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地理学报, 2022, 77(4): 1028-1038 doi: 10.11821/dlxb202204017

区域发展

基于最优供需分配的公共设施空间可达性分析

翟石艳,1, 何新新2, 孔云峰,1, 罗静静2, 宋根鑫2

1. 河南大学黄河中下游数字地理技术教育部重点实验室,开封 475000

2. 河南大学地理与环境学院,开封 475004

Measuring the spatial accessibility of public services by optimal supply-demand allocation

ZHAI Shiyan,1, HE Xinxin2, KONG Yunfeng,1, LUO Jingjing2, SONG Genxin2

1. Key Laboratory of Geospatial Technology for the Middle and Lower Yellow River Regions, Ministry of Education, Henan University, Kaifeng 475000, Henan, China

2. College of Geography and Environmental Science, Henan University, Kaifeng 475004, Henan, China

通讯作者: 孔云峰(1967-), 男, 教授, 博士生导师, 主要从事空间分析、空间优化等研究。E-mail: yfkong@henu.edu.cn

收稿日期: 2020-12-18   修回日期: 2021-08-18  

基金资助: 国家自然科学基金项目(41871307)
河南省高等学校重点科研项目(21A170007)

Received: 2020-12-18   Revised: 2021-08-18  

Fund supported: National Natural Science Foundation of China, No(41871307)
Key Scientific Research Project of Henan Institution of Higher Education, No(21A170007)

作者简介 About authors

翟石艳(1983-), 女, 副教授, 博士生导师, 主要从事空间分析、空间优化和空间行为决策等研究。E-mail: zsycenu@hotmail.com

摘要

空间可达性是衡量公共服务设施公平性的重要指标,在医疗、教育、休闲等公共服务的布局规划中得到广泛应用。然而已有设施服务可达性模型难以充分反映服务供需关系,计算指标也缺乏物理意义。本文提出新的最优供需分配的公共设施空间可达性计算方法（OSD）取代现有方法。该方法基于最优供需分配模型,将设施服务分配给需求者,根据分配结果计算空间可达性指标。给定服务设施与需求的空间分布,以最小化旅行成本为目标,顾及设施服务能力,采用经典的运输问题模型确定最优的服务供需分配方案,进而度量服务的空间可达性。以郑州市金水区社区卫生服务为例,求解25个中心与1333个居住小区的最优服务配置。使用最优配置结果确定每个设施的服务范围、每个居住小区使用服务的旅行时间,以及特定时间阈值的服务覆盖比率。与流行的两步移动搜索法相比,新方法的计算指标具有明确的物理意义。本文提出的可达性评价方法无需参数,计算高效,结果易于解释,在公共服务评价及设施布局规划方面具有应用潜力。

关键词： 公共服务; 空间可达性; 最优供需分配; 案例研究

Abstract

Spatial accessibility is an important index for measuring fairness in public services such as health care, education, and outdoor leisure. It has also been widely used in the layout planning of public facilities. However, the existing accessibility models cannot fully illustrate the direct relationship between service supply and demand, and thus, the accessibility indicators cannot be explained with explicit meanings. This article introduces a new method for measuring accessibility based on optimal supply and demand allocation, which is different from the supply-demand relationships in existing methods. Given the supply locations, the demand locations, their attributes, and the travel costs between them, the transportation problem model is used to allocate services to demands. The classical model can be easily solved with optimal total travel cost. Based on the model results, the spatial accessibility of services can be measured. The proposed method was tested on healthcare services in a district in Zhengzhou, Henan, China. There were 25 healthcare centres and 1333 residential communities in the study area. The instance model was effectively built by an open-source MIP solver in 2.2 seconds. The service area of each healthcare centre and the travel time from each community were obtained according to the modelling results. In addition, the coverage percentages with predefined service thresholds were calculated. The proposed method was also compared with popular methods, such as the two-step floating catchment area method (2SFCA), Gaussian 2SFCA, and gravity 2SFCA. Areas with poor healthcare services were identified by using the results from the three 2SFCA methods. However, the spatial variation in service accessibility might be very different when selecting different search thresholds and/or distance decay coefficients. The optimal supply-demand accessibility method (OSD) proposed in this study has advantages such as being parameter-free, easy to calculate, and easy to explain. The authors believe that the method has application potential to replace the existing methods in evaluating public services and facility planning.

