# Spatial Efficiency Evaluation for Outdoor Environment in High-Rise Residential Area: An Data Envelopment Analysis Approach

Authors: Qian Li; Xiaorui Zhang; Xiaoyong Wan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, the data envelopment analysis (DEA) approach is put forward to deal with the spatial efficiency evaluation issue for the outdoor environment in high-rise residential (HRR) areas in China. The principles of quantifiable and objectivity are utilized to empirically research for the area of 10 high-rise residential houses. For the outdoor environment in HRR areas, our aim is to take appropriate optimization suggestions that not only satisfy the different conditions of the outdoor environment but also improve the efficiency of site space. By resorting to the data envelopment analysis technique, some practical methods are acquired for evaluating the outdoor environment for HRR areas.


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## Body

## 1. Introduction

In recent years, the residential high-rise model has been emerging in many cities, which is an important manifestation of the rapid development of China’s construction and the continuous improvement of people’s living standards. Intensive building forms effectively improve the development efficiency of the land and restrain the excessive expansion of the city. However, compared with the traditional multistorey residential areas, the high-density living form has caused many problems in the living environment, which are more prominent in the following aspects: the reduction of living space efficiency, the reduction of communication activities, the indifference of neighborhood relations, the idling of space environment, the lack of community identity, and other phenomena. Many of the above behaviors and phenomena occur in the outdoor environment of high-rise residential areas. For such a complicated, organically combined system, it is closely centered around the building, to meet residents’ daily life and communication, recreation and entertainment, and sports behavior and activities space, and it not only includes the entity’s physical location but also includes the entity’s abstract mental environment, such as local culture, local residents’ sense of security, comfort, belonging in their daily lives, and rapport with his neighbors.The residential environment is an important part of the city and accounts for the largest share of the urban living environment [1], consisting mainly of dwellings, residential buildings, public services, and outdoor spaces where residents spend their time, satisfy their needs, and participate in various activities. The residential environment can be specified by spatial and social indicators [2]. Outdoor space, as an important vehicle for residential behavior, is related to the form, shape, plan, structure, and function of the built environment [3]. Several studies have reported that the greener space residents have within their residences, the more physically active they are [4–8]. Possible explanations for such findings include the use of different criteria to assess the usability of outdoor environments and the study of the diversity of designs and settings [9–12].Various sustainable city indicators are used internationally such as Monocle’s Quality of Life Survey, Quality of Life Index (QLI), Indicators for Sustainability, European Green City Index, City Blueprint, et al. [13] and several methods of rating buildings’ effects on the environment such as BREEM, CASBEE, DGNB, HQE, LEED, and so on. Among them, the rating system of LEED-ND fails to deal strictly with important livability factors such as socio-cultural and socio-economic factors [14]. Skalicky used the REL method to construct a livability-focused residential environmental quality evaluation system using various physical elements of urban form in the residential environment as research indicators [15]. The Swedish model uses the Residential Environmental Impact Scale (REIS) to provide a valid perceptual assessment of how the residential environment affects residents (elderly and disabled people over 65 years old), reflecting the positive and negative dimensions of utility exhibited by occupational therapists in the residential environment [16–19]. Korean apartment houses were analyzed for living environment satisfaction, the covariance structure was used to establish an equation model for living environment satisfaction, and the determinants of living environment satisfaction were analyzed in combination with SPSS and AMOS software to provide the necessary conditions for determining the factors influencing living environment satisfaction in the exterior space of the house [20]: Payamiazad considers spatial openness and perimeter of fences as practical indicators for evaluating the quality of outdoor spaces in residential areas.The research on the outdoor environment of high-rise settlements by domestic scholars is more diversified in terms of dimensions and more referential based on the combination of practice. Shen scientifically constructed a comprehensive evaluation system of the built environment walkability of settlements from the subject’s human needs based on a multidimensional theoretical perspective [21]. Song applied the fuzzy integrated evaluation method to study the environmental ecological performance of urban old settlements [22]. Feng applied Depthmap, T-sun, and ENVI-met software to simulate and evaluate the spatial accessibility and microclimate environment of the external environment of settlements [23]. Shi used the entropy value method to study the spatial characteristics of public open spaces and the activity of different age groups in high-rise settlements and explored the degree of influence of spatial characteristics on the activity [24]. Wang used the hierarchical analysis method (AHP) to construct an evaluation system for the livability of urban settlements [25]. Zhu proposed an overall optimization strategy based on the theory of outdoor activity places in high-rise settlements, outdoor activity space, place and environment components, and activity population, with residents’ participation as the core [26].However, most of the analysis on the utilization rate of outdoor environment space in high-rise residential areas is superficial and lack of in-depth research on the substantive connotation of spatial ineffectiveness. In this paper, on the premise of fully considering the objective physical environment characteristics of the space and the psychological perception and experience of users, the author combined the data envelope-analysis method to objectively evaluate the effectiveness of space use from the two dimensions of material and spirit, so as to provide reference for improving the efficiency of outdoor environmental space use in residential areas.The theoretical basis of DEA can be traced back to 1978 [27–29], and the method is based on linear programming and can be used for relative performance efficiency with multiple inputs and multiple outputs [30]. Because it does not require the identification of any type of relationship between inputs and outputs [31], nor does it require any specific statistical distribution for the data of input and output variables, it has now been applied to the field of urban space research. Questioning the decision-maker preference structure of the traditional data envelopment analysis model [32] decided to use an interactive decision technology for optimization, including DEA and multiobjective linear programming (MOLP). The Zionts–Wallenius (Z-W) method was used to reflect