Decision-Making Conflict Measurement of Old Neighborhoods Renovation Based on Mixed Integer Programming DEA-Discriminant Analysis (MIP DEA–DA) Models

Abstract

Renovating old neighborhoods for the benefit of people has become increasingly important in urban renewal. Nevertheless, old neighborhood renovations are currently considered a group decision-making issue under public participation, involving diverse decision-making subjects. Conflicts within a group are a common problem during group decision-making. In this paper, conflict is examined in the decision-making process for renovating old neighborhoods and novel ideas are provided for quantifying conflict. Public participation in old neighborhood renovations is assessed using conflict degree calculations in group decision-making. Based on the preferences of decision-making experts, a MIP DEA–DA (Mixed Integer Programming Data Envelopment Analysis–Discriminant Analysis) based partial binary tree cyclic clustering model is constructed for clustering experts, and an aggregated group conflict indicator and an aggregated conflict vector are computed, allowing for the quantification of conflict during the renovation process of the old neighborhood based on actual situations. Results indicate that there is primarily a conflict between the benefits of decision-making subject interests and the professionalism of decision-making renovations. This paper contributes to improving public participation, promoting the application of group decision-making theory in old neighborhood renovation, reducing conflict between decision-makers, and speeding up urban renewal.

1. Introduction

As cities continue to thrive and urbanize at an increasing rate, the number of older neighborhoods has increased dramatically over time [1]. Many old neighborhoods have been left unattended for long periods of time due to their facilities, presenting serious problems of aging infrastructure, difficult community governance issues, and a shortage of space-crowded resources [2], which cannot satisfy the residents’ ever-increasing needs.

The number of plans to renovate old neighborhoods in China has been increasing every year since 2019, gradually becoming an essential part of urban renewal [3]. Renovation projects in old neighborhoods have the primary goal of renewing the optimal use of land plots as well as promoting sustainable urban development [4]. An important factor in the renovation process is decision-making, which determines the overall direction. The renovation of old neighborhoods will require the participation of multiple decision-making bodies, and there are some contradictions between public and personal interests, information sharing, and information monopoly among different interest groups [5]. In addition, due to different interests, backgrounds, and knowledge reserved between the subjects, there will be differences in preferences between them in decision-making [6], resulting in unnecessary conflict in decision-making. Whenever a conflict is not resolved in a timely manner, it will lead to decision-making errors, resulting in ineffective decision-making as well as the loss of multi-interested parties, resulting in significant losses in resources, including human, material, and financial resources [7]. There are current studies related to the renovation of old neighborhoods that explore the impact on renovation priorities from the perspective of focusing on residents’ willingness to participate in the renovation [8]. Some scholars have explored the key barriers affecting the renewal of China’s old residential neighborhoods [9], and in addition, there are studies that explore the role of renovation policy changes in the age of sustainability from the perspective of the changes in renovation policy in the era of sustainable development [10]. However, the current research on the renovation of old neighborhoods is relatively one-sided; on the one hand, scholars pay more attention to the specific ways of renovating old neighborhoods, most of which are at the micro level, and there is a lack of research on the overall decision-making of the renovation of old neighborhoods. On the other hand, a large number of scholars have shifted their perspectives to the combination of the renovation of old neighborhoods and public participation and have paid increasing attention to public opinion, but have neglected the need to introduce the group decision-making method due to the increase in decision-making subjects. At present, China’s group decision-making method only exists in the theoretical maturity, but there is little concrete practice for the renovation of old neighborhoods. In addition, there are also few studies have paid attention to the perspective of conflict of interest between different subjects of interest in the process of renovation of old neighborhoods and explored the impact of multi-party conflict of interest on renovation decision-making from a holistic point of view. Therefore, in order to promote the sustainable development of the renovation of old neighborhoods, it is necessary to fundamentally solve the problem of decision-making conflicts in the process of renovation of old neighborhoods.

