0.2486
0.1929
0.4871
0.2382
0.0878
1.0000
0.1088
0.2088
0.1398
0.1078
0.2048
0.0447
0.5718
0.1010
1.0000
0.1335
0.0405
0.2386
0.3106
0.0177
2003
0.3754
0.3775
0.4421
0.4598
0.0961
0.0347
0.1912
0.0487
0.2748
0.1789
0.2572
0.1140
0.5935
0.1783
0.9396
0.1741
0.0989
0.3087
0.5085
0.0250
2004
0.5346
1.0000
0.4167
0.5401
0.1169
0.3915
0.1941
0.2276
0.2780
0.2616
0.3969
0.1911
0.4806
0.2535
0.6726
0.1751
0.1406
0.3923
0.8464
0.0213
2005
0.6168
0.6046
0.5918
0.6281
0.1248
0.6958
0.2088
0.2715
0.3634
0.4043
0.3861
0.2756
0.4204
0.2375
0.2708
0.1108
0.2142
0.4234
1.0000
0.0273
2006
0.7935
0.8251
0.6777
0.6882
0.8050
0.5245
0.2029
0.1504
0.3634
0.4188
0.4271
0.4224
0.3393
0.4586
0.3365
0.2561
0.3042
0.4430
0.8362
0.0278
2007
0.8480
0.9660
0.8351
0.7056
0.8344
0.5360
0.2765
0.1991
0.4565
0.4540
0.4337
0.5292
0.2861
0.5905
0.3396
0.4149
0.4704
0.4265
0.7713
0.1077
2008
0.9348
0.5646
0.9242
0.7949
0.7852
0.8167
0.3647
0.4542
0.6118
0.4155
0.5139
0.6257
0.2100
0.6958
0.2889
0.4317
0.5685
0.4015
0.6143
0.1062
2009
0.6154
0.4736
0.9181
0.7843
0.8338
0.7522
0.4294
0.6161
0.7516
0.4705
0.5081
0.6248
0.1509
0.7678
0.2873
0.4378
0.6478
0.4430
0.5461
0.1202
2010
0.6977
0.5010
0.9482
0.7856
0.8759
0.6693
0.5059
0.6919
0.8292
0.4891
0.5768
0.5847
0.0588
0.9124
0.2762
0.4351
0.7866
0.5070
0.4471
0.1949
2011
0.6670
0.3221
0.9069
0.8526
0.9162
0.5281
0.6147
0.7720
0.8758
0.4982
0.6396
0.5871
0.0501
1.0000
0.3496
0.4825
0.1624
0.6229
0.4027
0.2003
2012
0.7493
0.3294
0.9535
0.8643
0.9388
0.4847
0.6000
0.8674
0.9068
0.5377
0.8046
0.6113
0.0110
0.6863
0.0000
0.6432
0.2056
0.7785
0.3652
0.3074
2013
0.7987
0.3475
0.9351
0.8849
0.9602
0.5308
0.8088
0.9729
0.9379
0.5471
0.8536
0.6518
0.0000
0.7361
0.1110
0.7472
0.2914
0.8176
0.3072
0.4034
2014
0.8186
0.3363
0.9293
0.9136
0.9817
0.5898
1.0000
1.0000
0.9689
0.6132
0.9078
0.7595
0.0175
0.7895
0.3443
0.7759
0.3696
1.0000
0.2253
0.4405
2015
0.8369
0.2923
0.9769
0.9405
0.9859
0.5739
0.9971
0.8107
0.9845
0.6432
0.9661
0.8574
0.0312
0.8351
0.3379
0.8787
0.5771
0.9835
0.1433
0.6948
2016
1.0000
0.4077
1.0000
1.0000
1.0000
0.6353
1.0000
0.8023
1.0000
1.0000
1.0000
1.0000
0.0519
0.8695
0.3264
1.0000
1.0000
0.9304
0.0000
0.8269Table 4
The standardized data of urban economic social and environmental system.
Year
Gross regional product actual valueS11
The proportion of the tertiary industriesS12
Actual use of foreign investment amountS13
Fixed assets investment ratioS14
Per capita household disposable incomeS21
Per capita consumption expenditureS22
Number of employees in three industriesS23
Urbanization rateS24
Annual average inhalable particlesS31
Total wastewater dischargeS32
General industrial solid waste production volumeS33
Sulfur dioxide emissionsS34
2000
0.0000
0.2210
0.0000
0.7978
0.0000
0.0000
0.0000
0.0000
0.9459
0.0000
0.7017
1.0000
2001
0.0674
0.2918
0.0035
0.8216
0.0262
0.0144
0.0160
0.0574
0.9865
0.0044
0.6976
0.8959
2002
0.1294
0.3938
0.0195
0.8978
0.0451
0.0602
0.0997
0.1127
1.0000
0.0532
0.5834
0.8572
2003
0.1952
0.1303
0.0343
1.0000
0.0753
0.0884
0.1398
0.1680
0.6622
0.0555
0.7664
0.8161
2004
0.2817
0.0000
0.1175
0.8840
0.1127
0.1245
0.3908
0.2219
0.7703
0.1116
0.9274
0.0000
2005
0.3315
0.3173
0.1567
0.7514
0.1556
0.1596
0.4306
0.6778
0.6757
0.1498
0.8256
0.8511
2006
0.3833
0.4023
0.2482
0.8470
0.2052
0.2127
0.5000
0.7574
0.9324
0.2016
1.0000
0.7837
2007
0.4862
0.3598
0.2934
0.7235
0.2480
0.2297
0.5383
0.7757
0.7568
0.2383
0.8886
0.6771
2008
0.5498
1.0000
0.2478
0.1641
0.3063
0.2677
0.6019
0.8205
0.4054
0.3091
0.7260
0.5501
2009
0.5350
0.3088
0.2934
0.6932
0.3492
0.3158
0.6308
0.8322
0.3919
0.6672
0.8442
0.5303
2010
0.6347
0.0992
0.3835
0.5891
0.3990
0.3844
0.6863
0.9384
0.3919
0.6100
0.8805
0.5136
2011
0.7641
0.5099
0.4086
0.3365
0.4806
0.4533
0.7497
0.9691
0.2973
0.7277
0.6832
0.4370
2012
0.8096
0.4136
0.4698
0.3146
0.5566
0.5226
0.8123
0.9652
0.2297
0.6602
0.6536
0.4190
2013
0.8707
0.3739
0.5575
0.2521
0.6387
0.5974
0.8683
0.9767
0.2162
0.7161
0.5711
0.3884
2014
0.8795
0.4561
0.6461
0.2460
0.7152
0.6557
0.8945
0.9878
0.3243
0.7959
0.5390
0.3519
2015
0.9002
0.7932
0.9971
0.1736
0.9059
0.9458
0.9434
1.0000
0.1351
0.8091
0.1111
0.3175
2016
1.0000
0.5694
1.0000
0.0000
1.0000
1.0000
1.0000
0.9991
0.0000
1.0000
0.0000
0.1482


### 2.2.2. The ANP Is Used to Evaluate the Interdependence of the Subsystems of Urban Public Infrastructure

The ANP is a multiobjective decision-making method developed by Professor SaatyTL and is based on the AHP, which is especially applicable to complex system decision-making problems with internal interdependence and feedback from indicator elements [31]. The ANP method is used to evaluate the decision-making of the interdependence subsystem in the network to obtain weights of subsystem in the paper.


### 2.2.3. The Weight of the Evaluation Index of Each Subsystem Is Obtained by Using a Combination Weighting Method Based on Game Theory

Game theory is an important part of operations research and is used for studying competitive elements. Drawing on game theory, relevant experts have developed a game theory combination weighting method [32, 33] that uses comprehensive subjective and objective weighting to overcome the limitations of subjective weighting and objective weighting. The paper uses a combined weight method based on game theory to calculate the weight of the evaluation indicators of the urban public infrastructure system and the urban economic social and environmental system.The calculation process of combination weighting method based on game theory is as follows:(1)
Obtainn weights according to the n-type weighting method, and then construct a basic weight vector set U=u1,u2,…,un These n vectors are arbitrarily linearly combined to form a possible set of weights:(3)U=∑k=1nWkukTwk≥0,where u is a possible weight vector of a possible set of weight vectors and Wk is the weight coefficient.We use game theory to find the most satisfactoryu∗ in the possible vector set. The most satisfactory weight vector can be transformed by optimizing the linear combination weight coefficient Wk. The goal of optimization is to minimize the dispersion of u from each uk. That is,(4)min∑j=1nwj×ujT−ujT2,i=1,2,…,n.Based on the differential properties of the matrix, the first derivative condition of the optimization of equation (4) is(5)∑j=1nWj×uj×ujT=ui×ujT,i=1,2,…,n.Equation (5) corresponds to a linear system of equations:(6)u1⋅u1Tu1⋅u2T⋯u1⋅unTu2⋅u1Tu2⋅u2T⋯u2⋅unT⋯⋯⋯⋯un⋅u1Tun⋅u2T⋯un⋅unTw1w2⋯wn=u1⋅u1Tu1⋅u2T⋯un⋅unT.After obtainingw1,w2,…,wn using formula (6), we normalize it:(7)Wk∗=wk∑k=1nwk.Finally, the combined weight is(8)u∗=∑k=1nwkukT.