Keywords： public service; spatial accessibility; optimal service allocation; case study

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本文引用格式

翟石艳, 何新新, 孔云峰, 罗静静, 宋根鑫. 基于最优供需分配的公共设施空间可达性分析. 地理学报, 2022, 77(4): 1028-1038 doi:10.11821/dlxb202204017

ZHAI Shiyan, HE Xinxin, KONG Yunfeng, LUO Jingjing, SONG Genxin. Measuring the spatial accessibility of public services by optimal supply-demand allocation. Acta Geographica Sinice, 2022, 77(4): 1028-1038 doi:10.11821/dlxb202204017

1 引言

在社会经济加速转型发展中的中国,城市居民对生活质量的追求从基础的物质条件转向更深层次的生活品质。然而,随着城镇化的快速推进,城市人口快速增长,致使城市公共服务失衡,城市义务教育、基础医疗、公共交通、绿色空间等资源供应不足,且在城市内部存在空间供应和结构性差异等问题[1]。“基本公共服务均等化水平明显提高”是中国“十四五”时期经济社会发展主要目标之一,因此,评价及改进城市公共服务设施空间布局、服务质量和公平性具有现实意义。

可达性是评价公共服务公平性的一个重要指标。Hansen于1959首次提出可达性概念,即“网络中各节点相互作用的机会大小”,基于此研究可达性与城市土地利用之间的关系[2]。随着地理空间技术的快速发展,部分学者从空间角度对可达性的概念进行解读,定义“空间可达性”为“基于某种交通方式从一个地方到一个目的地的便捷性”[3,4]或“互动机会的潜力”[5,6]。这一概念逐渐成为衡量公共服务设施公平性的重要依据,被广泛的应用于医疗设施[7,8,9]、公共绿色空间[10,11,12,13,14]、教育设施[15]、交通设施[16,17,18]的公平性和空间布局的研究中[19,20,21,22,23]。目前,空间可达性测度方法主要包括供需比法、最近距离法、重力模型法、累积机会测度和两步移动搜索法（Two-step Floating Catchment Area Method, 2SFCA）等[19,20,21,22,23,24,25,26]。这些方法具有计算简单、使用方便的特点,得到了广泛应用,但是也存在各种不足。例如,供需比指标仅可以反映区域间的公共服务设施可达性的差异[27]。最近距离法假设居民总是选择最近的公共服务设施,忽略设施的服务能力[28]。2SFCA方法较好地考虑了供给和需求,但出行时间或距离阈值的设定对结果影响较大[8, 11, 26]。

因此,本文提出了一个基于最优供需分配的公共设施空间可达性测算方法（Optimal Supply-demand Accessibility Method, OSD）,试图取代现有方法。基本思路是：利用优化模型将设施服务分配给需求者,根据分配结果计算空间可达性指标,并与传统的2SFCA可达性计算结果进行对比分析。案例实验表明所提出的可达性计算方法计算效率高,统计指标物理意义明确,适用于公共服务设施的可达性分析与布局优化改进。

2 文献回顾

已有公共服务设施空间可达性分析方法主要包括供需比法、最近距离法、潜力模型法、2SFCA等。供需比法是测度一定地域范围内（如以地级市、县为单元）设施的供给量与需求量的比值[27]。该方法假定,一定地域范围内的设施只服务于该范围内的人,不考虑边界范围以外的人被服务情形。所得的比值越大,说明该地域范围内设施的总体可达性越好。该方法虽然综合考虑了供给方和需求方的量的大小,但无法揭示所研究单元内部可达性的空间差异,而且也无法体现供给双方跨地域单元的相互作用。主要评估大尺度省、市、县之间以及城乡之间的公共服务资源分布差异。

最近距离法是通过计算研究对象到最近设施的距离来测度研究可达性。该方法假设人们总是喜欢访问离自己最近的设施[28]。常用的距离度量包括欧式距离、路网距离和时间距离。其中,欧式距离是研究对象和设施点之间的直线距离。该方法计算比较简单,思路清晰,但是无法反映供应和需求之间的匹配比,也无法反映其他设施点对研究对象的可达性。