the DM’s preference in evaluating efficiency in a general combination-oriented CCR model. The authors of [33] reestablished an equivalent model between the DEA model and the MOLP model according to their respective characteristics. A case study explained how to use the MOLP method to evaluate the efficiency of DEA. The authors of [34] established an equivalent model between MOLP and DEA using the directional distance function, which was effectively used to support the interaction process and performance measurement aimed at establishing future performance goals. The authors of [35] introduced an innovative method of decision unit ordering in DEA–cross-efficiency evaluation and proposed the proportional weight assignment technique, which selected the weights proportional to the corresponding input or output as the secondary objective of DEA–cross-efficiency evaluation. The authors of [36] focused on the group consensus ranking in the field of performance evaluation and selection, proposed a group consistency ranking method based on DEA and simulation, scored each candidate’s efficiency preference, and provided rich and effective information for decision-makers. The authors of [37] proposed a fuzzy random DEA model with bad output, applied three fuzzy DEA models of probability-possibility constraint, probability-necessity constraint, and probability-confidence constraint, and considered the influence of bad products on the performance of decision-making units. The efficiency score of all decision-making units is between 0 and 1, which effectively solves the fuzziness and randomness in the original DEA model. Zhang et al. used the DEA method to study the efficiency of spatial development in the Beijing–Tianjin region from the perspective of regional urbanization [38]; Zhang studied the efficiency of typical old settlements and guaranteed settlements in Nanjing to reflect on their socio-spatial performance and construction methods [39]; Tian and Chen used DEA to explore the relationship between the compactness of urban spatial structure and urban land use efficiency in Nanjing [40]; Li selected the PCA-DEA model to construct a performance evaluation system for the refined governance of old neighborhoods [41]. Wang based on the correlation between AHP and DEA constructed urban community fitness environment evaluation index system and hierarchical structure model, and comprehensive evaluation [42]. Lv focuses on the strategy of new rural residential design matching with the rural industrial model and carries out empirical research on the efficiency of new rural residential use based on the DEA model [43]; Wu applies the DEA method to evaluate the investment efficiency of urban residential neighborhoods and constructs the overall value of urban settlements with the goal of improving residential neighborhood harmony system [44, 45]. From the above analysis, it can be seen that data envelopment analysis (DEA) is perfectly suitable for residential space efficiency analysis [46].DEA uses a mathematical planning model [47] to evaluate the relative effectiveness of multiple input-output decision units to obtain a data envelope. Decision units that fall on the boundary are considered as valid combinations with an efficiency value of 1, while decision units that do not fall on the boundary are invalid decision units with an efficiency value between 0 and 1 [48]. This model has been developed into CCR (Charnes, Cooper, and Rhodes), BCC (Banker, Charnes, and Cooper), FG, and ST models. Considering that the spatial efficiency of the outdoor environment in high-rise residential areas is not a purely economic problem, but a relatively complex practical problem, according to the controllability of input factors in the production process, this study adopts the input-oriented model with different input combinations under the given conditions of output; that is, the CCR and BCC models of DEA are used for calculation and analysis. CCR model is used to evaluate whether the decision unit achieves scale and technical validity [49], and the BCC model is used to evaluate whether the decision unit achieves technical validity [50]. Among them, the CCR model is used to evaluate whether the DMU achieves both scale effectiveness and technical effectiveness, while the BCC model can be used to evaluate the technical effectiveness of the DMU.Compared with other statistical methods, the data envelope analysis method has significant advantages, which also means that it is more suitable for the analysis and study of spatial efficiency of the outdoor environment in high-rise residential areas. Its advantages are as follows:(1)
Data envelopment analysis method has the ability to adapt to multiinput and multioutput complex structures. The research object of this paper—the outdoor environment system of a high-rise residential area—is a complex system with many input and output indexes, and the mutual influence between indexes is objective. Therefore, it is necessary to adopt the method combining subjective and objective analysis to evaluate the investment efficiency.(2)
When measuring the relative effectiveness of decision-making units, the DEA method is not affected by the dimensionalization of input and output indicators and data. The influencing factors of spatial efficiency evaluation of the outdoor environment in high-rise residential areas are not single but need to be considered from multiple aspects, and multiple input-output indicators are summarized.(3)
Taking the input and output weight of the decision-making unit as the variable, the model adopts the optimization method to determine the weight internally and carries out an evaluation from the perspective that is most conducive to the decision-making unit. It is unnecessary to determine the various relationships between the input and output, thus avoiding the subjectivity brought by determining the weight of each index.(4)
It is a nonparametric analysis method, based on sample data, which can directly find the best efficiency value from the actual observation data of each decision unit;This method does not need to presuppose an input-output function relation in advance, so it can effectively avoid the influence of human subjective factors and has stronger objectivity.(5)
For the noneffective decision-making unit in the conclusion, the “projection principle” can not only indicate the adjustment direction of the index but also provide the corresponding adjustment quantity, which can be used for multidimensional comparison, and the spatial efficiency can be judged as invalid residential areas for quantitative adjustment and design.This paper is based on the relative efficiency of multiple-input multiple-output analysis and data envelopment analysis, with an area of 10 typical high-rise residential houses as the research object, based on the investigation data, taking specific service efficiency of spatial data for the analysis of factors, summarize and put forward the high-rise residential outdoor space habitability of quantitative design issues related conclusions. The distinctive contributions of this paper are categorized into the following three aspects. (1) A unified framework is established that is capable of coping with the design issue presence of outdoor space habitability and high-rise residential buildings. (2) A new analysis tool is introduced where multiple-input multiple-output analysis and data envelopment analysis are comprehensively applied to the relative efficiency. (3) A practical evaluation method is proposed for the environmental use efficiency of completed settlements.