## 2.2.1. Preliminary Processing of Data

The relevant data on the supply level of various subsystems of Beijing’s infrastructure and on the development level of Beijing’s urban economic, social, and environmental system (2000–2016) were acquired from the Beijing Statistical Yearbook (Beijing Bureau of Statistics 2000–2016) through simple processing. All per capita indicators (includingI11, I12, I14, I22, I41, I43, and I51) represent the corresponding statistical data divided by the number of permanent residents in that year. All density indicators (including I21, I42, and I44) represent the corresponding statistical data divided by the area of Beijing. The indicator I21 is the sum of railway mileage and highway mileage divided by the area of Beijing. The data are standardized using formulas (1) and (2) (as shown in Tables 3 and 4).Positive indicator:(1)Yij=Xij−minXjmaxXj−minXj.Negative indicator:(2)Yij=maxXj−XijmaxXj−minXj,where Xij represents the observed value of the jth index in the ith year; maxXj represents the maximum observation; minXj represents the minimum observation; and Yij is a standardized value.Table 3
The standardized data of urban public infrastructure system.
year
Per capita social electricity consumptionI11
Per capita energy consumptionI12
Natural gas ratioI13
Per capita heating areaI14
Railway and highway facilities densityI21
Per capita urban road areaI22
Number of public transportation linesI23
Bus operation passenger trafficI24
Per capita park green areaI31
Sewage treatment capacityI32
Total number of urban vehicle sanitation special equipmentI33
Domestic garbage removal volumeI34
Per capita annual water supplyI41
Tap water supply pipe densityI42
Per capita tap water comprehensive production capacityI43
Sewage pipe densityI44
Per capita postal and telecommunications volumeI51
Mobile phone penetration rateI52
Fixed line mainline penetration rateI53
Bureau number of post and telecommunicationsI54
2000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.3735
0.0000
0.0000
0.0000
0.0000
0.0000
1.0000
0.0000
0.6906
0.0000
0.0000
0.0000
0.0375
0.0000
2001
0.0364
0.0355
0.4163
0.1323
0.0329
0.4355
0.0000
0.1052
0.0637
0.0310
0.1859
0.0238
0.8757
0.0573
0.6780
0.0515
0.0002
0.1214
0.2048
0.0047
2002
0.2486
0.1929
0.4871
0.2382
0.0878
1.0000
0.1088
0.2088
0.1398
0.1078
0.2048
0.0447
0.5718
0.1010
1.0000
0.1335
0.0405
0.2386
0.3106
0.0177
2003
0.3754
0.3775
0.4421
0.4598
0.0961
0.0347
0.1912
0.0487
0.2748
0.1789
0.2572
0.1140
0.5935
0.1783
0.9396
0.1741
0.0989
0.3087
0.5085
0.0250
2004
0.5346
1.0000
0.4167
0.5401
0.1169
0.3915
0.1941
0.2276
0.2780
0.2616
0.3969
0.1911
0.4806
0.2535
0.6726
0.1751
0.1406
0.3923
0.8464
0.0213
2005
0.6168
0.6046
0.5918
0.6281
0.1248
0.6958
0.2088
0.2715
0.3634
0.4043
0.3861
0.2756
0.4204
0.2375
0.2708
0.1108
0.2142
0.4234
1.0000
0.0273
2006
0.7935
0.8251
0.6777
0.6882
0.8050
0.5245
0.2029
0.1504
0.3634
0.4188
0.4271
0.4224
0.3393
0.4586
0.3365
0.2561
0.3042
0.4430
0.8362
0.0278
2007
0.8480
0.9660
0.8351
0.7056
0.8344
0.5360
0.2765
0.1991
0.4565
0.4540
0.4337
0.5292
0.2861
0.5905
0.3396
0.4149
0.4704
0.4265
0.7713
0.1077
2008
0.9348
0.5646
0.9242
0.7949
0.7852
0.8167
0.3647
0.4542
0.6118
0.4155
0.5139
0.6257
0.2100
0.6958
0.2889
0.4317
0.5685
0.4015
0.6143
0.1062
2009
0.6154
0.4736
0.9181
0.7843
0.8338
0.7522
0.4294
0.6161
0.7516
0.4705
0.5081
0.6248
0.1509
0.7678
0.2873
0.4378
0.6478
0.4430
0.5461
0.1202
2010
0.6977
0.5010
0.9482
0.7856
0.8759
0.6693
0.5059
0.6919
0.8292
0.4891
0.5768
0.5847
0.0588
0.9124
0.2762
0.4351
0.7866
0.5070
0.4471
0.1949
2011
0.6670
0.3221
0.9069
0.8526
0.9162
0.5281
0.6147
0.7720
0.8758
0.4982
0.6396
0.5871
0.0501
1.0000
0.3496
0.4825
0.1624
0.6229
0.4027
0.2003
2012
0.7493
0.3294
0.9535
0.8643
0.9388
0.4847
0.6000
0.8674
0.9068
0.5377
0.8046
0.6113
0.0110
0.6863
0.0000
0.6432
0.2056
0.7785
0.3652
0.3074
2013
0.7987
0.3475
0.9351
0.8849
0.9602
0.5308
0.8088
0.9729
0.9379
0.5471
0.8536
0.6518
0.0000
0.7361
0.1110
0.7472
0.2914
0.8176
0.3072
0.4034
2014
0.8186
0.3363
0.9293
0.9136
0.9817
0.5898
1.0000
1.0000
0.9689
0.6132
0.9078
0.7595
0.0175
0.7895
0.3443
0.7759
0.3696
1.0000
0.2253
0.4405
2015
0.8369
0.2923
0.9769
0.9405
0.9859
0.5739
0.9971
0.8107
0.9845
0.6432
0.9661
0.8574
0.0312
0.8351
0.3379
0.8787
0.5771
0.9835
0.1433
0.6948
2016
1.0000
0.4077
1.0000
1.0000
1.0000
0.6353
1.0000
0.8023
1.0000
1.0000
1.0000
1.0000
0.0519
0.8695
0.3264
1.0000
1.0000
0.9304
0.0000
0.8269Table 4
The standardized data of urban economic social and environmental system.
Year
Gross regional product actual valueS11
The proportion of the tertiary industriesS12
Actual use of foreign investment amountS13
Fixed assets investment ratioS14
Per capita household disposable incomeS21
Per capita consumption expenditureS22
Number of employees in three industriesS23
Urbanization rateS24
Annual average inhalable particlesS31
Total wastewater dischargeS32
General industrial solid waste production volumeS33
Sulfur dioxide emissionsS34
2000
0.0000
0.2210
0.0000
0.7978
0.0000
0.0000
0.0000
0.0000
0.9459
0.0000
0.7017
1.0000
2001
0.0674
0.2918
0.0035
0.8216
0.0262
0.0144
0.0160
0.0574
0.9865
0.0044
0.6976
0.8959
2002
0.1294
0.3938
0.0195
0.8978
0.0451
0.0602
0.0997
0.1127
1.0000
0.0532
0.5834
0.8572
2003
0.1952
0.1303
0.0343
1.0000
0.0753
0.0884
0.1398
0.1680
0.6622
0.0555
0.7664
0.8161
2004
0.2817
0.0000
0.1175
0.8840
0.1127
0.1245
0.3908
0.2219
0.7703
0.1116
0.9274
0.0000
2005
0.3315
0.3173
0.1567
0.7514
0.1556
0.1596
0.4306
0.6778
0.6757
0.1498
0.8256
0.8511
2006
0.3833
0.4023
0.2482
0.8470
0.2052
0.2127
0.5000
0.7574
0.9324
0.2016
1.0000
0.7837
2007
0.4862
0.3598
0.2934
0.7235
0.2480
0.2297
0.5383
0.7757
0.7568
0.2383
0.8886
0.6771
2008
0.5498
1.0000
0.2478
0.1641
0.3063
0.2677
0.6019
0.8205
0.4054
0.3091
0.7260
0.5501
2009
0.5350
0.3088
0.2934
0.6932
0.3492
0.3158
0.6308
0.8322
0.3919
0.6672
0.8442
0.5303
2010
0.6347
0.0992
0.3835
0.5891
0.3990
0.3844
0.6863
0.9384
0.3919
0.6100
0.8805
0.5136
2011
0.7641
0.5099
0.4086
0.3365
0.4806
0.4533
0.7497
0.9691
0.2973
0.7277
0.6832
0.4370
2012
0.8096
0.4136
0.4698
0.3146
0.5566
0.5226
0.8123
0.9652
0.2297
0.6602
0.6536
0.4190
2013
0.8707
0.3739
0.5575
0.2521
0.6387
0.5974
0.8683
0.9767
0.2162
0.7161
0.5711
0.3884
2014
0.8795
0.4561
0.6461
0.2460
0.7152
0.6557
0.8945
0.9878
0.3243
0.7959
0.5390
0.3519
2015
0.9002
0.7932
0.9971
0.1736
0.9059
0.9458
0.9434
1.0000
0.1351
0.8091
0.1111
0.3175
2016
1.0000
0.5694
1.0000
0.0000
1.0000
1.0000
1.0000
0.9991
0.0000
1.0000
0.0000
0.1482


## 2.2.2. The ANP Is Used to Evaluate the Interdependence of the Subsystems of Urban Public Infrastructure

The ANP is a multiobjective decision-making method developed by Professor SaatyTL and is based on the AHP, which is especially applicable to complex system decision-making problems with internal interdependence and feedback from indicator elements [31]. The ANP method is used to evaluate the decision-making of the interdependence subsystem in the network to obtain weights of subsystem in the paper.


## 2.2.3. The Weight of the Evaluation Index of Each Subsystem Is Obtained by Using a Combination Weighting Method Based on Game Theory

Game theory is an important part of operations research and is used for studying competitive elements. Drawing on game theory, relevant experts have developed a game theory combination weighting method [32, 33] that uses comprehensive subjective and objective weighting to overcome the limitations of subjective weighting and objective weighting. The paper uses a combined weight method based on game theory to calculate the weight of the evaluation indicators of the urban public infrastructure system and the urban economic social and environmental system.The calculation process of combination weighting method based on game theory is as follows:(1)
Obtainn weights according to the n-type weighting method, and then construct a basic weight vector set U=u1,u2,…,un These n vectors are arbitrarily linearly combined to form a possible set of weights:(3)U=∑k=1nWkukTwk≥0,where u is a possible weight vector of a possible set of weight vectors and Wk is the weight coefficient.We use game theory to find the most satisfactoryu∗ in the possible vector set. The most satisfactory weight vector can be transformed by optimizing the linear combination weight coefficient Wk. The goal of optimization is to minimize the dispersion of u from each uk. That is,(4)min∑j=1nwj×ujT−ujT2,i=1,2,…,n.Based on the differential properties of the matrix, the first derivative condition of the optimization of equation (4) is(5)∑j=1nWj×uj×ujT=ui×ujT,i=1,2,…,n.Equation (5) corresponds to a linear system of equations:(6)u1⋅u1Tu1⋅u2T⋯u1⋅unTu2⋅u1Tu2⋅u2T⋯u2⋅unT⋯⋯⋯⋯un⋅u1Tun⋅u2T⋯un⋅unTw1w2⋯wn=u1⋅u1Tu1⋅u2T⋯un⋅unT.After obtainingw1,w2,…,wn using formula (6), we normalize it:(7)Wk∗=wk∑k=1nwk.Finally, the combined weight is(8)u∗=∑k=1nwkukT.