This paper innovatively applies the conflict degree calculation method in group decision-making to old neighborhood renovation, which improves the decision-making process of urban old neighborhood renovation based on group decision-making theory. Meanwhile, due to the diversified interests involved in the renovation process, this paper clusters the decision-making subjects of the masses, the government, as well as the investment and construction units based on the partial binary tree MIP DEA–DA cyclic model according to the preference information provided by the decision-making experts, and groups members who have similar interests together. To quantify the level of conflict among decision-making subjects in the old neighborhood renovation decision-making process, the group decision-making method was used. We calculated the degree of conflict indicators between clusters and the conflict vector between each cluster and the whole decision-makers group and identified the areas where conflict occurred during the renovation process. The results show that there is a conflict between the balance of benefits for the subject of interest and the professionalism of the decision-making. In the process of renovation, stakeholders from all walks of life, including residents, government agencies, and investment companies, are involved. In addition, the consciousness and values of each decision-making subject affect the renovation process as well. Considering the diversity of subjects, this paper clusters decision-making subjects while adopting the group decision-making method, which enables residents, government, and investment and construction companies to participate more deeply in the renovation process, in addition to promoting group decision-making theory in the renovation of old neighborhoods, it also offers new approaches to quantifying decision-making conflicts in old neighborhoods, to resolve the underlying conflicts in the renovation decision-making process. Thus, the problem of conflict in the renovation decision-making process will be solved at the root, and the scientific and democratic development of the decision-making process related to the renovation of old communities and urban renewal will be promoted.

2. Literature Review

2.1. Renovation of Old Neighborhoods

An old neighborhood is defined as a residential neighborhood that was constructed earlier, has lower construction standards, an aging infrastructure, poorly maintained facilities, and a lack of long-term management mechanisms [11]. In the context of an accelerating urbanization cycle and the continuous innovation of urban planning and construction concepts, China’s urban renewal has moved from incremental development to stock optimization and faces many challenges [12]. Due to the high proportion of old neighborhoods in urban areas and the necessity of improving the quality of urban areas [13], the renovation of old neighborhoods has gradually become a research hotspot in urban renewal in recent years. There are many advantages of renovating an old neighborhood over tearing it down and rebuilding it in place. For example, in China, where there is a shortage of modern housing, renovating older neighborhoods can help alleviate this phenomenon and has the advantage of costing less than new construction [14]. In most cases, the communities of these old neighborhoods usually have typically developed a stable and mature atmosphere, and renovation can contribute to maintaining their atmosphere and even enhancing their sense of belonging [8].

At present, many scholars have carried out rich research on old neighborhoods from multiple perspectives. Through case studies, He [15] analyzes the current environmental pollution problems associated with domestic building renovation based on relevant management studies and proposes “green” renovation, a method of improving the building’s energy efficiency and extending its service life. As well as addressing the pollution problem, it is important to renovate old neighborhoods so that they can regain their contemporary significance, which is not just about increasing their functions and beautifying their surroundings, but also about achieving long-term sustainability [16]. Additionally, In order to improve the image of a city in general, an urban planner must approach the renovation of old neighborhoods in a scientific and structural manner [17]. Li [6] summarizes the main factors influencing renovation decisions by analyzing the current situation of older homes in several major cities in China. Since urban areas are complex dynamic systems combining physical, ecological, economic, and environmental aspects, it is particularly important to provide decision-makers with comprehensive assessment tools during the renovation process [18]. In transforming evaluation studies, Luo [19] evaluated and analyzed the impact of carbon emissions on the entire life cycle of the old residential renovation district, and improved and expanded the accounting scope of residential renovation from the perspective of space and time scale. Further, Ding [4] suggests that a key premise for the renovation of older neighborhoods is to increase residents’ willingness to participate in the process. In general, “neighborhood effects” are believed to influence housing renovations, and Helms [20] found strong evidence of positive feedback between rehabilitation activities and neighborhood quality through modeling and data analysis. Resident satisfaction is also one of the key indicators of the final results of the old neighborhood renovation projects [21]. During the old neighborhood renovation project, residents need to be mobilized to participate in the initiative and enthusiasm of the renovation and encouraged to participate voluntarily [22]. This can, to some extent, compensate for the shortcomings of the old neighborhood renovation under the administration’s leadership [23].

To cope with the enormous demands generated by the exponential growth of the population, it is imperative that we promote the urban renewal process of existing living environments to prevent aging and deterioration of buildings [24]. Balaban [25] proposed that the construction industry is usually the engine of economic growth and that the focus of urban construction needs to fall not only on basic functions but also on listening to the public and incorporating public aspirations. As stated by Lidegaard [26], the government must determine the renovation methods in accordance with the goal of renovation and the needs of the main interests involved in the renovation in order to establish the right position, actively facilitate public participation, and reasonably determine the renovation methods. For the cause of renovation of old neighborhoods, Bottero [27] defines it as a multifaceted behavioral vision that is designed to identify and solve existing problems in the city to achieve sustainable economic and social development. In addition, interested parties in the renovation of old neighborhoods also exercise different rights and pursue different benefits in the process of renovation, all of which need to be considered in a comprehensive manner.