## 2. Evaluation Method–Data Envelopment Analysis Model

### 2.1. Problem Formulation

Suppose there aren decision-making units (DMU), and each decision-making unit has m inputs xij (j = 1, 2, …, m), and there are s producing yir (r = 1, …, s) (xij ≥ 0, yir ≥ 0), and the relaxation variable s −  and s + are introduced. Then, the relative efficiency of the DMU is as following DEA-CRR measure evaluation model:(1)Minσ,s.t.∑i=1nxijλi+s−=θx0jj=1,2,…,m,∑i=1nyijλi+s+=y0rr=1,2,…,s,∧≥0,s+≥0,s−≥0,0<σ≤1,where λi is the weight variable of each decision unit on an index, and σ is the comprehensive technology and scale efficiency of the decision unit. The CCR model is obtained under the assumption that DMU is constant returns to scale (CRS). σ is used to calculate by DEA-CCR, as CRSTE for short.Remark 1.
If constraints∑i=1nλi=1 are added to the above formula, it becomes the DEA-BCC model, which is obtained under the assumption that DMU is variable returns to scale (VRS). σ is the pure technical efficiency of DMU (VRS).Remark 2.
VRS refers to the distance between the investigated DMU and the effective production front when the scale return is variable. Scale efficiency (SE) (S E = CRSTE/VRSIE) refers to the distance between the effective production frontier with constant return on scale and variable return on scale. CRS includes VRS and SE. S E = 1 indicates that the DMU is located at the most appropriate scale efficiency level. S E < 1 indicates that the DMU is in a state of scale inefficiency.


### 2.2. Establishment of Indicator System

The input index and output index are both included in the DEA evaluation index system. When the DEA evaluation model is set up to make the decision-making unit number and maintain the proper proportion relation between evaluation index number, it is generally believed that the number of decision-making units should be greater than or equal to 2 times the total number of input and output indicators.As a practical problem, the spatial efficiency of the outdoor environment in high-rise residential areas needs to be closely combined with the actual research content and research results. The index system adopts the model combining analysis method and comprehensive method to consider the data’s accessibility, operability, and pertinence. This paper selects indicators from two aspects of economic input and design input and selects output indicators from the user’s participation and perception. The number of decision-making units selected in this study is the outdoor environment of 10 high-rise residential areas, so 3 input indexes and 2 output indexes are set. The final results are shown in Table1.(1)
In terms of investment, the capital factor is the most basic component. Considering the various early inputs of high-rise residential areas, most of them are visually represented in real life as signs of housing price, and the specific values are based on the average market price. The outdoor environment space system in the residential area includes greening, road, hard site, facilities, and other components. Among them, greening helps to form the environmental image of space path, node, area, sign, and boundary and better create the landscape environment inside the residential area. The hard ground takes up a relatively large proportion of the area, providing residents with daily outdoor activities. The area includes the central scenic area, the leisure area, and fitness area and facilities between groups, children’s entertainment venues, and the interoperable garden paths between groups. The facilities sketch is a small building facility in the outdoor environment of a residential area for people’s viewing, decoration, and lighting and for the convenience of garden management. It is the leading element for people to carry out activities on the site. All the above elements will have an impact on the efficiency of space use. Therefore, the volume rate and the green rate of high-rise residential outdoor environments are selected for quantitative evaluation based on the comprehensive technical and economic indicators in residential planning and design.(2)
In terms of output, as an important place of urban residents’ daily life behavior and the carrier of undertaking various life service facilities, the direct output of outdoor environment production in residential areas should include meeting all the needs and services of residents’ daily life, which is directly reflected in the participation and satisfaction degree of residents in the residential areas. The specific data can be weighted and quantitatively evaluated according to the form of field research and questionnaire survey for the spatial frequency and resident satisfaction of the participants.Table 1
Evaluation index system.
DMUInput indicatorsOutput indicatorsRoom rate ($/m2)Plot ratioGreen rate (%)Frequency of use (person-time)SatisfactionLuyang districtBojing architecture53002.541403.1North American impression garden46004.5543383.33Yaohai districtZhongxiang Shengjing Garden45003.0741412.67Royal Lagoon64004.348453.3Baohe districtBerlin Spring45002.6836653.7Wanzhen Free Garden68002.8749.2673.57Shushan districtCamphor villas62001.8542673.9Xinhua University spring56002.0250423.4New station districtGreen and garden31002.3831.2372.53Jingrong city light45002.4442583.6