## 2.3. Research Model Construction

### 2.3.1. Model Used to Evaluate the Supply Level of the Urban Public Infrastructure System

The model used to evaluate the supply level of the urban public infrastructure system (see Figure3) is divided into three levels. The first level is the rule layer, which includes three aspects: economic growth, social welfare, and environmental pressure on urban public infrastructure systems. The second layer of the model is the network layer. Considering the purpose of the research, five subsystems of the public infrastructure system are included in the model, namely, water resources and water supply and drainage system (I1), transportation system (I2), postal power system (I3), environmental facilities system (I4), and energy facilities system (I5), which are consistent with Table 1. The third layer of the model is the system indicator layer, which consists of the evaluation indicators for the supply level of each subsystem.Figure 3
Model used to evaluate the management of the supply level of the urban public infrastructure system.The main process used to evaluate the supply level of the urban public infrastructure system is described below:Step 1.
According to the urban development orientation, the established stock of the urban public infrastructure and the interdependence of the public infrastructure determine the structural relationship of the urban public infrastructure system and act as the basis for constructing the structure of the urban public infrastructure system ANP management activities. This process needs to be conducted by experts.Step 2.
The ANP super matrix and the weighted super matrix are calculated. The evaluation criteria used for the ANP act as the basis for evaluating the relationships in the system and reflect the overall goal of the evaluation. Let the evaluation criteria in the ANP structure beCi,i=1,2,…,m. The system’s network layer has subsystems Ij,j=1,2,…,n. For evaluating the supply level of urban public infrastructure systems and interdependence, Ci is used as a criterion to judge the interdependence of urban public infrastructure subsystems. On this basis, the judgment matrix is constructed to form the feature vector W1j,W2j,…,Wij. When the feature vector passes the consistency test, it is expressed as a matrix form, and a local weight vector matrix Wij is generated. m supermatrices W are formed under the influence of the control criterion index Ci; however, W is not a normalized matrix. To ensure the calculation results are objective and comparable, the supermatrix columns are normalized. The corresponding weighting factor Yij,i,j=1,2,3,…,n is set, and the superweighted matrix is Wij¯=yijwij.
To accurately reflect the interaction between the subsystems, the stability of the supermatrix must be processed. Stability processing is performed on the superweighting matrixWij¯, generating an ANP limit matrix W∞. The processing method is as follows:(9)W∞=limk=⟶∞1m∑i=1mW¯k.
Equation (9) is convergent and unique, and the value of the corresponding column in the original matrix is the weight of the stability of the urban public infrastructure subsystem.Step 3.
After the subsystem weights are obtained, the weights of the internal indicators of each subsystem are obtained using AHP.Step 4.
The entropy method is used to obtain the evaluation index weight of each subsystem [34, 35].Step 5.
Using a combination weighting method based on game theory, the weights of the evaluation indexes of each subsystem are obtained.Step 6.
The supply level of each subsystem and the overall system are obtained.
The supply level of each subsystem and the system supply level are obtained by using index weights are calculated using a combination weighting method based on game theory.(10)Ssubsystem,i,r=∑j=1nujr∗Yij,where Ssubsystem,i,r represents the supply level of each subsystem; 1, 2, ..., 5 represent 5 different subsystems; i represents the year; and ujr∗ represents the weight of the combination index (which is based on game theory) for each subsystem.
The supply level of the urban public infrastructure system is obtained by using the following formula:(11)Ssystem,i=∑j=1nuj∗Yij,where ujr∗ represents the supply level of the urban public infrastructure system and u∗ represents the weight of the combination index (which is based on game theory) of the urban public infrastructure system.


### 2.3.2. Coupling Coordination Analysis

(1) Coupling Coordination Model (CCM). Coupling is a term used in physics to indicate the degree to which two or more systems interact and affect each other to reach a degree of synergy. Coupling is used to measure the degree of agreement in terms of the level of system development. Liao was the first to develop a model to evaluate the degree of coupling coordination within a system or between different systems and further differentiated the levels of coupling coordination degree [36]. This paper constructs a CCM of the urban public infrastructure system and the urban economic social and environmental system to analyze the degree of coordinated development between the urban public infrastructure system and the level of urban development. The calculation process of the CCM is as follows:(1)
The evaluation value of the urban public infrastructure system and the urban economic, social, and environmental system is calculated, which are represented byfx and gy, respectively. These are shown in Table 5.(2)
The coupling degree of the urban public infrastructure system and the urban economic social and environmental system is calculated:(12)C=fx∗gyfx+gy/221/2,T=afx+bgy,Dx,y=C⋅T,where D represents the degree of the coupling between the two systems, D∈0,1, C represents the degree of coordination between the two systems, C∈0,1, T represents the comprehensive coordination index of the two systems, and a and b represent the contribution of the two systems. Let fx.Table 5
Evaluation of the supply level of urban infrastructure system and the demand level of the urban economic, social, and environmental system.
Years
Urban infrastructure system supply level (fx)
Urban economic, social, and environmental system evaluation value (gy)
2000
0.1383
0.1590
2001
0.1675
0.1882
2002
0.2247
0.2310
2003
0.2413
0.2474
2004
0.3200
0.2683
2005
0.3204
0.3344
2006
0.4425
0.3979
2007
0.5045
0.4131
2008
0.5320
0.4138
2009
0.5528
0.4670
2010
0.5862
0.4884
2011
0.5519
0.5416
2012
0.5568
0.5556
2013
0.6076
0.5870
2014
0.6606
0.6210
2015
0.6860
0.6924
2016
0.6372
0.7097(2) Dynamic Coupling Coordination Model (DCCM). The DCCM is used to determine the coupling degree of the composite system [37]. Li and Ding was the first to propose a DCCM and used it to evaluate the coordination of resource environment systems and social economic systems and then calculated the coordination degree of the composite system [38].To clarify the relationship between the urban public infrastructure system and the urban economic, social, and environmental system, this paper assumes that both of these systems and their relationship form a composite system. Based on theory of the evolution of subsystems in a general system, we construct a DCCM to analyze the evolution state and coupling state of the composite systems. The changes between the urban public infrastructure system and the urban economic, social, and environmental system are nonlinear, so the evolution equation can be expressed as(13)dxtdt=fx1,x2,…,xn;i=1,2,…,n,where f is a nonlinear function of xi. The motion stability of a nonlinear system depends on the characteristic root property of the first approximation system. The high-order term can guarantee the stability of motion and be used to obtain an approximate linear system.(14)dxtdt=∑i=1naixi;i=1,2,…,n.Using the above method, we establish the general function of the urban public infrastructure system and the urban economic, social, and environmental system.(15)fx=∑i=1naixi∗;i=1,2,…,n,gy=∑i=1nbiyi∗;i=1,2,…,n.We assume that the urban public infrastructure system and the urban economic, social, and environmental system and their relationship are in the same system, which has two elementsFx and Fy. According to Beta Langfi’s general system theory, the evolution equation of a composite system is expressed as(16)Fx=dfxdt=T1fx+T2gy,VFx=dFxdt,(17)Gy=dgxdt=U1fx+U2gy,VGy=dGydt.F
x and Gy are the evolutionary states of the urban public infrastructure system and the urban economic, social, and environmental system under their own interaction and external conditions. VFx and VGy are the respective evolutionary speeds of the urban public infrastructure system and the urban economic, social, and environmental system. In a composite system, because the entire system contains only two elements Fx and Gy, any changes in the subsystem that occur when Fx and Gy interact will cause the entire system to change. The evolution rate V of a composite system can be seen as a function of VFx and VGy, so VGy. Therefore, VFx and VGy are used as control variables; the coupling relationship between the composite system and the two subsystems can be obtained by studying the variation in V. Because the evolution of the entire system satisfies the S-type development mechanism, we assume that the dynamic coupling and coordination relationship between the urban public infrastructure system and the urban economic, social, and environmental system is cyclical. Since a change in V is caused by VFx and VGy, we analyze V on the two-dimensional coordinate plane (VFx, VGy). The change in V is an ellipse in the coordinate system (the change in the urban economic, social, and environmental system is not as rapid as that of the urban public infrastructure system, and the amplitude is relatively small), as shown in Figure 4.Figure 4
Dynamic coupling degree between urban public infrastructure and the urban economic, social, and environmental system (created by author based on Li Chongming’s study [38]).In this case, the variableV is an ellipse. The variable β represents the angle between VFx and VGy. According to the value of β, we can determine the value of the dynamic coupling coordination degree of the composite system. Based on relevant research [38], this paper proposes the development stage and state of the dynamic coupling coordination degree of the composite system, as shown in Table 6.(18)β=arctanVFxVGy.Table 6
Dynamic coupling and coordination of the composite system.
Range ofβ
Stage
System development stage
System status
−
90
∘
≺
β
≤
0
∘
I
Low-level symbiosis
At this stage of development, the development speed of urban public infrastructure systems is very low, and the impact of urban public infrastructure systems on urban economic, social, and environmental systems is almost 0.
0
∘
≺
β
≺
45
∘
II
Primary coordinated development stage
V
F
x
<
V
G
y The development speed of urban public infrastructure system is lower than that of urban economic, social, and environmental system. Urban public infrastructure system can not promote the development of urban economic, social, and environmental system, showing that urban infrastructure system can not carry the development of urban economic, social, and environmental system.
β
=
45
∘
Coordinated development stage
V
F
x
=
V
G
y The speed of urban public infrastructure system is equal to the urban economic, social, and environmental system, and the urban public infrastructure system and the urban economic, social, and environmental system are coordinated.
45
∘
≺
β
≤
90
∘
Coordinated development stage
V
F
x
>
V
G
y With the development of the urban public infrastructure system, the two systems began to interact with each other. The restrictions of the urban economic, social, and environmental system on the urban public infrastructure system began to appear, but it was not obvious.
90
∘
≺
β
≤
180
∘
III
Extreme development stage
With the accelerated development of urban economic, social, and environmental systems, urban public infrastructure systems have increased the demand for urban development, and the contradiction between infrastructure systems and urban systems has emerged, and began to limit the improvement of urban development level.
−
180
∘
≺
β
≤
−
90
∘
IV
High-level coordinated development stage
The composite system gradually transforms into a common development stage, and finally reaches the high-level coordinated development of the urban public infrastructure system and the urban economic, social, and environmental system
The authors edited the table by themselves according to Li Chongming’s literature [38].