In addition to this, in terms of the content of the renovation, the perspective of foreign scholars has spread out from the conventional retrofitting of elevators, sponge renovation, and other infrastructure content. Allik and Kearns [28] examine the integration of multiple sporting events with the redevelopment of dilapidated areas from the perspective of the overall regeneration of the region. The demolition of large areas is often part of urban renewal projects, which can destroy social networks and local identities [29]. Yung [30] shifts the perspective to the relationship between old neighborhood renovation projects and heritage preservation and suggests the social role they play, arguing that the two need to be organically integrated in the urban renewal process. The density of older people is generally higher in older neighborhoods; Yung [31] examined elderly residents’ needs in older neighborhoods and conducted a study on aging in public spaces in older neighborhoods, suggesting that city planners and designers should attach importance to and improve the social welfare of older people and active aging. Serrano-Jiménez [32] introduces a novel approach to enhance the decision-making process for revitalizing sustainable older communities, enabling them to effectively adapt to the specific requirements of their population, particularly older individuals, and serve as a catalyst for promoting active population aging. It can be seen that foreign scholars on the renovation of old neighborhoods started earlier and gradually in-depth research while paying more attention to the core concept of human-centered renovation, emphasizing that the community serves people, and that people are one of the most important factors in renovating old neighborhoods. Therefore, when making decisions on the renewal of old neighborhoods, we need to consider the diversity of the objects of renovation, listen to the views of different participating subjects, and coordinate the interests of all the participating subjects so as to maintain sustainable economic and social development. A review of the literature also suggests that there is less research into conflicting values and perceptions of multiple participants when deciding on the renewal of older neighborhoods. A lack of research in this area is compensated for by this paper which quantifies preference conflicts within an old neighborhood renovation process using group decision-making and identifies the sources of those conflicts so that future old neighborhood renovation projects can be theoretically supported.

2.2. Group Decision-Making

The decision-making problem on the renovation of old neighborhoods under public participation has been transformed into a group decision-making problem because its subject tends to be diversified and is no longer a single decision by the government. Xu [33] was the first to propose the theory of group decision-making in the field of national institutional reform, and subsequently, different models of group decision-making have been proposed to be applied to different types of practical decision-making problems. Over time, the decision-making subject in the process of global democratization to diversification, so the composition of the group and decision-making mode has also changed; in the pluralistic and complex decision-making group structure, increasing numbers of scholars have conducted in-depth research on group decision-making. It is mainly embodied in decision-making expert clustering [34], decision attribute weighting [35], and consensus-reaching processes [36], and the research focus is also mainly embodied in multi-attribute large-group decision-making methods [37], dynamic group decision-making methods [38], and multi-objective group decision-making methods [39]. Liu [40] examined the process of continuous improvement of the DEA–DA method, developed a more rigorous MIP DEA–DA clustering model for expert clustering, and laid the foundation for the subsequent quantification of the degree of conflict between clusters and enabling experts to reach consensus. For the renovation of old communities in China, the fuzzy Delphi method was applied to develop the criteria for outdoor environment renovation, and the optimal decision-making model was developed and applied based on the I-S model and zero-one integer programming [41]. The development of group decision-making is no longer a mere comparison of options but more focused on the dynamic interaction method of consensus formation and agreement among decision-making subjects. Clearly, group decision-making has evolved beyond purely comparing options to forming consensus and agreement based on dynamic interaction methods. However, these decision-making issues not only have many aspects to consider but also require the reconciliation of multiple relationships and interests. The most prominent feature of public participation in decision-making for the renovation of old neighborhoods is that the decision-making body is “large in scale and strong in conflict”. Only by identifying the problems of the public participation mechanism can we set a benchmark for the decision-makers and make the decision more scientific and rational.

For the conflicts generated during the group decision-making process, the issues associated with conflict measurement, conflict coordination, and consensus building among decision-makers have gradually become a hot research topic among scholars. Zeitzoff [42] measures conflict intensity using Twitter and other social media data sources. Concerning conflict causes, Osei-Kyei [43] investigates and evaluates the root causes of project conflict and summarizes project-appropriate conflict resolution mechanisms. Liu [44] proposed a large-scale group decision-making model for social network environments that is capable of defining the notion of conflict level, incorporating evaluative information and trust relationships between decision-makers in large groups, quantifying the level of collective conflict, and using correlation modeling to de-conflict among decision-makers. It can be seen that group decision-making has been relatively mature from theory to method; however, the conflict within the decision-making group is a long-standing problem. As a major livelihood project, quantifying the conflict between groups in the decision-making process and identifying the conflict points is essential, which can ensure the smooth progress of decision-making as well as the scientific and democratic nature of decision-making.