## 2.1. Problem Formulation

Suppose there aren decision-making units (DMU), and each decision-making unit has m inputs xij (j = 1, 2, …, m), and there are s producing yir (r = 1, …, s) (xij ≥ 0, yir ≥ 0), and the relaxation variable s −  and s + are introduced. Then, the relative efficiency of the DMU is as following DEA-CRR measure evaluation model:(1)Minσ,s.t.∑i=1nxijλi+s−=θx0jj=1,2,…,m,∑i=1nyijλi+s+=y0rr=1,2,…,s,∧≥0,s+≥0,s−≥0,0<σ≤1,where λi is the weight variable of each decision unit on an index, and σ is the comprehensive technology and scale efficiency of the decision unit. The CCR model is obtained under the assumption that DMU is constant returns to scale (CRS). σ is used to calculate by DEA-CCR, as CRSTE for short.Remark 1.
If constraints∑i=1nλi=1 are added to the above formula, it becomes the DEA-BCC model, which is obtained under the assumption that DMU is variable returns to scale (VRS). σ is the pure technical efficiency of DMU (VRS).Remark 2.
VRS refers to the distance between the investigated DMU and the effective production front when the scale return is variable. Scale efficiency (SE) (S E = CRSTE/VRSIE) refers to the distance between the effective production frontier with constant return on scale and variable return on scale. CRS includes VRS and SE. S E = 1 indicates that the DMU is located at the most appropriate scale efficiency level. S E < 1 indicates that the DMU is in a state of scale inefficiency.


## 2.2. Establishment of Indicator System

The input index and output index are both included in the DEA evaluation index system. When the DEA evaluation model is set up to make the decision-making unit number and maintain the proper proportion relation between evaluation index number, it is generally believed that the number of decision-making units should be greater than or equal to 2 times the total number of input and output indicators.As a practical problem, the spatial efficiency of the outdoor environment in high-rise residential areas needs to be closely combined with the actual research content and research results. The index system adopts the model combining analysis method and comprehensive method to consider the data’s accessibility, operability, and pertinence. This paper selects indicators from two aspects of economic input and design input and selects output indicators from the user’s participation and perception. The number of decision-making units selected in this study is the outdoor environment of 10 high-rise residential areas, so 3 input indexes and 2 output indexes are set. The final results are shown in Table1.(1)
In terms of investment, the capital factor is the most basic component. Considering the various early inputs of high-rise residential areas, most of them are visually represented in real life as signs of housing price, and the specific values are based on the average market price. The outdoor environment space system in the residential area includes greening, road, hard site, facilities, and other components. Among them, greening helps to form the environmental image of space path, node, area, sign, and boundary and better create the landscape environment inside the residential area. The hard ground takes up a relatively large proportion of the area, providing residents with daily outdoor activities. The area includes the central scenic area, the leisure area, and fitness area and facilities between groups, children’s entertainment venues, and the interoperable garden paths between groups. The facilities sketch is a small building facility in the outdoor environment of a residential area for people’s viewing, decoration, and lighting and for the convenience of garden management. It is the leading element for people to carry out activities on the site. All the above elements will have an impact on the efficiency of space use. Therefore, the volume rate and the green rate of high-rise residential outdoor environments are selected for quantitative evaluation based on the comprehensive technical and economic indicators in residential planning and design.(2)
In terms of output, as an important place of urban residents’ daily life behavior and the carrier of undertaking various life service facilities, the direct output of outdoor environment production in residential areas should include meeting all the needs and services of residents’ daily life, which is directly reflected in the participation and satisfaction degree of residents in the residential areas. The specific data can be weighted and quantitatively evaluated according to the form of field research and questionnaire survey for the spatial frequency and resident satisfaction of the participants.Table 1
Evaluation index system.
DMUInput indicatorsOutput indicatorsRoom rate ($/m2)Plot ratioGreen rate (%)Frequency of use (person-time)SatisfactionLuyang districtBojing architecture53002.541403.1North American impression garden46004.5543383.33Yaohai districtZhongxiang Shengjing Garden45003.0741412.67Royal Lagoon64004.348453.3Baohe districtBerlin Spring45002.6836653.7Wanzhen Free Garden68002.8749.2673.57Shushan districtCamphor villas62001.8542673.9Xinhua University spring56002.0250423.4New station districtGreen and garden31002.3831.2372.53Jingrong city light45002.4442583.6


## 3. Data Set and Application Analysis

### 3.1. Data Set Statistics

#### 3.1.1. Decision Unit Selection

DMU, the decision unit, is the research object. Each decision unit needs to have the same DEA evaluation objectives and the same input and output indicators. The decision-making unit in this study is the outdoor environment area of 10 high-rise residential areas in a certain region, and its use efficiency can be represented as the output result under the input level of production factors of a certain scale. If the output level is high, the efficiency is high; if the output level is low, the efficiency is low.


#### 3.1.2. Data Acquisition

The input index (housing price, floor area ratio, and green rate) data in this paper can be reflected by the market price and design technical index of each residential area. The data of output indicators (spatial frequency of participants and residents’ satisfaction) are obtained through field research and questionnaire survey to obtain the participants’ and residents’ satisfaction of residents per unit area, as shown in Table2. In the process of field research to ensure the basic premise, namely every day at the same time, choose the double cease day time Monday to Friday, roughly the same weather season, statistical residential outdoor environment in a certain field of residents’ activities in the area of related data and information, including the number of people to participate in activities and the classification, the specific activities of human behavior type, based on the data, make a careful arrangement and analysis. At the same time, in each high-rise residential area, 30 residents were asked to fill in the satisfaction questionnaire to reflect the actual willingness of residents to use the environment. (Due to certain limitations in the selection of season, weather, and time, relevant conclusions are based on this survey and are not universal in any practical scope).Table 2
Data investigation and analysis table of 10 decision units in an area.
DMUObservation area and timeThe surrounding environment and landscape facilities in the spaceBehavioral activity graphBojing architecturePavilion, wooden seats, small bridge, children’s play facilities, waterscape, flower frame, tree arrayNorth American impression gardenKindergarten, tree circle, flower pool, pavilion, seat, square, basketball court, children’s playground, fitness squareZhongxiang Shengjing GardenKindergarten, central waterscape, stone bridge, small wooden bridge, fitness facilities, children’s play facilities, wooden seatsRoyal LagoonCentral square, swimming pool, gallery frame, seats, pavilion, badminton court, children’s play area, fitness areaBerlin springKindergarten, golden cafe, swimming pool, fountain, wooden seat, children’s play area, fitness facilities0WanZhen Free Garden1Kindergartens, supermarkets, post offices, chess and card rooms, squares, fountains, flower racks, seats, fitness facilities, large artificial waterscape2Camphor villas3Club (1958 International Art Hall), wooden seats, flower pool, tree array, small stone bridge, badminton court4Xinhua University Spring5Child care training center, beauty club, central waterscape, wooden platform, small stone bridge, seats, fountain, colonnade6Green and garden7Property management office, central waterscape, pavilion, tree array, sculpture, natural revetment8Jingrong city light9Basketball court, hard square, fitness facilities, wooden seats