## 2.3.1. Model Used to Evaluate the Supply Level of the Urban Public Infrastructure System

The model used to evaluate the supply level of the urban public infrastructure system (see Figure3) is divided into three levels. The first level is the rule layer, which includes three aspects: economic growth, social welfare, and environmental pressure on urban public infrastructure systems. The second layer of the model is the network layer. Considering the purpose of the research, five subsystems of the public infrastructure system are included in the model, namely, water resources and water supply and drainage system (I1), transportation system (I2), postal power system (I3), environmental facilities system (I4), and energy facilities system (I5), which are consistent with Table 1. The third layer of the model is the system indicator layer, which consists of the evaluation indicators for the supply level of each subsystem.Figure 3
Model used to evaluate the management of the supply level of the urban public infrastructure system.The main process used to evaluate the supply level of the urban public infrastructure system is described below:Step 1.
According to the urban development orientation, the established stock of the urban public infrastructure and the interdependence of the public infrastructure determine the structural relationship of the urban public infrastructure system and act as the basis for constructing the structure of the urban public infrastructure system ANP management activities. This process needs to be conducted by experts.Step 2.
The ANP super matrix and the weighted super matrix are calculated. The evaluation criteria used for the ANP act as the basis for evaluating the relationships in the system and reflect the overall goal of the evaluation. Let the evaluation criteria in the ANP structure beCi,i=1,2,…,m. The system’s network layer has subsystems Ij,j=1,2,…,n. For evaluating the supply level of urban public infrastructure systems and interdependence, Ci is used as a criterion to judge the interdependence of urban public infrastructure subsystems. On this basis, the judgment matrix is constructed to form the feature vector W1j,W2j,…,Wij. When the feature vector passes the consistency test, it is expressed as a matrix form, and a local weight vector matrix Wij is generated. m supermatrices W are formed under the influence of the control criterion index Ci; however, W is not a normalized matrix. To ensure the calculation results are objective and comparable, the supermatrix columns are normalized. The corresponding weighting factor Yij,i,j=1,2,3,…,n is set, and the superweighted matrix is Wij¯=yijwij.
To accurately reflect the interaction between the subsystems, the stability of the supermatrix must be processed. Stability processing is performed on the superweighting matrixWij¯, generating an ANP limit matrix W∞. The processing method is as follows:(9)W∞=limk=⟶∞1m∑i=1mW¯k.
Equation (9) is convergent and unique, and the value of the corresponding column in the original matrix is the weight of the stability of the urban public infrastructure subsystem.Step 3.
After the subsystem weights are obtained, the weights of the internal indicators of each subsystem are obtained using AHP.Step 4.
The entropy method is used to obtain the evaluation index weight of each subsystem [34, 35].Step 5.
Using a combination weighting method based on game theory, the weights of the evaluation indexes of each subsystem are obtained.Step 6.
The supply level of each subsystem and the overall system are obtained.
The supply level of each subsystem and the system supply level are obtained by using index weights are calculated using a combination weighting method based on game theory.(10)Ssubsystem,i,r=∑j=1nujr∗Yij,where Ssubsystem,i,r represents the supply level of each subsystem; 1, 2, ..., 5 represent 5 different subsystems; i represents the year; and ujr∗ represents the weight of the combination index (which is based on game theory) for each subsystem.
The supply level of the urban public infrastructure system is obtained by using the following formula:(11)Ssystem,i=∑j=1nuj∗Yij,where ujr∗ represents the supply level of the urban public infrastructure system and u∗ represents the weight of the combination index (which is based on game theory) of the urban public infrastructure system.


## 2.3.2. Coupling Coordination Analysis

(1) Coupling Coordination Model (CCM). Coupling is a term used in physics to indicate the degree to which two or more systems interact and affect each other to reach a degree of synergy. Coupling is used to measure the degree of agreement in terms of the level of system development. Liao was the first to develop a model to evaluate the degree of coupling coordination within a system or between different systems and further differentiated the levels of coupling coordination degree [36]. This paper constructs a CCM of the urban public infrastructure system and the urban economic social and environmental system to analyze the degree of coordinated development between the urban public infrastructure system and the level of urban development. The calculation process of the CCM is as follows:(1)
The evaluation value of the urban public infrastructure system and the urban economic, social, and environmental system is calculated, which are represented byfx and gy, respectively. These are shown in Table 5.(2)
The coupling degree of the urban public infrastructure system and the urban economic social and environmental system is calculated:(12)C=fx∗gyfx+gy/221/2,T=afx+bgy,Dx,y=C⋅T,where D represents the degree of the coupling between the two systems, D∈0,1, C represents the degree of coordination between the two systems, C∈0,1, T represents the comprehensive coordination index of the two systems, and a and b represent the contribution of the two systems. Let fx.Table 5
Evaluation of the supply level of urban infrastructure system and the demand level of the urban economic, social, and environmental system.
Years
Urban infrastructure system supply level (fx)
Urban economic, social, and environmental system evaluation value (gy)
2000
0.1383
0.1590
2001
0.1675
0.1882
2002
0.2247
0.2310
2003
0.2413
0.2474
2004
0.3200
0.2683
2005
0.3204
0.3344
2006
0.4425
0.3979
2007
0.5045
0.4131
2008
0.5320
0.4138
2009
0.5528
0.4670
2010
0.5862
0.4884
2011
0.5519
0.5416
2012
0.5568
0.5556
2013
0.6076
0.5870
2014
0.6606
0.6210
2015
0.6860
0.6924
2016
0.6372
0.7097(2) Dynamic Coupling Coordination Model (DCCM). The DCCM is used to determine the coupling degree of the composite system [37]. Li and Ding was the first to propose a DCCM and used it to evaluate the coordination of resource environment systems and social economic systems and then calculated the coordination degree of the composite system [38].To clarify the relationship between the urban public infrastructure system and the urban economic, social, and environmental system, this paper assumes that both of these systems and their relationship form a composite system. Based on theory of the evolution of subsystems in a general system, we construct a DCCM to analyze the evolution state and coupling state of the composite systems. The changes between the urban public infrastructure system and the urban economic, social, and environmental system are nonlinear, so the evolution equation can be expressed as(13)dxtdt=fx1,x2,…,xn;i=1,2,…,n,where f is a nonlinear function of xi. The motion stability of a nonlinear system depends on the characteristic root property of the first approximation system. The high-order term can guarantee the stability of motion and be used to obtain an approximate linear system.(14)dxtdt=∑i=1naixi;i=1,2,…,n.Using the above method, we establish the general function of the urban public infrastructure system and the urban economic, social, and environmental system.(15)fx=∑i=1naixi∗;i=1,2,…,n,gy=∑i=1nbiyi∗;i=1,2,…,n.We assume that the urban public infrastructure system and the urban economic, social, and environmental system and their relationship are in the same system, which has two elementsFx and Fy. According to Beta Langfi’s general system theory, the evolution equation of a composite system is expressed as(16)Fx=dfxdt=T1fx+T2gy,VFx=dFxdt,(17)Gy=dgxdt=U1fx+U2gy,VGy=dGydt.F
x and Gy are the evolutionary states of the urban public infrastructure system and the urban economic, social, and environmental system under their own interaction and external conditions. VFx and VGy are the respective evolutionary speeds of the urban public infrastructure system and the urban economic, social, and environmental system. In a composite system, because the entire system contains only two elements Fx and Gy, any changes in the subsystem that occur when Fx and Gy interact will cause the entire system to change. The evolution rate V of a composite system can be seen as a function of VFx and VGy, so VGy. Therefore, VFx and VGy are used as control variables; the coupling relationship between the composite system and the two subsystems can be obtained by studying the variation in V. Because the evolution of the entire system satisfies the S-type development mechanism, we assume that the dynamic coupling and coordination relationship between the urban public infrastructure system and the urban economic, social, and environmental system is cyclical. Since a change in V is caused by VFx and VGy, we analyze V on the two-dimensional coordinate plane (VFx, VGy). The change in V is an ellipse in the coordinate system (the change in the urban economic, social, and environmental system is not as rapid as that of the urban public infrastructure system, and the amplitude is relatively small), as shown in Figure 4.Figure 4
Dynamic coupling degree between urban public infrastructure and the urban economic, social, and environmental system (created by author based on Li Chongming’s study [38]).In this case, the variableV is an ellipse. The variable β represents the angle between VFx and VGy. According to the value of β, we can determine the value of the dynamic coupling coordination degree of the composite system. Based on relevant research [38], this paper proposes the development stage and state of the dynamic coupling coordination degree of the composite system, as shown in Table 6.(18)β=arctanVFxVGy.Table 6
Dynamic coupling and coordination of the composite system.
Range ofβ
Stage
System development stage
System status
−
90
∘
≺
β
≤
0
∘
I
Low-level symbiosis
At this stage of development, the development speed of urban public infrastructure systems is very low, and the impact of urban public infrastructure systems on urban economic, social, and environmental systems is almost 0.
0
∘
≺
β
≺
45
∘
II
Primary coordinated development stage
V
F
x
<
V
G
y The development speed of urban public infrastructure system is lower than that of urban economic, social, and environmental system. Urban public infrastructure system can not promote the development of urban economic, social, and environmental system, showing that urban infrastructure system can not carry the development of urban economic, social, and environmental system.
β
=
45
∘
Coordinated development stage
V
F
x
=
V
G
y The speed of urban public infrastructure system is equal to the urban economic, social, and environmental system, and the urban public infrastructure system and the urban economic, social, and environmental system are coordinated.
45
∘
≺
β
≤
90
∘
Coordinated development stage
V
F
x
>
V
G
y With the development of the urban public infrastructure system, the two systems began to interact with each other. The restrictions of the urban economic, social, and environmental system on the urban public infrastructure system began to appear, but it was not obvious.
90
∘
≺
β
≤
180
∘
III
Extreme development stage
With the accelerated development of urban economic, social, and environmental systems, urban public infrastructure systems have increased the demand for urban development, and the contradiction between infrastructure systems and urban systems has emerged, and began to limit the improvement of urban development level.
−
180
∘
≺
β
≤
−
90
∘
IV
High-level coordinated development stage
The composite system gradually transforms into a common development stage, and finally reaches the high-level coordinated development of the urban public infrastructure system and the urban economic, social, and environmental system
The authors edited the table by themselves according to Li Chongming’s literature [38].