3. Methodology

3.1. Clustering Method for Decision Makers of Old Neighborhood Renovation Based on the Partial Binary Tree DEA–DA Cyclic Classification Model

There are many stakeholders involved in the renovation of the old neighborhood, and its decision-making process involves decision-makers from different fields at different levels, which is a large group decision-making process. Large group decision-making usually involves multiple participants and multiple decision-making factors and requires comprehensive consideration of the stakeholders, and the needs of each participant have to be considered from multiple perspectives to reach an optimal outcome. Clustering is a method of grouping similar objects together and can be used in large-group decision-making to divide participants or decision factors into several categories so that these categories can be better compared and analyzed. Through cluster analysis, participants or decision factors in large groups can be divided into categories according to their similarity, resulting in high similarity and less difference between participants or decision factors within the same category, and greater difference between different categories. This allows for discussion and decision-making on a smaller scale, reducing the complexity and uncertainty of decision-making.

Group decision-making has evolved with numerous clustering methods, and there is a need to find a clustering method that is applicable to the decision-making for the renovation of old neighborhoods. This paper clusters the decision-making members of the old neighborhood renovation with the basis of the partial binary tree DEA–DA clustering method, which combines the advantages of both DEA (Data Envelopment Analysis) and DA (Discriminant Analysis) methods.

This method is capable of handling high-dimensional and large-scale data and reduces the dimensionality of the data by selecting a representative subset, thus allowing for large-scale datasets to be handled. In addition to this, it is also capable of generating highly interpretable clustering results, and the decision units are divided into several clusters with similar characteristics by generating a partial binary tree. The model is not only capable of generating highly interpretable clustering results but also capable of assigning different decision units to the corresponding clusters, which makes the clustering results more practical.

Let $E=\left(e_1,e_2,\cdots e_n\right)\left(n\ge 20\right)$ represent the set of experts in decision-making for the renovation of old neighborhoods; $U=\{u_1,u_2,\dots ,u_l\}$ represents the set of decision attributes. The expert preference vector $B_k=(b_{k1},b_{k2},\cdots ,b_{kl})(k=1,2,\cdots ,n)$ represents a vector of expert $e_k$ evaluations of the importance of each decision attribute, where $b_{kj}$ denotes the $k$th expert’s rating of the importance of the $j$th pair of decision attributes in influencing the final decision goal.

In the old neighborhoods’ renovation decision-making, the decision-making members may involve different interest subjects and interest relationships; cluster analysis can put the decision-making members with similar interests in the same group to improve the efficiency and accuracy of decision-making. At the same time, the clustering model based on the partial binary tree is also able to consider the weight difference between the members, which can be clustered more accurately and improve the relevance and effectiveness of decision-making.

Therefore, the DEA–DA clustering model based on the partial binary tree can be an important tool for decision-making in the renovation of old neighborhoods. In this paper, we first cluster the experts with the help of the DEA–DA model of a partial binary tree and then use the conflict measurement model to calculate the degree of conflict level indicator between clusters and the conflict vector between each cluster and the whole group of decision-makers, to analyze and quantify the degree of aggregated conflict and to provide the relevant suggestions. The specific conflict quantification process is shown in Figure 1 below.

3.1.1. MIP DEA–DA Model

DEA–DA provides more decision support and guidance to decision-makers by applying data analysis techniques such as clustering and regression on the DEA output. Its basic framework is shown in Figure 2.

The earliest DEA–DA model began to classify the research objects into two groups, Group1 and Group2, and based on this, used cluster analysis and discriminant analysis to reclassify these two groups of research objects so that the members with smaller differences form a group, thus helping decision makers to better reduce conflicts and make effective decisions.

However, the low accuracy and many limitations of the DEA–DA method lead to the fact that the accuracy of this method needs to be improved. In 2001, Sueyoshi [45] proposed an improved DEA–DA method to solve this problem effectively. Based on this improved model, Sueyoshi further refined the method by reconstructing the DEA–DA model based on mixed integer programming (MIP) by taking the number of misjudged objects as the objective function. In this paper, we propose to use the MIP DEA–DA model as a basis for categorizing large groups of decision-making experts in the transaction model of engineering projects, which is briefly described in the following.