### 3.2. Application Results and Analysis

The quantified input and output index data of 10 decision-making units are brought into DEA-CCR and BCC models to obtain the results of data envelopment analysis. In this paper, DEAP2.1 software is used for calculation, and the calculation results are shown in Table3.Table 3
Data envelope analysis results of outdoor environmental spatial efficiency of 10 high-rise residential areas in a certain area.
DMUCRSVRSSERTSResultBojing Architecture0.7870.8690.905IRSDEA is invalidNorth American Impression Garden0.880.8820.998IRSDEA is invalidZhongxiang Shengjing Garden0.7220.7820.923IRSDEA is invalidRoyal Lagoon0.6690.7160.934IRSDEA is invalidBerlin Spring111—DEA is efficientWanZhen Free Garden0.8040.9120.882DRSDEA is invalidCamphor Villas111—DEA is efficientXinhua University Spring0.9030.9770.923IRSDEA is invalidGreen and garden0.99310.993IRSDEA is invalidJingrong City light111—DEA is efficientThe average0.87609140.956——Note: CRS is the comprehensive efficiency; VRS is technical efficiency; SE=CRS/VRS is the scale efficiency; RTS is revenue of scale; IRS for increasing returns to scale; DRS is diminishing returns to scale; - represents the constant return to scale.


#### 3.2.1. DEA Effective Analysis

According to the data in the table, among the 10 decision-making units, 3 units, namely, DEA, are at the forefront of inputting and output efficiency, accounting for 30% of the total. The comprehensive efficiency, technical efficiency, and scale efficiency of these high-rise residential areas are all 1, and the relaxation variable S is all 0, indicating that these three high-rise residential areas have achieved effective allocation, utilization, and scale aggregation of early-stage invested resources.


#### 3.2.2. DEA Invalid Analysis

The remaining 7 are invalid for DEA, accounting for 70% of the total. Also, it is divided into two types of analysis.(1)
One is a kind of green and garden, according to its pure technical efficiency (VRS) is 1, the comprehensive efficiency and scale efficiency are less than 1, which means the high-rise residential house for DEA efficient cause is not due to the inefficiency of its scale, its size, and the input and output do not match, need to increase or reduce the size, while technology effectively shows the output of the high-rise residential house has reached maximum, compared to the input, or on the configuration and utilization of resources, is effective. Because of factors such as residential location, early-stage capital investment, early-stage outdoor environment design, and others, as well as the usage frequency and satisfaction found in the late-stage survey. The two are basically unified. The specific reason for the invalid DEA of Greenheart Park lies in its scale inefficiency. Therefore, it is necessary to optimize the reasonable arrangement and division of the function and structure of the internal space of the outdoor environment of the residential area and take the activity willingness of residents as the premise of space design to fundamentally improve the scale efficiency, so as to achieve comprehensive efficiency.(2)
The other is the remaining 6 decision units, such as Bojing Architecture, Zhongxiang, and Shengjing Garden, that is, the pure technical efficiency, scale efficiency, and comprehensive efficiency are all less than 1, indicating that these high-rise residential areas are not effective in terms of technology and scale. Their average comprehensive efficiency is 0.794, their average pure technical efficiency is 0.856, and their average scale efficiency is 0.928. It can be seen that the technical efficiency is low, and the scale efficiency is high, but since the average scale efficiency has reached 0.928, if the comprehensive efficiency is achieved by simply expanding the scale, the improvement effect is not high. Therefore, in the high-rise residential outdoor environment in the process of development, scale increases inputs, attention should be paid to control the size of the outdoor environment space expanding without purpose, think the large area space place is good, pay attention to optimize the structure of input and output of the environment, promote the rationality of the design of space planning, enhance the degree of effective utilization of resources, increase unit investment on technology, to improve technical efficiency and scale efficiency at the same time, in order to reach the ultimate goal of comprehensive efficiency effectively.In particular, the comprehensive efficiency of the royal lagoon is only 0.669, and the pure technical efficiency is only 0.716, while the average comprehensive efficiency of the out-door environment in high-rise residential areas is 0.876, and the average pure technical efficiency is 0.914. Obviously, the value of the royal lagoon is far lower than the overall average level, and its scale efficiency is 0.934, close to the mean value of the overall scale efficiency of 0.956. This kind of similar situation needs to be paid enough attention. At the later stages of the outdoor environment space at the royal lagoon, efforts should be made to increase investment in technology rather than simply increasing the outdoor environment space. These efforts should be based on the original space and should optimize and adjust the entire space structure, dividing it into multiple functional uses of space to meet the needs of the users.