## 3. Empirical Research

### 3.1. Calculation of the Evaluation Value of the Supply Level of the Urban Public Infrastructure System

#### 3.1.1. ANP Analysis of the Supply Weights of the Urban Public Infrastructure Subsystem

Under the constraints of the comprehensive development level of the economic, social, and environmental subsystems in cities, this paper uses Super Decisions software to evaluate the supply level of the urban public infrastructure. Taking into account the strong professionalism of the research questions, this paper selects eight experts, including senior management personnel and senior engineers, who have 10–20 years of experience in the urban public infrastructure field, to score the judgment matrix reflecting the supply structure management of the urban public infrastructure system. The experts evaluated based on a comparison of contribution rates, the principle of the universality rate and the principle of irreplaceability. The results are shown in Tables7–14.Table 7
ANP judgment matrix of the system (based onC1).
I
1
I
2
I
3
I
4
I
5
I
1
1
1/2
2
5
3
I
2
2
1
3
6
4
I
3
1/2
1/3
1
4
2
I
4
1/5
1/6
1/4
1
1/3
I
5
1/3
1/4
1/2
3
1Table 8
ANP judgment matrix of the system (based onC2).
I
1
I
2
I
3
I
4
I
5
I
1
1
1/2
2
5
3
I
2
2
1
3
6
4
I
3
1/2
1/3
1
4
2
I
4
1/5
1/6
1/4
1
1/3
I
5
1/3
1/4
1/2
3
1Table 9
ANP judgment matrix of the system (based onC3).
I
1
I
2
I
3
I
4
I
5
1
2
1/2
3
5
1/2
1
1/3
2
4
2
3
1
4
7
1/3
1/2
1/4
1
3
1/5
1/4
1/7
1/3
1Table 10
ANP judgment matrix of the system (based onI1).
I
2
I
3
I
4
I
5
I
2
1
4
2
6
I
3
1/4
1
1/2
3
I
4
1/2
2
1
5
I
5
1/6
1/3
1/5
1Table 11
ANP judgment matrix of the system (based onI2).
I
1
I
3
I
4
I
5
I
1
1
3
2
5
I
3
1/3
1
1/2
3
I
4
1/2
2
1
4
I
5
1/5
1/3
1/4
1Table 12
ANP judgment matrix of the system (based onI3).
I
1
I
2
I
4
I
5
I
1
1
1/2
1
3
I
2
2
1
2
4
I
4
1
1/2
1
3
I
5
1/3
1/4
1/3
1Table 13
ANP judgment matrix of the system (based onI4).
I
1
I
2
I
3
I
5
I
1
1
1/2
3
5
I
2
2
1
4
6
I
3
1/3
1/4
1
3
I
5
1/5
1/6
1/3
1Table 14
ANP judgment matrix of the system (based onI5).
I
1
I
2
I
3
I
4
I
1
1
1/2
3
5
I
2
2
1
4
7
I
3
1/3
1/4
1
3
I
4
1/5
1/7
1/3
1In the ANP model of the overall supply level of the urban public infrastructure system, evaluation criteria for economic growth (C1), social welfare (C2), and environmental pressure (C3) evaluation criteria are used to identify the interactions among the subsystems of urban public infrastructure in the network layer, and the corresponding judgment matrix is established, as shown in the table below. The relationship between the judgment indicators in the table is based on a 9-point method; 1–9 indicates that the influence of one subsystem on the other subsystem has gradually increased. The score of the judgment matrix is taken as the mean of the expert score. The consistency test result of the judgment matrix is 0.056 (<0.1), indicating that the judgment matrix passes the consistency test.When all the judgment matrices pass the consistency test, the ANP supermatrix, the weighted supermatrix, and the limit matrix of the urban public infrastructure system supply level evaluation model are generated using Super Decisions software. Since the limits in the limit matrix converge and are unique, the weights of the interdependent subsystems in the model of the supply level of urban public infrastructure are obtained, which isWi = (0.5396, 0.2970, 0.1634). The weights of the subsystems in the network layer are (0.2075, 0.3270, 0.1386, 0.2040, and 0.1229).


#### 3.1.2. Entropy Method

Through the collection, processing, and standardization of the statistical yearbook data used in the evaluation indexes of the five subsystems in Beijing, the entropy weight of the evaluation index is calculated using the entropy method (see Table1).


#### 3.1.3. The Weights in Each Indicator System Are Calculated Using a Combined Weight Method Based on Game Theory

The weight of the evaluation index of each subsystem obtained by the entropy method isuIEW. The weight of the evaluation index obtained by the ANP method is uANP. Using the combined weight method based on game theory combined with the objective evaluation of the entropy method and the subjective evaluation of ANP, the weight of the optimal urban public infrastructure evaluation index is expressed as ucw∗ (as shown in Table 1).


#### 3.1.4. The Evaluation Value of the Supply Level of the Typical Urban Public Infrastructure System Is Obtained

The evaluation value of the urban infrastructure system supply level is obtained by using equation (11) (Table 5). The evaluation value of the urban infrastructure system from 2000 to 2016 gradually increased the supply level of the urban infrastructure systems from 0.1383 to 0.6372, with a growth rate of 78.3%.


### 3.2. Calculation of the Evaluation Value of the Urban Economic, Social, and Environmental System

Using the entropy method and the AHP method on the urban economic, social, and environmental system indicators shown in Table2, the index weights are obtained by using a combined weight method based on game theory. Then, the evaluation value of the demand level of the economic, social, and environmental system in Beijing is obtained (Table 5). The demand level of the urban economic, social, and environmental system has gradually increased from 2000 to 2016, from 0.1590 to 0.7097, with a growth rate of 77.6%.The time series of the evaluation values of the supply level of the urban public infrastructure system and the demand level of the urban economic, social, and environmental system from 2000 to 2016 can be divided into four stages. The evaluation value of the urban public infrastructure system from 2000 to 2005 is very close to the demand level of the urban economic, social, and environmental system. From 2006 to 2011, the supply level of the urban infrastructure system is greater than the demand level of the urban economic, social, and environmental system. From 2011 to 2015, the supply level of the urban public infrastructure system is close to the evaluation level of the demand level of the urban economic, social, and environmental system. In 2016, the demand level of the urban economic, social, and environmental system is greater than the supply level of the urban public infrastructure system. During the 11th Five-Year Plan period, the average annual growth rate of urban infrastructure fixed asset investment reached 27%, which is much higher than 7.29% in the 15th period and 7.35% in the 12th Five-Year Plan period. As a result of the global financial crisis, Beijing’s GDP had negative growth in 2009, which explains why the supply of the urban public infrastructure system from 2006 to 2010 is higher than the demand for the urban economic, social, and environmental system.


### 3.3. Analysis of the Coupling Degree of the Level of Supply of the Urban Public Infrastructure System and the Urban Economic, Social, and Environmental System

Table15 shows that the degree of coordination between the urban public infrastructure system and the urban economic, social, and environmental system increases from 0.9976 to 0.9986 from 2000 to 2016, and the change is relatively stable. However, the coupling coordination degree D increases from 0.3851 to 0.8200, and a significant change occurs. The third column of Table 15 shows that the coupling and coordination degree has gone through three stages from 2000 to 2016, namely, the unbalanced development stage from 2000 to 2003, the microbalance stage from 2004 to 2005, and the balanced development stage from 2006 to 2016.Table 15
Degree of coordination between the urban public infrastructure system and urban economic, social, and environmental system.
Years
Coordination degreeC
Coupling coordination degreeD
Coordination level
2000
0.9976
0.3851
Mild imbalance development
2001
0.9983
0.4214
Slightly unbalanced development
2002
0.9999
0.4773
Slightly unbalanced development
2003
0.9999
0.4943
Slightly unbalanced development
2004
0.9961
0.5413
Slightly unbalanced development
2005
0.9998
0.5721
Slightly unbalanced development
2006
0.9986
0.6478
Mild imbalance development
2007
0.9950
0.6757
Mild imbalance development
2008
0.9922
0.6850
Mild imbalance development
2009
0.9965
0.7128
Moderately balanced development
2010
0.9958
0.7315
Moderately balanced development
2011
1.0000
0.7394
Moderately balanced development
2012
1.0000
0.7458
Moderately balanced development
2013
0.9999
0.7728
Moderately balanced development
2014
0.9995
0.8003
Highly balanced development
2015
1.0000
0.8302
Highly balanced development
2016
0.9986
0.8200
Highly balanced developmentTo determine the influence of the evaluation values of two systems on the coupling coordination degreeD, this paper establishes a multivariate linear regression model that uses the coupling coordination degree D as the explained variable and the evaluation values of the urban public infrastructure system and the urban economic, social, and environmental system as explanatory variables. The model fitting results shown in Tables 16 and 17 are obtained by SPSS software. The fitting results show that the R2 fit of the model is 0.995, which is very close to 1, indicating that the model has a high degree of fit. The model is shown in Table 17. When the supply level of the urban infrastructure system increases by one unit, the coupling coordination degree D increases by 0.555. When the demand for the urban economic, social, and environmental system increases by one unit, the coupling coordination degree D increases by 0.244.Table 16
Summary of multiple linear regression models.
Model
R
R Side
Adjust theR side
Standard estimated error
1
0.997a
0.995
0.994
0.0111601
aPredictor (constant): gy, fx.Table 17
Multiple linear regression coefficientsa.
Model
Nonstandardized coefficient
Standard coefficient
t
Sig.
B
Standard error
Trial version
1 (constant)
0.293
0.008
38.834
0.000
f
x
0.555
0.054
0.707
10.224
0.000
g
y
0.244
0.057
0.299
4.322
0.001
aDependent variable: D.