潜力模型法考虑了研究目标点和设施间的空间相互作用[2]。该方法假设目标点的可达性与设施的服务能力（供给规模,如医生数量、医院病床数）成正比,与两者间的距离成反比。早期,该模型忽略了需求点的属性,如人口规模。另外,距离衰减系数的设置对计算结果影响较大。Joseph等[27]和Shen[29]在重力模型基础上,加入了对需求方的考虑,所构建的可达性模型可以反映供应方和需求方之间的关系。程敏等[30]对研究医疗设施可达性应用中,对设施周围服务人口与设施服务能力之间的竞争纳入考虑范围。

2SFCA包括简单2SFCA、增强E2SFCA、可变搜寻区V2SFCA、广义G2SFCA、重力2SFCA等,综合考虑供给和需求,通过设定出行时间或距离阈值,计算特定地点可访问设施（或资源）的数量。阈值内的设施越多,可达性越好。简单2SFCA用二分法来处理距离衰减,仅考虑了阈值范围内的供给点,阈值范围以外的供给点未予以考虑。并假设阈值范围内的供给点,无论距离远近,对需求点的可达性有着同等的影响。之后,部分学者采用出行时间分段赋予权重[31,32]、可变搜寻区[33,34]、核密度函数[35]和高斯函数[36]等来刻画距离衰减问题。重力2SFCA将各种距离衰减函数（幂函数型、指数型和对数型等）引入到2SFCA中[37,38]。2SFCA的问题在于阈值时间或距离设定,以及距离衰减参数β的取值。参数β,阈值时间或距离设定不同,结果差异较大[31-32, 39-40]。

旅行时间/距离是空间可达性计算的关键因子之一。通常受到两个主要因素的影响,一是不同交通模式的选择,二是道路的实际通行速度。传统情况下,距离常采用静态的供给点和需求点间的起止点（Origin-destination, OD）距离来衡量。随着大量开源的交通时空大数据的涌现,部分学者[39,40]结合互联网地图服务平台所公开的实时电子网络地图数据和道路导航数据,利用实时交通时间来代替静态网络距离,更加准确的刻画旅行时间,进而对传统的空间可达性方法进行修订。Rong等[39]发展了基于地图API的潜力模型,刻画居民点和医疗设施点间的实时交通距离,评估郑州市医疗设施空间布局的合理性。王奇[40]以上海市为例,探讨了在互联网地图服务支持下,城市多交通模式和不同交通出行方式下的城市医疗服务的可达性。但是,通常存在与公共服务设施和需求方相关的高质量时空大数据难以获取等方面的问题。

综上,已有模型对于供需关系的表达不够充分。供需比法仅计算一定区域内整体服务供需比,不能反映局部差异。最近距离法考虑距离影响,而不涉及供需。潜力模型法和2SFCA在衡量公共设施空间可达性时,仅考虑局部空间范围内的服务供给和需求,局部空间的大小受时间或距离阈值设定影响很大。现实中,区域内不同区位实际获取服务的距离差异较大,固定时间或距离阈值,难以反映真实服务供需关系。因此,有必要提出一个能充分反映供需关系,且物理意义明确的公共设施可达性计算方法,用于公共服务设施的可达性分析与布局优化改进。

3 最优供需分配模型

本文基于最优供需分配结果计算空间可达性。在一个地理区域内,令集合V ={1, 2, …, m}表示m个需求点,每个需求点上的属性di表示需求量;集合U ={1, 2, …, n}表示n个供给点,每个供给点上的属性qj表示服务供给量;变量cij表示需求点i和供给点j之间的距离。引入决策变量xij表示设施j为需求点i提供的服务量,可构建最优供需分配线性规划模型：

Minimize∑i∈V∑j∈Ucijxij
Subject to∑j∈Uxij=di,∀i∈V
∑i∈Vxij≤qj,∀j∈U
xij=0,1,2,…,∀i∈V,∀j∈U