### 3.3. Suggestions for Space Efficiency Optimization

From the average value of outdoor environmental space efficiency in high-rise residential areas, the average comprehensive efficiency, technical efficiency, and scale efficiency of the 10 decision-making units are 0.876, 0.914, and 0.956, respectively. On the whole, the average value of comprehensive efficiency and technical efficiency is relatively low. This shows that in terms of the spatial optimization and transformation of the outdoor environment of high-rise residential areas in this region, if it is only to increase the size of the scale space, its role is no longer significant. More attention should be paid to the improvement of the technical level at the present stage, such as the selection of the spatial location of the outdoor environment, the targeted layout of the landscape facilities in the environment, the rational division of multiple functions of a single space, the guidance of the surrounding pedestrian system, and other aspects. At the same time, according to the actual conditions of the outdoor environment of each high-rise residential area, the space scale investment should be reasonably increased, so as to achieve the effectiveness of the comprehensive efficiency of the outdoor environment space of high-rise residential area.


## 3.1. Data Set Statistics

### 3.1.1. Decision Unit Selection

DMU, the decision unit, is the research object. Each decision unit needs to have the same DEA evaluation objectives and the same input and output indicators. The decision-making unit in this study is the outdoor environment area of 10 high-rise residential areas in a certain region, and its use efficiency can be represented as the output result under the input level of production factors of a certain scale. If the output level is high, the efficiency is high; if the output level is low, the efficiency is low.


### 3.1.2. Data Acquisition

The input index (housing price, floor area ratio, and green rate) data in this paper can be reflected by the market price and design technical index of each residential area. The data of output indicators (spatial frequency of participants and residents’ satisfaction) are obtained through field research and questionnaire survey to obtain the participants’ and residents’ satisfaction of residents per unit area, as shown in Table2. In the process of field research to ensure the basic premise, namely every day at the same time, choose the double cease day time Monday to Friday, roughly the same weather season, statistical residential outdoor environment in a certain field of residents’ activities in the area of related data and information, including the number of people to participate in activities and the classification, the specific activities of human behavior type, based on the data, make a careful arrangement and analysis. At the same time, in each high-rise residential area, 30 residents were asked to fill in the satisfaction questionnaire to reflect the actual willingness of residents to use the environment. (Due to certain limitations in the selection of season, weather, and time, relevant conclusions are based on this survey and are not universal in any practical scope).Table 2
Data investigation and analysis table of 10 decision units in an area.
DMUObservation area and timeThe surrounding environment and landscape facilities in the spaceBehavioral activity graphBojing architecturePavilion, wooden seats, small bridge, children’s play facilities, waterscape, flower frame, tree arrayNorth American impression gardenKindergarten, tree circle, flower pool, pavilion, seat, square, basketball court, children’s playground, fitness squareZhongxiang Shengjing GardenKindergarten, central waterscape, stone bridge, small wooden bridge, fitness facilities, children’s play facilities, wooden seatsRoyal LagoonCentral square, swimming pool, gallery frame, seats, pavilion, badminton court, children’s play area, fitness areaBerlin springKindergarten, golden cafe, swimming pool, fountain, wooden seat, children’s play area, fitness facilities0WanZhen Free Garden1Kindergartens, supermarkets, post offices, chess and card rooms, squares, fountains, flower racks, seats, fitness facilities, large artificial waterscape2Camphor villas3Club (1958 International Art Hall), wooden seats, flower pool, tree array, small stone bridge, badminton court4Xinhua University Spring5Child care training center, beauty club, central waterscape, wooden platform, small stone bridge, seats, fountain, colonnade6Green and garden7Property management office, central waterscape, pavilion, tree array, sculpture, natural revetment8Jingrong city light9Basketball court, hard square, fitness facilities, wooden seats


## 3.1.1. Decision Unit Selection

DMU, the decision unit, is the research object. Each decision unit needs to have the same DEA evaluation objectives and the same input and output indicators. The decision-making unit in this study is the outdoor environment area of 10 high-rise residential areas in a certain region, and its use efficiency can be represented as the output result under the input level of production factors of a certain scale. If the output level is high, the efficiency is high; if the output level is low, the efficiency is low.