### 3.4. Analysis of the Dynamic Coupling Coordination Degree between the Urban Public Infrastructure System and the Urban Economic, Social, and Environmental System

According to the evaluation value of the supply level of the urban infrastructure system (fx) and the urban economic, social, and environmental system (gy) shown in Table 5, we can obtain the evaluation curves and fitting functions of the supply level of the urban infrastructure system (fx) and the evaluation value of the urban economic, social, and environmental system (gy). All of these are presented in Figure 5. The dynamic degree of coupling coordination between the urban public infrastructure system and the urban economic, social, and environmental system is obtained by using equation (16)–18 (Figure 6).Figure 5
Trend function of the Beijing urban infrastructure system and the urban economic, social, and environmental system.Figure 6
Degree of the dynamic coupling coordination between the urban infrastructure system and the urban economic, social, and environmental system in Beijing.Figure6 shows that the degree of the dynamic coupling coordination between the urban public infrastructure system and the urban economic, social, and environmental system in Beijing can be divided into three stages. The first phase in 2000 shows that the degree of the dynamic coupling coordination between the two systems is 32.15° and that the two systems are in the primary coordinated development stage. From 2001 to 2009, the degree of the dynamic coupling coordination between the two systems ranges from 48.96° to 83.55°; this is the second stage. At this stage, the two systems are in a symbiotic and coordinated development stage. The two systems begin to interact with each other, and the supply level of the urban public infrastructure system increases the demand level of the urban economic, social, and environmental system. Because of the influence of the global financial crisis in 2008, from 2010 to 2016, the degree of the dynamic coupling coordination of two systems changed from 81.77° to 56.26°, which represents a low level of coexistence and the state of low interaction between the two systems.


### 3.5. Analysis of the Influencing Factors of the Degree of Coupling between the Urban Public Infrastructure System and the Urban Economic, Social, and Environmental System

The degree of the coupling between the urban public infrastructure system and the urban economic, social, and environmental system is affected by many factors. This paper mainly examines the impact of 20 evaluation indicators of the urban public infrastructure system and 12 evaluation indicators of the urban economic, social, and environmental system. Using SPSS software, a linear regression analysis was performed between coupling degreeD and 36 evaluation indicators of the two systems.As shown in Table18, the degree of the coupling coordination between the urban infrastructure system and the urban economic, social, and environmental system is most affected by urban sewage treatment capacity. In Table 18, it can be seen that in the 20 indicators in the urban public infrastructure system, the top five influencing factors on the degree of the coupling coordination between two systems are sewage treatment capacity, natural gas ratio, and per capita heating area, total number of urban sanitation special vehicles and equipment, household garbage clearance volume, and bureau number of post and telecommunications. The urban infrastructure system indicators that have a weaker impact on the degree of coupling coordination between the two systems are mainly railway density and highway facilities, the amount of postal and telecommunications services per capita, and the number of public transportation lines and passenger traffic on public transportation, per capita annual water supply. Table 19 shows that in the urban economic, social, and environmental system, the most influential indicator of the degree of coupling coordination are the regional GDP actual value. The next indicators are three social development indicators: the number of employees in the industry, the per capita household disposable income, and per capita consumption expenditures. The last indicator is actual utilization of foreign direct investment.Table 18
Impact of the 20 indicators of the urban public infrastructure system on the degree of coupling coordination.
Indicator layer
Linear regression equation
R
2
Sort
Urban public infrastructure system
I
11: per capita social electricity consumption
Y
=
0.382
+
0.432
x
0.807
8
I
12: per capita energy consumption
Y
=
0.604
+
0.103
x
0.042
I
13: natural gas ratio
Y
=
0.310
+
0.468
x
0.896
2
I
14: per capita heating area
Y
=
0.341
+
0.468
x
0.942
2
I
21: railway road facility density
Y
=
0.449
+
0.329
x
0.911
13
I
22: per capita urban road area
Y
=
0.524
+
0.233
x
0.163
I
23: number of public transportation lines
Y
=
0.483
+
0.361
x
0.686
11
I
24: bus operation passenger volume
Y
=
0.476
+
0.361
x
0.789
11
I
31: per capita park green area
Y
=
0.425
+
0.390
x
0.941
10
I
32: sewage treatment capacity
Y
=
0.428
+
0.534
x
0.862
1
I
33: total number of urban sanitation special vehicle and equipment
Y
=
0.406
+
0.457
x
0.898
3
I
34: household garbage clearance volume
Y
=
0.439
+
0.454
x
0.951
4
I
41: per capita annual water supply
Y
=
0.784
−
0.441
x
0.943
14
I
42: density of water supply pipe
Y
=
0.432
+
0.404
x
0.898
9
I
43: per capita comprehensive water production capacity
Y
=
0.821
−
0.401
x
0.597
I
44: sewage pipe density
Y
=
0.468
+
0.434
x
0.858
7
I
51: per capita total postal and telecommunications volume
Y
=
0.527
+
0.358
x
0.532
12
I
52: mobile phone penetration rate
Y
=
0.421
+
0.442
x
0.841
6
I
53: mailine penetration rate
Y
=
0.677
−
0.060
x
0.015
I
54: bureau number of post and telecommunications
Y
=
0.558
+
0.444
x
0.612
5
Y represents the coupling coordination degree D, and x represents the corresponding evaluation indicator of urban public infrastructure system in the table.Table 19
Influence of 12 indicators of urban economic social and environmental system on coupling coordination degree.
Indicator layer
Linear regression equation
R
2
Sort
S
11: regional GDP actual value
Y
=
0.425
+
0.434
x
0.950
1
S
12: the proportion of the tertiary industry
Y
=
0.538
+
0.288
x
0.241
S
13: actual utilization of foreign direct investment
Y
=
0.511
+
0.402
x
0.779
5
S
14: fixed assets investment ratio
Y
=
0.854
−
0.365
x
0.669
8
S
21: per capita household disposable income
Y
=
0.498
+
0.416
x
0.818
3
S
22: per capita consumption expenditure
Y
=
0.504
+
0.413
x
0.785
4
S
23: number of employees in three industries
Y
=
0.416
+
0.429
x
0.978
2
S
24: urbanization rate
Y
=
0.411
+
0.361
x
0.934
7
S
31: annual average of inhalable particles
Y
=
0.858
−
0.387
x
0.783
9
S
32: total wastewater discharge
Y
=
0.488
+
0.388
x
0.860
6
S
33: general industrial solid waste production
Y
=
0.808
−
0.236
x
0.195
S
34: sulfur dioxide emissions
Y
=
0.846
−
0.349
x
0.476
Y represents the coupling coordination degree D, and x represents the corresponding evaluation indicator of the urban economic, social, and environmental system in the table.


## 3.1. Calculation of the Evaluation Value of the Supply Level of the Urban Public Infrastructure System

### 3.1.1. ANP Analysis of the Supply Weights of the Urban Public Infrastructure Subsystem

Under the constraints of the comprehensive development level of the economic, social, and environmental subsystems in cities, this paper uses Super Decisions software to evaluate the supply level of the urban public infrastructure. Taking into account the strong professionalism of the research questions, this paper selects eight experts, including senior management personnel and senior engineers, who have 10–20 years of experience in the urban public infrastructure field, to score the judgment matrix reflecting the supply structure management of the urban public infrastructure system. The experts evaluated based on a comparison of contribution rates, the principle of the universality rate and the principle of irreplaceability. The results are shown in Tables7–14.Table 7
ANP judgment matrix of the system (based onC1).
I
1
I
2
I
3
I
4
I
5
I
1
1
1/2
2
5
3
I
2
2
1
3
6
4
I
3
1/2
1/3
1
4
2
I
4
1/5
1/6
1/4
1
1/3
I
5
1/3
1/4
1/2
3
1Table 8
ANP judgment matrix of the system (based onC2).
I
1
I
2
I
3
I
4
I
5
I
1
1
1/2
2
5
3
I
2
2
1
3
6
4
I
3
1/2
1/3
1
4
2
I
4
1/5
1/6
1/4
1
1/3
I
5
1/3
1/4
1/2
3
1Table 9
ANP judgment matrix of the system (based onC3).
I
1
I
2
I
3
I
4
I
5
1
2
1/2
3
5
1/2
1
1/3
2
4
2
3
1
4
7
1/3
1/2
1/4
1
3
1/5
1/4
1/7
1/3
1Table 10
ANP judgment matrix of the system (based onI1).
I
2
I
3
I
4
I
5
I
2
1
4
2
6
I
3
1/4
1
1/2
3
I
4
1/2
2
1
5
I
5
1/6
1/3
1/5
1Table 11
ANP judgment matrix of the system (based onI2).
I
1
I
3
I
4
I
5
I
1
1
3
2
5
I
3
1/3
1
1/2
3
I
4
1/2
2
1
4
I
5
1/5
1/3
1/4
1Table 12
ANP judgment matrix of the system (based onI3).
I
1
I
2
I
4
I
5
I
1
1
1/2
1
3
I
2
2
1
2
4
I
4
1
1/2
1
3
I
5
1/3
1/4
1/3
1Table 13
ANP judgment matrix of the system (based onI4).
I
1
I
2
I
3
I
5
I
1
1
1/2
3
5
I
2
2
1
4
6
I
3
1/3
1/4
1
3
I
5
1/5
1/6
1/3
1Table 14
ANP judgment matrix of the system (based onI5).
I
1
I
2
I
3
I
4
I
1
1
1/2
3
5
I
2
2
1
4
7
I
3
1/3
1/4
1
3
I
4
1/5
1/7
1/3
1In the ANP model of the overall supply level of the urban public infrastructure system, evaluation criteria for economic growth (C1), social welfare (C2), and environmental pressure (C3) evaluation criteria are used to identify the interactions among the subsystems of urban public infrastructure in the network layer, and the corresponding judgment matrix is established, as shown in the table below. The relationship between the judgment indicators in the table is based on a 9-point method; 1–9 indicates that the influence of one subsystem on the other subsystem has gradually increased. The score of the judgment matrix is taken as the mean of the expert score. The consistency test result of the judgment matrix is 0.056 (<0.1), indicating that the judgment matrix passes the consistency test.When all the judgment matrices pass the consistency test, the ANP supermatrix, the weighted supermatrix, and the limit matrix of the urban public infrastructure system supply level evaluation model are generated using Super Decisions software. Since the limits in the limit matrix converge and are unique, the weights of the interdependent subsystems in the model of the supply level of urban public infrastructure are obtained, which isWi = (0.5396, 0.2970, 0.1634). The weights of the subsystems in the network layer are (0.2075, 0.3270, 0.1386, 0.2040, and 0.1229).