式（1）为目标函数,表示最小化旅行成本;式（2）保证对于任一需求点,到所有供给点访问的需求量等于该点总需求量;式（3）保证每一个设施分配的服务量不超过其供给量。

式（1）~（4）本质上是一个运输问题模型。运输问题是把某种商品从若干个产地运至若干个销地而使总运费最小的一类问题,是运筹学中的经典问题之一。该问题具有特殊的结构,即设施与需求构成二部图、约束条件矩阵是全么模矩阵,从而使该问题的求解较为容易。首先,可以使用线性规划方法进行求解,其最优解能够保证变量xij为整数;同时,也可以采用更简单的表上作业法进行求解,如西北角法（左上角法）、元素差额法（VAM）。基于这些方法获得初始解,进一步采用闭回路法[41]、位势法[42]获得最优解。

基于模型求解结果进行空间可达性测算。运输问题模型以最优的方式将设施服务分配给需求者。根据模型计算获得的供需关系,可以灵活地统计公共设施服务指标,并制作各种专题地图。服务指标主要包括：人均使用服务旅行时间、各需求点使用服务旅行时间、各设施使用者的平均旅行时间、特定半径内的服务覆盖率等。这些指标不仅可以评价各区位获取服务的便捷程度,也可以判断设施服务能力是否充足、设施区位是否合理。

应当注意,以上模型假设设施服务能力大于总需求。若这一条件无法满足时,模型无解。针对这一情况,可考虑两种解决方案。一是假设服务设施超负荷运行,按比例调整供给点的服务能力,使服务供应量等于或大于总需求;二是假设部分需求不能得到满足,调整模型求解,模型修订如下：

Minimize∑i∈V∑j∈Ucijxij
Subject to∑j∈Uxij≤di,∀i∈V
∑i∈Vxij=qj,∀j∈U
xij=0,1,2,…,∀i∈V,∀j∈U

作者在Python 3.8环境中,开发了基于最优供需分配的公共设施空间可达性计算工具。该工具读取在GIS中准备好的数据,建立案例模型,然后使用开源CBC MIP求解器（ https://github.com/coin-or/cbc）进行模型求解。该工具及实验数据见 https://github.com/trirumisu/OSD。

4 案例分析

4.1 案例数据

本文以郑州市金水区为例,利用基层社区卫生服务中心和小区尺度人口数据,采用基于最优供需分配和2SFCA方法分别进行可达性计算,并对比两者结果。本文暂没考虑除了基层社区卫生服务中心之外的高等级医疗资源。郑州市金水区总面积约242 km2,全区辖24个街道,1333个小区,2020年全区人口约161万,25个社区卫生服务中心。通过实地调查,获得每个社区卫生服务中心服务范围和小区尺度人口数据。收集案例区2020年街道边界、社区卫生服务中心空间分布、居住小区空间分布等数据,在ArcGIS中整理为地理图层（图1）。

图1

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图1   案例区示意图

Fig. 1   The study area in Zhengzhou, China


根据郑州市金水区25个社区卫生服务中心实际服务情况,计算服务中心和实际服务小区的OD时间成本矩阵,统计步行15 min阈值内各小区点获取基层医疗服务空间可达性（图2）。研究发现,只有距离服务中心较近的小区居民可在15 min内到达社区卫生服务中心,占比46.57%,超过一半的居民（53.43%）无法到达。

图2

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图2   2020年郑州市金水区医疗服务现状

Fig. 2   Healthcare centres and their service areas in Jinshui District, Zhengzhou, 2020


参考社区卫生服务中心服务能力标准（2018年）[43],建筑标准为1400 m2,服务人口3万~5万人;建筑标准为1700 m2,服务人口为5万~7万人;建筑标准为2000 m2,服务人口7万~10万人。结合案例区卫生服务中心服务能力各项指标（建筑面积、床位设置、人员配备等）,如果将每个服务中心的服务人口均提高至服务能力上限,可计算每个服务中心的供需匹配差额（图3）。10个服务中心存在明显供给量不足,个别中心的供给量缺口较大,如22号、4号、9号和12号中心,其中9号服务中心缺口甚至达到60%。14个服务中心存在明显供给量冗余,其中25号和17号服务中心的冗余分别达到57.1%和46.5%。