## 3.1.2. Data Acquisition

The input index (housing price, floor area ratio, and green rate) data in this paper can be reflected by the market price and design technical index of each residential area. The data of output indicators (spatial frequency of participants and residents’ satisfaction) are obtained through field research and questionnaire survey to obtain the participants’ and residents’ satisfaction of residents per unit area, as shown in Table2. In the process of field research to ensure the basic premise, namely every day at the same time, choose the double cease day time Monday to Friday, roughly the same weather season, statistical residential outdoor environment in a certain field of residents’ activities in the area of related data and information, including the number of people to participate in activities and the classification, the specific activities of human behavior type, based on the data, make a careful arrangement and analysis. At the same time, in each high-rise residential area, 30 residents were asked to fill in the satisfaction questionnaire to reflect the actual willingness of residents to use the environment. (Due to certain limitations in the selection of season, weather, and time, relevant conclusions are based on this survey and are not universal in any practical scope).Table 2
Data investigation and analysis table of 10 decision units in an area.
DMUObservation area and timeThe surrounding environment and landscape facilities in the spaceBehavioral activity graphBojing architecturePavilion, wooden seats, small bridge, children’s play facilities, waterscape, flower frame, tree arrayNorth American impression gardenKindergarten, tree circle, flower pool, pavilion, seat, square, basketball court, children’s playground, fitness squareZhongxiang Shengjing GardenKindergarten, central waterscape, stone bridge, small wooden bridge, fitness facilities, children’s play facilities, wooden seatsRoyal LagoonCentral square, swimming pool, gallery frame, seats, pavilion, badminton court, children’s play area, fitness areaBerlin springKindergarten, golden cafe, swimming pool, fountain, wooden seat, children’s play area, fitness facilities0WanZhen Free Garden1Kindergartens, supermarkets, post offices, chess and card rooms, squares, fountains, flower racks, seats, fitness facilities, large artificial waterscape2Camphor villas3Club (1958 International Art Hall), wooden seats, flower pool, tree array, small stone bridge, badminton court4Xinhua University Spring5Child care training center, beauty club, central waterscape, wooden platform, small stone bridge, seats, fountain, colonnade6Green and garden7Property management office, central waterscape, pavilion, tree array, sculpture, natural revetment8Jingrong city light9Basketball court, hard square, fitness facilities, wooden seats


## 3.2. Application Results and Analysis

The quantified input and output index data of 10 decision-making units are brought into DEA-CCR and BCC models to obtain the results of data envelopment analysis. In this paper, DEAP2.1 software is used for calculation, and the calculation results are shown in Table3.Table 3
Data envelope analysis results of outdoor environmental spatial efficiency of 10 high-rise residential areas in a certain area.
DMUCRSVRSSERTSResultBojing Architecture0.7870.8690.905IRSDEA is invalidNorth American Impression Garden0.880.8820.998IRSDEA is invalidZhongxiang Shengjing Garden0.7220.7820.923IRSDEA is invalidRoyal Lagoon0.6690.7160.934IRSDEA is invalidBerlin Spring111—DEA is efficientWanZhen Free Garden0.8040.9120.882DRSDEA is invalidCamphor Villas111—DEA is efficientXinhua University Spring0.9030.9770.923IRSDEA is invalidGreen and garden0.99310.993IRSDEA is invalidJingrong City light111—DEA is efficientThe average0.87609140.956——Note: CRS is the comprehensive efficiency; VRS is technical efficiency; SE=CRS/VRS is the scale efficiency; RTS is revenue of scale; IRS for increasing returns to scale; DRS is diminishing returns to scale; - represents the constant return to scale.


### 3.2.1. DEA Effective Analysis

According to the data in the table, among the 10 decision-making units, 3 units, namely, DEA, are at the forefront of inputting and output efficiency, accounting for 30% of the total. The comprehensive efficiency, technical efficiency, and scale efficiency of these high-rise residential areas are all 1, and the relaxation variable S is all 0, indicating that these three high-rise residential areas have achieved effective allocation, utilization, and scale aggregation of early-stage invested resources.


### 3.2.2. DEA Invalid Analysis

The remaining 7 are invalid for DEA, accounting for 70% of the total. Also, it is divided into two types of analysis.(1)
One is a kind of green and garden, according to its pure technical efficiency (VRS) is 1, the comprehensive efficiency and scale efficiency are less than 1, which means the high-rise residential house for DEA efficient cause is not due to the inefficiency of its scale, its size, and the input and output do not match, need to increase or reduce the size, while technology effectively shows the output of the high-rise residential house has reached maximum, compared to the input, or on the configuration and utilization of resources, is effective. Because of factors such as residential location, early-stage capital investment, early-stage outdoor environment design, and others, as well as the usage frequency and satisfaction found in the late-stage survey. The two are basically unified. The specific reason for the invalid DEA of Greenheart Park lies in its scale inefficiency. Therefore, it is necessary to optimize the reasonable arrangement and division of the function and structure of the internal space of the outdoor environment of the residential area and take the activity willingness of residents as the premise of space design to fundamentally improve the scale efficiency, so as to achieve comprehensive efficiency.(2)
The other is the remaining 6 decision units, such as Bojing Architecture, Zhongxiang, and Shengjing Garden, that is, the pure technical efficiency, scale efficiency, and comprehensive efficiency are all less than 1, indicating that these high-rise residential areas are not effective in terms of technology and scale. Their average comprehensive efficiency is 0.794, their average pure technical efficiency is 0.856, and their average scale efficiency is 0.928. It can be seen that the technical efficiency is low, and the scale efficiency is high, but since the average scale efficiency has reached 0.928, if the comprehensive efficiency is achieved by simply expanding the scale, the improvement effect is not high. Therefore, in the high-rise residential outdoor environment in the process of development, scale increases inputs, attention should be paid to control the size of the outdoor environment space expanding without purpose, think the large area space place is good, pay attention to optimize the structure of input and output of the environment, promote the rationality of the design of space planning, enhance the degree of effective utilization of resources, increase unit investment on technology, to improve technical efficiency and scale efficiency at the same time, in order to reach the ultimate goal of comprehensive efficiency effectively.In particular, the comprehensive efficiency of the royal lagoon is only 0.669, and the pure technical efficiency is only 0.716, while the average comprehensive efficiency of the out-door environment in high-rise residential areas is 0.876, and the average pure technical efficiency is 0.914. Obviously, the value of the royal lagoon is far lower than the overall average level, and its scale efficiency is 0.934, close to the mean value of the overall scale efficiency of 0.956. This kind of similar situation needs to be paid enough attention. At the later stages of the outdoor environment space at the royal lagoon, efforts should be made to increase investment in technology rather than simply increasing the outdoor environment space. These efforts should be based on the original space and should optimize and adjust the entire space structure, dividing it into multiple functional uses of space to meet the needs of the users.