Phase I: Classification and Cross-identification

$$\min \hspace{0.33em}s$$

$$\mathrm{s}.\mathrm{t}.\hspace{0.33em}{\displaystyle \sum _{j=1}^l\left({\lambda }_j^{+}-{\lambda }_j^{-}\right)b_{kj}}-d+s\ge 0,k\in G_1,$$

$${\displaystyle \sum _{j=1}^l\left({\lambda }_j^{+}-{\lambda }_j^{-}\right)b_{kj}}-d-s\le 0,k\in G_2,$$

$${\displaystyle \sum _{j=1}^l\left({\lambda }_j^{+}+{\lambda }_j^{-}\right)}=1,$$

$${\zeta }_j^{+}\ge {\lambda }_j^{+}\ge \epsilon {\zeta }_j^{+}\hspace{0.33em}\mathrm{and}\hspace{0.33em}{\zeta }_j^{-}\ge {\lambda }_j^{-}\ge \epsilon {\zeta }_j^{-},j=1,2,\dots ,l,$$

(1)

$${\zeta }_j^{+}+\hspace{0.33em}{\zeta }_j^{-}\le 1,j=1,2,\dots ,l,$$

$${\displaystyle \sum _{j=1}^l\left({\zeta }_j^{+}+\hspace{0.33em}{\zeta }_j^{-}\right)}=l,$$

$$d,s:unconstrai ned,{\zeta }_j^{+}=0/1,{\zeta }_j^{-}=0/1$$

$$\mathit{Other}\mathit{variables}\ge 0.$$

$b_{\mathit{kj}}—\mathit{Evaluation}\mathit{value}\mathit{for}\mathit{the}\mathit{kth}\mathit{observation}\mathit{for}\mathit{the}\mathit{jth}\mathit{attribute};{\lambda }^{+}$,

${\lambda }_j({\lambda }_j={\lambda }_j^{+}-{\lambda }_j^{-})—\mathit{Weight}\mathit{of}\mathit{the}\mathit{jth}\mathit{attribute}$;

$\left|{\lambda }_j\right|(\left|{\lambda }_j\right|={\lambda }_j^{+}+{\lambda }_j^{-})—\mathit{Absolute}\mathit{value}\mathit{of}\mathit{attribute}\mathit{weights}$;

$d—\mathit{Discriminant}\mathit{score}$;

$s—\mathit{Distance}\mathit{of}\mathit{the}\mathit{discriminant}\mathit{function}\mathit{from}\mathit{the}\mathit{discriminant}d$.

According to the first stage of the discriminative results, the optimal solution is ${\lambda }_j^{+*},{\lambda }_j^{-*},d^{*},s^{*}$. The following discriminatory criteria were then adopted: For any observation object, if $s^{*}<0$, then there is no intersection; If $s^{*}\ge 0$, then there are the following:

(1)

If ${\displaystyle \sum _{j=1}^l({\lambda }_j^{+}-{\lambda }_j^{+})b_{kj}}>d^{*}+s^{*}$, there is $B_k\in G_1$;

(2)

If ${\displaystyle \sum _{j=1}^l({\lambda }_j^{+}-{\lambda }_j^{+})b_{kj}}<d^{*}-s^{*}$, there is $B_k\in G_2$;

(3)

If $d^{*}-s^{*}\le {\displaystyle \sum _{j=1}^l({\lambda }_j^{+}-{\lambda }_j^{+})b_{kj}}\le d^{*}+s^{*}$, there is $B_k\in G_1\cap G_2$.

Through the above process, the following subsets exist in $G=G_1\cup G_2$.

(a)

$R_1=\left\{k\in G_1\left|{\displaystyle \sum _{j=1}^l\left({{\lambda }_j^{+}}^{*}-{{\lambda }_j^{-}}^{*}\right)b_{kj}}>d^{*}+s^{*}\right.\right\},$

(b)

$R_2=\left\{k\in G_2\left|{\displaystyle \sum _{j=1}^l\left({{\lambda }_j^{+}}^{*}-{{\lambda }_j^{-}}^{*}\right)b_{kj}}<d^{*}-s^{*}\right.\right\},$

(c)

$D_1=G_1-R_1,$

(d)

$D_2=G_2-R_2,$

(e)

$D=D_1\cup D_2=\left\{k\in G\left|d^{*}-s^{*}\le {\displaystyle \sum _{j=1}^l\left({{\lambda }_j^{+}}^{*}-{{\lambda }_j^{-}}^{*}\right)b_{kj}}\le d^{*}+s^{*}\right.\right\}.$