图3

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图3   郑州市金水区医疗服务冗余或短缺

Fig. 3   Service redundancy or shortages for healthcare centres in Jinshui District, Zhengzhou


4.2 基于最优供需分配的可达性分析

基于最优供需分配评估服务可达性,具体步骤为：① 在ArcGIS中准备案例数据,包括社区卫生服务中心空间位置及其服务能力,居住小区空间位置及其人口规模。② 计算供应点和需求点的曼哈顿距离,获得OD时间成本矩阵。③ 构建最优供需分配模型,使用CBC开源线性规划求解器对模型求解。④ 计算可达性指标,包括每个需求点使用服务旅行时间、特定半径内（≤ 5 min、5~10 min、10~15 min、>15 min）的服务覆盖率等。

实验计算环境：HP桌面计算机,配置Inter® CoreTM i7-7700 CPU和8 GB内存,Window 10操作系统,安装Python 3.8、ArcGIS 10.2和CBC开源线性规划求解器。CBC模型求解时间为2.2 s,获得最优解。

案例区郑州金水区只有46.57%的居民可以在15 min步行范围内到达社区卫生服务中心（表1）,该比例远远低于国家所规定的80%的居民“15 min到达医疗点”的目标[44]。利用本文所提出的基于供需最优分配的就医可达性计算方法,分别按照现有每个服务中心的服务能力和最大上限的服务能力,对服务中心的覆盖范围和居民就医的服务中心进行重新分配,可达性计算结果如图4所示。由表1和图4可知,依据服务中心的现有服务能力,重新匹配服务覆盖范围,15 min内到达服务中心的居民数明显提高,占比53.22%;若依据国家社区卫生服务中心建设标准,将服务中心的现有服务能力提高至上限,15 min内到达服务中心的居民数达到54.86%。相对服务中心现状而言,基于空间服务最优分配的可达性方法,有效缓解了供需平衡问题。

表1   郑州市金水区不同时间阈值下医疗服务覆盖统计

Tab. 1  The coverage percentages of healthcare services within different time thresholds in Jinshui District, Zhengzhou

分类(min)实际服务能力最优分配最大服务能力最优分配服务现状
人数(人)占比(%)人数(人)占比(%)人数(人)占比(%)
≤ 51340388.291414428.751204197.45
5~1036800922.7638578923.8632168919.89
10~1535849422.1735984322.2531097319.23
> 1575653846.7873000545.1486399853.43

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图4

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图4   郑州市金水区医疗服务最优分配结果

Fig. 4   The optimal allocation of healthcare services in Jinshui District, Zhengzhou


4.3 与两步移动搜索法对比

本文采用2SFCA计算需求点的可达性,首先,以服务中心为核心,计算15 min步行阈值范围内所有居民点,计算搜寻区内的卫生服务中心和人口数之比;第二步,以每个小区为中心,搜寻阈值时间范围内的服务中心数量,将搜寻区内的各服务中心的比率加和,得到每个小区的就医可达性。

基于简单2SFCA、高斯2SFCA和重力衰减2SFCA分别计算步行15 min（步行速度=72 m/min）时间阈值内各小区点获取基层医疗服务空间可达性。对重力衰减2SFCA模型而言,距离衰减参数β对其可达性结果影响较大。参考文献[40, 45-46],本文对β参数取1、1.2、1.5、1.8、2这5种情况。为便于分析,本文根据自然断点法,将可达性计算结果分为差（≤ 0.4）、较差（0.4~0.8）、一般（0.8~1.2）、较好（1.2~1.6）和好（> 1.6）,并对以上6种模型的结果进行对比分析。图5显示不同方法计算的可达性结果呈现出类似的空间特征,案例区“15 min”基层医疗服务可达性呈现出明显的圈层结构。人口密集的文化路、丰产路、京八路、花园路街道所在的老城区可达性较好,而外围街道的医疗可达性较差。尤其是郑东新区基层医疗服务中心较少,小区“15 min”步行获取基层医疗服务可达性较差,部分小区的可达性为0。