### 3.1.2. Entropy Method

Through the collection, processing, and standardization of the statistical yearbook data used in the evaluation indexes of the five subsystems in Beijing, the entropy weight of the evaluation index is calculated using the entropy method (see Table1).


### 3.1.3. The Weights in Each Indicator System Are Calculated Using a Combined Weight Method Based on Game Theory

The weight of the evaluation index of each subsystem obtained by the entropy method isuIEW. The weight of the evaluation index obtained by the ANP method is uANP. Using the combined weight method based on game theory combined with the objective evaluation of the entropy method and the subjective evaluation of ANP, the weight of the optimal urban public infrastructure evaluation index is expressed as ucw∗ (as shown in Table 1).


### 3.1.4. The Evaluation Value of the Supply Level of the Typical Urban Public Infrastructure System Is Obtained

The evaluation value of the urban infrastructure system supply level is obtained by using equation (11) (Table 5). The evaluation value of the urban infrastructure system from 2000 to 2016 gradually increased the supply level of the urban infrastructure systems from 0.1383 to 0.6372, with a growth rate of 78.3%.


## 3.1.1. ANP Analysis of the Supply Weights of the Urban Public Infrastructure Subsystem

Under the constraints of the comprehensive development level of the economic, social, and environmental subsystems in cities, this paper uses Super Decisions software to evaluate the supply level of the urban public infrastructure. Taking into account the strong professionalism of the research questions, this paper selects eight experts, including senior management personnel and senior engineers, who have 10–20 years of experience in the urban public infrastructure field, to score the judgment matrix reflecting the supply structure management of the urban public infrastructure system. The experts evaluated based on a comparison of contribution rates, the principle of the universality rate and the principle of irreplaceability. The results are shown in Tables7–14.Table 7
ANP judgment matrix of the system (based onC1).
I
1
I
2
I
3
I
4
I
5
I
1
1
1/2
2
5
3
I
2
2
1
3
6
4
I
3
1/2
1/3
1
4
2
I
4
1/5
1/6
1/4
1
1/3
I
5
1/3
1/4
1/2
3
1Table 8
ANP judgment matrix of the system (based onC2).
I
1
I
2
I
3
I
4
I
5
I
1
1
1/2
2
5
3
I
2
2
1
3
6
4
I
3
1/2
1/3
1
4
2
I
4
1/5
1/6
1/4
1
1/3
I
5
1/3
1/4
1/2
3
1Table 9
ANP judgment matrix of the system (based onC3).
I
1
I
2
I
3
I
4
I
5
1
2
1/2
3
5
1/2
1
1/3
2
4
2
3
1
4
7
1/3
1/2
1/4
1
3
1/5
1/4
1/7
1/3
1Table 10
ANP judgment matrix of the system (based onI1).
I
2
I
3
I
4
I
5
I
2
1
4
2
6
I
3
1/4
1
1/2
3
I
4
1/2
2
1
5
I
5
1/6
1/3
1/5
1Table 11
ANP judgment matrix of the system (based onI2).
I
1
I
3
I
4
I
5
I
1
1
3
2
5
I
3
1/3
1
1/2
3
I
4
1/2
2
1
4
I
5
1/5
1/3
1/4
1Table 12
ANP judgment matrix of the system (based onI3).
I
1
I
2
I
4
I
5
I
1
1
1/2
1
3
I
2
2
1
2
4
I
4
1
1/2
1
3
I
5
1/3
1/4
1/3
1Table 13
ANP judgment matrix of the system (based onI4).
I
1
I
2
I
3
I
5
I
1
1
1/2
3
5
I
2
2
1
4
6
I
3
1/3
1/4
1
3
I
5
1/5
1/6
1/3
1Table 14
ANP judgment matrix of the system (based onI5).
I
1
I
2
I
3
I
4
I
1
1
1/2
3
5
I
2
2
1
4
7
I
3
1/3
1/4
1
3
I
4
1/5
1/7
1/3
1In the ANP model of the overall supply level of the urban public infrastructure system, evaluation criteria for economic growth (C1), social welfare (C2), and environmental pressure (C3) evaluation criteria are used to identify the interactions among the subsystems of urban public infrastructure in the network layer, and the corresponding judgment matrix is established, as shown in the table below. The relationship between the judgment indicators in the table is based on a 9-point method; 1–9 indicates that the influence of one subsystem on the other subsystem has gradually increased. The score of the judgment matrix is taken as the mean of the expert score. The consistency test result of the judgment matrix is 0.056 (<0.1), indicating that the judgment matrix passes the consistency test.When all the judgment matrices pass the consistency test, the ANP supermatrix, the weighted supermatrix, and the limit matrix of the urban public infrastructure system supply level evaluation model are generated using Super Decisions software. Since the limits in the limit matrix converge and are unique, the weights of the interdependent subsystems in the model of the supply level of urban public infrastructure are obtained, which isWi = (0.5396, 0.2970, 0.1634). The weights of the subsystems in the network layer are (0.2075, 0.3270, 0.1386, 0.2040, and 0.1229).


## 3.1.2. Entropy Method

Through the collection, processing, and standardization of the statistical yearbook data used in the evaluation indexes of the five subsystems in Beijing, the entropy weight of the evaluation index is calculated using the entropy method (see Table1).


## 3.1.3. The Weights in Each Indicator System Are Calculated Using a Combined Weight Method Based on Game Theory

The weight of the evaluation index of each subsystem obtained by the entropy method isuIEW. The weight of the evaluation index obtained by the ANP method is uANP. Using the combined weight method based on game theory combined with the objective evaluation of the entropy method and the subjective evaluation of ANP, the weight of the optimal urban public infrastructure evaluation index is expressed as ucw∗ (as shown in Table 1).


## 3.1.4. The Evaluation Value of the Supply Level of the Typical Urban Public Infrastructure System Is Obtained

The evaluation value of the urban infrastructure system supply level is obtained by using equation (11) (Table 5). The evaluation value of the urban infrastructure system from 2000 to 2016 gradually increased the supply level of the urban infrastructure systems from 0.1383 to 0.6372, with a growth rate of 78.3%.


## 3.2. Calculation of the Evaluation Value of the Urban Economic, Social, and Environmental System

Using the entropy method and the AHP method on the urban economic, social, and environmental system indicators shown in Table2, the index weights are obtained by using a combined weight method based on game theory. Then, the evaluation value of the demand level of the economic, social, and environmental system in Beijing is obtained (Table 5). The demand level of the urban economic, social, and environmental system has gradually increased from 2000 to 2016, from 0.1590 to 0.7097, with a growth rate of 77.6%.The time series of the evaluation values of the supply level of the urban public infrastructure system and the demand level of the urban economic, social, and environmental system from 2000 to 2016 can be divided into four stages. The evaluation value of the urban public infrastructure system from 2000 to 2005 is very close to the demand level of the urban economic, social, and environmental system. From 2006 to 2011, the supply level of the urban infrastructure system is greater than the demand level of the urban economic, social, and environmental system. From 2011 to 2015, the supply level of the urban public infrastructure system is close to the evaluation level of the demand level of the urban economic, social, and environmental system. In 2016, the demand level of the urban economic, social, and environmental system is greater than the supply level of the urban public infrastructure system. During the 11th Five-Year Plan period, the average annual growth rate of urban infrastructure fixed asset investment reached 27%, which is much higher than 7.29% in the 15th period and 7.35% in the 12th Five-Year Plan period. As a result of the global financial crisis, Beijing’s GDP had negative growth in 2009, which explains why the supply of the urban public infrastructure system from 2006 to 2010 is higher than the demand for the urban economic, social, and environmental system.


## 3.3. Analysis of the Coupling Degree of the Level of Supply of the Urban Public Infrastructure System and the Urban Economic, Social, and Environmental System

Table15 shows that the degree of coordination between the urban public infrastructure system and the urban economic, social, and environmental system increases from 0.9976 to 0.9986 from 2000 to 2016, and the change is relatively stable. However, the coupling coordination degree D increases from 0.3851 to 0.8200, and a significant change occurs. The third column of Table 15 shows that the coupling and coordination degree has gone through three stages from 2000 to 2016, namely, the unbalanced development stage from 2000 to 2003, the microbalance stage from 2004 to 2005, and the balanced development stage from 2006 to 2016.Table 15
Degree of coordination between the urban public infrastructure system and urban economic, social, and environmental system.
Years
Coordination degreeC
Coupling coordination degreeD
Coordination level
2000
0.9976
0.3851
Mild imbalance development
2001
0.9983
0.4214
Slightly unbalanced development
2002
0.9999
0.4773
Slightly unbalanced development
2003
0.9999
0.4943
Slightly unbalanced development
2004
0.9961
0.5413
Slightly unbalanced development
2005
0.9998
0.5721
Slightly unbalanced development
2006
0.9986
0.6478
Mild imbalance development
2007
0.9950
0.6757
Mild imbalance development
2008
0.9922
0.6850
Mild imbalance development
2009
0.9965
0.7128
Moderately balanced development
2010
0.9958
0.7315
Moderately balanced development
2011
1.0000
0.7394
Moderately balanced development
2012
1.0000
0.7458
Moderately balanced development
2013
0.9999
0.7728
Moderately balanced development
2014
0.9995
0.8003
Highly balanced development
2015
1.0000
0.8302
Highly balanced development
2016
0.9986
0.8200
Highly balanced developmentTo determine the influence of the evaluation values of two systems on the coupling coordination degreeD, this paper establishes a multivariate linear regression model that uses the coupling coordination degree D as the explained variable and the evaluation values of the urban public infrastructure system and the urban economic, social, and environmental system as explanatory variables. The model fitting results shown in Tables 16 and 17 are obtained by SPSS software. The fitting results show that the R2 fit of the model is 0.995, which is very close to 1, indicating that the model has a high degree of fit. The model is shown in Table 17. When the supply level of the urban infrastructure system increases by one unit, the coupling coordination degree D increases by 0.555. When the demand for the urban economic, social, and environmental system increases by one unit, the coupling coordination degree D increases by 0.244.Table 16
Summary of multiple linear regression models.
Model
R
R Side
Adjust theR side
Standard estimated error
1
0.997a
0.995
0.994
0.0111601
aPredictor (constant): gy, fx.Table 17
Multiple linear regression coefficientsa.
Model
Nonstandardized coefficient
Standard coefficient
t
Sig.
B
Standard error
Trial version
1 (constant)
0.293
0.008
38.834
0.000
f
x
0.555
0.054
0.707
10.224
0.000
g
y
0.244
0.057
0.299
4.322
0.001
aDependent variable: D.