## 3.2.1. DEA Effective Analysis

According to the data in the table, among the 10 decision-making units, 3 units, namely, DEA, are at the forefront of inputting and output efficiency, accounting for 30% of the total. The comprehensive efficiency, technical efficiency, and scale efficiency of these high-rise residential areas are all 1, and the relaxation variable S is all 0, indicating that these three high-rise residential areas have achieved effective allocation, utilization, and scale aggregation of early-stage invested resources.


## 3.2.2. DEA Invalid Analysis

The remaining 7 are invalid for DEA, accounting for 70% of the total. Also, it is divided into two types of analysis.(1)
One is a kind of green and garden, according to its pure technical efficiency (VRS) is 1, the comprehensive efficiency and scale efficiency are less than 1, which means the high-rise residential house for DEA efficient cause is not due to the inefficiency of its scale, its size, and the input and output do not match, need to increase or reduce the size, while technology effectively shows the output of the high-rise residential house has reached maximum, compared to the input, or on the configuration and utilization of resources, is effective. Because of factors such as residential location, early-stage capital investment, early-stage outdoor environment design, and others, as well as the usage frequency and satisfaction found in the late-stage survey. The two are basically unified. The specific reason for the invalid DEA of Greenheart Park lies in its scale inefficiency. Therefore, it is necessary to optimize the reasonable arrangement and division of the function and structure of the internal space of the outdoor environment of the residential area and take the activity willingness of residents as the premise of space design to fundamentally improve the scale efficiency, so as to achieve comprehensive efficiency.(2)
The other is the remaining 6 decision units, such as Bojing Architecture, Zhongxiang, and Shengjing Garden, that is, the pure technical efficiency, scale efficiency, and comprehensive efficiency are all less than 1, indicating that these high-rise residential areas are not effective in terms of technology and scale. Their average comprehensive efficiency is 0.794, their average pure technical efficiency is 0.856, and their average scale efficiency is 0.928. It can be seen that the technical efficiency is low, and the scale efficiency is high, but since the average scale efficiency has reached 0.928, if the comprehensive efficiency is achieved by simply expanding the scale, the improvement effect is not high. Therefore, in the high-rise residential outdoor environment in the process of development, scale increases inputs, attention should be paid to control the size of the outdoor environment space expanding without purpose, think the large area space place is good, pay attention to optimize the structure of input and output of the environment, promote the rationality of the design of space planning, enhance the degree of effective utilization of resources, increase unit investment on technology, to improve technical efficiency and scale efficiency at the same time, in order to reach the ultimate goal of comprehensive efficiency effectively.In particular, the comprehensive efficiency of the royal lagoon is only 0.669, and the pure technical efficiency is only 0.716, while the average comprehensive efficiency of the out-door environment in high-rise residential areas is 0.876, and the average pure technical efficiency is 0.914. Obviously, the value of the royal lagoon is far lower than the overall average level, and its scale efficiency is 0.934, close to the mean value of the overall scale efficiency of 0.956. This kind of similar situation needs to be paid enough attention. At the later stages of the outdoor environment space at the royal lagoon, efforts should be made to increase investment in technology rather than simply increasing the outdoor environment space. These efforts should be based on the original space and should optimize and adjust the entire space structure, dividing it into multiple functional uses of space to meet the needs of the users.


## 3.3. Suggestions for Space Efficiency Optimization

From the average value of outdoor environmental space efficiency in high-rise residential areas, the average comprehensive efficiency, technical efficiency, and scale efficiency of the 10 decision-making units are 0.876, 0.914, and 0.956, respectively. On the whole, the average value of comprehensive efficiency and technical efficiency is relatively low. This shows that in terms of the spatial optimization and transformation of the outdoor environment of high-rise residential areas in this region, if it is only to increase the size of the scale space, its role is no longer significant. More attention should be paid to the improvement of the technical level at the present stage, such as the selection of the spatial location of the outdoor environment, the targeted layout of the landscape facilities in the environment, the rational division of multiple functions of a single space, the guidance of the surrounding pedestrian system, and other aspects. At the same time, according to the actual conditions of the outdoor environment of each high-rise residential area, the space scale investment should be reasonably increased, so as to achieve the effectiveness of the comprehensive efficiency of the outdoor environment space of high-rise residential area.


## 4. Conclusion

In this paper, we have provided a new evaluation method, data envelopment analysis, for the study of spatial efficiency of HRR areas, which has improved the accuracy and objectivity of the evaluation. Furthermore, this paper has quantified the index contents from two aspects of objective material space analysis and subjective spiritual satisfaction feedback and provided a more convincing result for the spatial research of residential living environment by data envelopment model analysis. In the end, the validity of our approach has been verified by taking the residential area of a city in Anhui province in China. It is worth noting that the relationship between the spatial efficiency of the outdoor environment and spatial planning is of higher complexity, which needs to be supported by more samples and data findings. Because of the limited number of decision units and geographical area, a general evaluation of the use of high-rise outdoor space in a wider area could not be carried out, which could be studied in the near future [51].



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*Source: 4141412-2022-08-12.xml*