图5

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图5   基于2SFCA“15 min”步行搜索半径下郑州市金水区医疗服务可达性分布

Fig. 5   Maps of healthcare services accessibility assessed by 2SFCA within a "15-minute" walking search radius


表2显示,基于简单2SFCA,“15 min”步行搜索半径下基层医疗服务空间可达性良好的居民,占比为30.89%;高斯2SFCA显示,医疗服务空间可达性良好的居民,占比为32.29%。重力2SFCA计算结果随β取值不同而差异较大。随β取值增加,医疗服务空间可达性“较好”和“好”这两个级别的居民占比呈逐渐下降趋势,分别为28.57%（β = 1）、26.26%（β = 1.2）、22.37%（β = 1.5）、17.56%（β = 1.8）和15.75%（β = 2）。

对比分析基于最优供需分配（表1）和2SFCA方法的可达性计算结果（表2）,发现距离阈值和衰减参数设置不同,基于简单2SFCA、高斯2SFCA和重力衰减2SFCA的可达性结果差异较大,且只能粗略反映出需求点可达性“好”“较好”“一般”“较差”“差”的占比,物理意义不明确。基于最优供需分配的可达性,可明确反映出“15 min到达医疗点”的人数及空间分布,社区卫生中心设施供给与需求空间分布差异性,以及供给能力和需求的差异性,便于识别空间上设施可达性较差区域。

表2   基于2SFCA“15 min”步行搜索半径下医疗服务可达性统计(%)

Tab. 2  Statistics of healthcare service accessibility within a "15-minute" walking search radius by 2SFCA

可达性计算方法差(≤ 0.4)较差(0.4~0.8)一般(0.8~1.2)较好(1.2~1.4)好(> 1.4)
简单2SFCA42.824.1222.167.9422.95
高斯2SFCA51.516.639.577.9724.32
重力2SFCA(β=1)43.4317.2610.756.9521.62
重力2SFCA(β=1.2)46.0119.138.607.4418.82
重力2SFCA(β=1.5)52.9315.639.067.3415.03
重力2SFCA(β=1.8)59.9713.548.933.9313.63
重力2SFCA(β=2)63.6014.096.563.1012.65

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5 结论与讨论

本文在综合考虑公共服务设施供需不平衡前提下,采用经典的运输问题模型确定最优的服务供需分配方案,进而度量服务的空间可达性,并以郑州市金水区社区卫生服务为例,采用两步移动搜索法和最优供需分配模型,求解可达性。

案例研究表明：① 本文所提方法计算简单,求解高效,使用CBC求解器,模型计算时间约为2.2 s。也可采用CPLEX、GUROBI等商用软件计算,计算时间约为0.2~0.3 s。② 与传统的公共服务设施空间可达性度量方法（潜力模型和累积模型等）相比,最优供需分配模型在衡量设施空间可达性方面具有诸多优势。该模型运算不涉及参数选择,其计算结果物理意义明确;可灵活地用于统计公共设施服务指标（如人均使用服务旅行时间、各需求点使用服务旅行时间、各设施使用者的平均旅行时间、特定半径内的服务覆盖率等）,制作各种专题地图。有效衡量公共服务设施供给与需求空间分布差异性以及供给能力和需求的差异性,识别空间上设施可达性较差区域,能够弥补现有方法的不足。

本文方法能够用于基础医疗、义务教育、绿色空间等公共服务空间可达性评估。公共服务体系中,存在动态需求、设施变动（增减或者重新选址）和交通条件变化等情景。服务体系发生变化时,可使用本文方法重新评估其可达性,判断公共服务水平是否有所改进及是否有必要调整。也应当注意本文案例数据的局限性。在“15 min社区生活圈”内,现有社区卫生服务覆盖人口的比率明显低于国家所计划的目标,原因可能是数据采集不完备、人口数据不够精确等。为更好地满足社区卫生服务覆盖及便捷性要求,可通过以下几种方式改进：扩充现有设施的容量、增加新设施或调整现有设施位置等。本文可达性模型未考虑需求点居民的需求偏好,未来的设施布局规划及可达性评估研究,应考虑居民的需求偏好,打造高质量和人性化的公共服务体系。

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