## 3.4. Analysis of the Dynamic Coupling Coordination Degree between the Urban Public Infrastructure System and the Urban Economic, Social, and Environmental System

According to the evaluation value of the supply level of the urban infrastructure system (fx) and the urban economic, social, and environmental system (gy) shown in Table 5, we can obtain the evaluation curves and fitting functions of the supply level of the urban infrastructure system (fx) and the evaluation value of the urban economic, social, and environmental system (gy). All of these are presented in Figure 5. The dynamic degree of coupling coordination between the urban public infrastructure system and the urban economic, social, and environmental system is obtained by using equation (16)–18 (Figure 6).Figure 5
Trend function of the Beijing urban infrastructure system and the urban economic, social, and environmental system.Figure 6
Degree of the dynamic coupling coordination between the urban infrastructure system and the urban economic, social, and environmental system in Beijing.Figure6 shows that the degree of the dynamic coupling coordination between the urban public infrastructure system and the urban economic, social, and environmental system in Beijing can be divided into three stages. The first phase in 2000 shows that the degree of the dynamic coupling coordination between the two systems is 32.15° and that the two systems are in the primary coordinated development stage. From 2001 to 2009, the degree of the dynamic coupling coordination between the two systems ranges from 48.96° to 83.55°; this is the second stage. At this stage, the two systems are in a symbiotic and coordinated development stage. The two systems begin to interact with each other, and the supply level of the urban public infrastructure system increases the demand level of the urban economic, social, and environmental system. Because of the influence of the global financial crisis in 2008, from 2010 to 2016, the degree of the dynamic coupling coordination of two systems changed from 81.77° to 56.26°, which represents a low level of coexistence and the state of low interaction between the two systems.


## 3.5. Analysis of the Influencing Factors of the Degree of Coupling between the Urban Public Infrastructure System and the Urban Economic, Social, and Environmental System

The degree of the coupling between the urban public infrastructure system and the urban economic, social, and environmental system is affected by many factors. This paper mainly examines the impact of 20 evaluation indicators of the urban public infrastructure system and 12 evaluation indicators of the urban economic, social, and environmental system. Using SPSS software, a linear regression analysis was performed between coupling degreeD and 36 evaluation indicators of the two systems.As shown in Table18, the degree of the coupling coordination between the urban infrastructure system and the urban economic, social, and environmental system is most affected by urban sewage treatment capacity. In Table 18, it can be seen that in the 20 indicators in the urban public infrastructure system, the top five influencing factors on the degree of the coupling coordination between two systems are sewage treatment capacity, natural gas ratio, and per capita heating area, total number of urban sanitation special vehicles and equipment, household garbage clearance volume, and bureau number of post and telecommunications. The urban infrastructure system indicators that have a weaker impact on the degree of coupling coordination between the two systems are mainly railway density and highway facilities, the amount of postal and telecommunications services per capita, and the number of public transportation lines and passenger traffic on public transportation, per capita annual water supply. Table 19 shows that in the urban economic, social, and environmental system, the most influential indicator of the degree of coupling coordination are the regional GDP actual value. The next indicators are three social development indicators: the number of employees in the industry, the per capita household disposable income, and per capita consumption expenditures. The last indicator is actual utilization of foreign direct investment.Table 18
Impact of the 20 indicators of the urban public infrastructure system on the degree of coupling coordination.
Indicator layer
Linear regression equation
R
2
Sort
Urban public infrastructure system
I
11: per capita social electricity consumption
Y
=
0.382
+
0.432
x
0.807
8
I
12: per capita energy consumption
Y
=
0.604
+
0.103
x
0.042
I
13: natural gas ratio
Y
=
0.310
+
0.468
x
0.896
2
I
14: per capita heating area
Y
=
0.341
+
0.468
x
0.942
2
I
21: railway road facility density
Y
=
0.449
+
0.329
x
0.911
13
I
22: per capita urban road area
Y
=
0.524
+
0.233
x
0.163
I
23: number of public transportation lines
Y
=
0.483
+
0.361
x
0.686
11
I
24: bus operation passenger volume
Y
=
0.476
+
0.361
x
0.789
11
I
31: per capita park green area
Y
=
0.425
+
0.390
x
0.941
10
I
32: sewage treatment capacity
Y
=
0.428
+
0.534
x
0.862
1
I
33: total number of urban sanitation special vehicle and equipment
Y
=
0.406
+
0.457
x
0.898
3
I
34: household garbage clearance volume
Y
=
0.439
+
0.454
x
0.951
4
I
41: per capita annual water supply
Y
=
0.784
−
0.441
x
0.943
14
I
42: density of water supply pipe
Y
=
0.432
+
0.404
x
0.898
9
I
43: per capita comprehensive water production capacity
Y
=
0.821
−
0.401
x
0.597
I
44: sewage pipe density
Y
=
0.468
+
0.434
x
0.858
7
I
51: per capita total postal and telecommunications volume
Y
=
0.527
+
0.358
x
0.532
12
I
52: mobile phone penetration rate
Y
=
0.421
+
0.442
x
0.841
6
I
53: mailine penetration rate
Y
=
0.677
−
0.060
x
0.015
I
54: bureau number of post and telecommunications
Y
=
0.558
+
0.444
x
0.612
5
Y represents the coupling coordination degree D, and x represents the corresponding evaluation indicator of urban public infrastructure system in the table.Table 19
Influence of 12 indicators of urban economic social and environmental system on coupling coordination degree.
Indicator layer
Linear regression equation
R
2
Sort
S
11: regional GDP actual value
Y
=
0.425
+
0.434
x
0.950
1
S
12: the proportion of the tertiary industry
Y
=
0.538
+
0.288
x
0.241
S
13: actual utilization of foreign direct investment
Y
=
0.511
+
0.402
x
0.779
5
S
14: fixed assets investment ratio
Y
=
0.854
−
0.365
x
0.669
8
S
21: per capita household disposable income
Y
=
0.498
+
0.416
x
0.818
3
S
22: per capita consumption expenditure
Y
=
0.504
+
0.413
x
0.785
4
S
23: number of employees in three industries
Y
=
0.416
+
0.429
x
0.978
2
S
24: urbanization rate
Y
=
0.411
+
0.361
x
0.934
7
S
31: annual average of inhalable particles
Y
=
0.858
−
0.387
x
0.783
9
S
32: total wastewater discharge
Y
=
0.488
+
0.388
x
0.860
6
S
33: general industrial solid waste production
Y
=
0.808
−
0.236
x
0.195
S
34: sulfur dioxide emissions
Y
=
0.846
−
0.349
x
0.476
Y represents the coupling coordination degree D, and x represents the corresponding evaluation indicator of the urban economic, social, and environmental system in the table.


## 4. Conclusion

Research on the relationship between the urban public infrastructure system and the urban economic, social, and environmental system is important for the effective management of integrated urban public infrastructure systems and the long-term sustainable development of the city. Taking Beijing as an example, this paper empirically studies the degree of coupling coordination between two system using data from 2000 to 2016. This study finds the following:(1)
From 2000 to 2016, the comprehensive evaluation value of Beijing’s urban public infrastructure system and the urban economic, social, and environmental system gradually increased, and the two values are relatively close. From 2006 to 2011, due to the impact of the financial crisis on Beijing’s economic development level and the increase of urban infrastructure investment during the Eleventh Five-Year Plan in Beijing, the supply level of Beijing’s infrastructure system was higher than that of the urban economic, social, and environmental system. The supply level of urban public infrastructure system was slightly lower than the level of urban economic and social environment development in 2016.(2)
The analysis of the degree of coupling coordination between the Beijing infrastructure system and the urban economic, social, and environmental system shows that it is gradually increasing; the supply level of the urban public infrastructure system has a significant impact on the coupling coordination degree of the two systems.(3)
The analysis of the degree of dynamic coupling coordination between the Beijing urban public infrastructure system and the urban economic, social, and environmental system shows that these two systems were in dynamic coupling coordination states from 2000 to 2009. The global financial crisis affected the degree of dynamic coupling coordination between the two systems, and the two systems entered a low-level symbiotic coupling stage from 2010 to 2016 mainly because the growth rate of urban public infrastructure system supply is negative. This indicates that the dynamic coupling coordination development between two systems can be realized only by accelerating the improvement of urban infrastructure supply in Beijing.(4)
To promote the coupling degree of the two systems in Beijing, the government should give priority to an improvement in the supply level of urban environmental infrastructure, especially an improvement in sewage treatment capacity, the number of urban sanitation special vehicles and equipment, and household garbage clearance volume. Furthermore, improving Beijing’s GDP and the social development level, such as the number of employees in three industries in Beijing, has a significant impact on the coupling coordination degree.On the basis of previous studies, this paper actively discusses the integration and management of urban public infrastructure systems and the coordinated development of urban public infrastructure systems and urban economic, social, and environmental systems. We hope that this research has a positive effect on urban public infrastructure systems by helping them meet their development needs. However, restrictions regarding the availability of relevant statistical data on Beijing affected the evaluation of the supply level of the urban public infrastructure system in Beijing to some extent. This is a problem that should be emphasized in future research.



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*Source: 8206902-2019-09-25.xml*