• where $D=D_1\cup D_2$ is the cross-misclassification part of the first stage, for which the following discriminant formula needs to be applied to further categorize the second stage.

$$\min \hspace{0.33em}{\displaystyle \sum _{k\in D_1}{y}_k}+{\displaystyle \sum _{k\in D_2}{y}_k}$$

$$\mathrm{s}.\mathrm{t}.\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\displaystyle \sum _{j=1}^l\left({\lambda }_j^{+}-{\lambda }_j^{-}\right)b_{kj}}-c+M{y}_k\ge 0,k\in D_1,$$

$${\displaystyle \sum _{j=1}^l\left({\lambda }_j^{+}-{\lambda }_j^{-}\right)b_{kj}}-c-M{y}_k\le -\epsilon ,k\in D_2,$$

$${\displaystyle \sum _{j=1}^l\left({\lambda }_j^{+}+{\lambda }_j^{-}\right)}=1,$$

$${\zeta }_j^{+}\ge {\lambda }_j^{+}\ge \epsilon {\zeta }_j^{+}\hspace{0.33em}and\hspace{0.33em}{\zeta }_j^{-}\ge {\lambda }_j^{-}\ge \epsilon {\zeta }_j^{-},$$

(2)

$${\zeta }_j^{+}+\hspace{0.33em}{\zeta }_j^{-}\le 1,\hspace{0.33em}j=1,2,\dots ,l,$$

$${\displaystyle \sum _{j=1}^l\left({\zeta }_j^{+}+\hspace{0.33em}{\zeta }_j^{-}\right)}=p,$$

$c:unconstrai ned,{\zeta }_j^{+}=0/1,{\zeta }_j^{-}=0/1,$

$y_k=0/1\hspace{0.33em},\mathit{other}\mathit{variables}\ge 0.$

$M—\mathit{arbitrarily}\mathit{large}\mathit{number}$;

$\epsilon —\mathit{Any}\mathit{small}\mathit{number}\mathit{greater}\mathit{than}\mathit{zero}$;

$c—\mathit{Discriminant}\mathit{score}$;

$p—A\mathit{specific}\mathit{number}\mathit{less}\mathit{than}\mathit{or}\mathit{equal}\mathit{to}\mathit{the}\mathit{number}\mathit{of}\mathit{elements}\mathit{in}D.$

If the number of elements in $D$ is greater than the number of attributes $l$, then $p=l$. According to the second stage of the discriminative results, the optimal solution is ${\lambda }_j^{+*},{\lambda }_j^{-*},c^{*}$.

The discriminant criterion for $B_k={(b_{k1},b_{k2},\cdots ,b_{kl})}^T$ is as follows:

(1)

If ${\displaystyle \sum _{j=1}^l({\lambda }_j^{+}-{\lambda }_j^{+})b_{kj}}\ge c^{*}$, there is $B_k\in G_1$;

(2)

If ${\displaystyle \sum _{j=1}^l({\lambda }_j^{+}-{\lambda }_j^{+})b_{kj}}\le c^{*}-\epsilon$, there is $B_k\in G_2$.

3.1.2. Partial Binary Tree DEA–DA Cyclic Model

Due to the relative limitations of the MIP DEA–DA model in terms of the observed objects and the stability of the discriminative results, Liu [46] introduced the partial binary tree and the iterative method for expert clustering on this basis.

Definition 1 (Degree of aggregation). 

For any two single-valued vectors $V^i=({\nu }_1^i,{\nu }_2^i,\cdots ,{\nu }_n^i)$ and $V^j=({\nu }_1^j,{\nu }_2^j,\cdots ,{\nu }_n^j)$ in n-dimensional space, the degree of aggregation $r(V^i,V^j)$ of the vector $V^i,V^j$ is as follows.

$$r\left({\mathrm{V}}^i,{\mathrm{V}}^j\right)=\frac{\left(\left|{\mathrm{V}}^i-{\overline{\mathrm{V}}}^i\right|\right)\cdot {\left(\left|{\mathrm{V}}^j-{\overline{\mathrm{V}}}^j\right|\right)}^T}{{‖{\mathrm{V}}^i-{\overline{\mathrm{V}}}^i‖}_2\cdot {‖{\mathrm{V}}^j-{\overline{\mathrm{V}}}^j‖}_2}$$

(3)

The above formula $\left|·\right|$ represents the absolute value of an element of the vector, ${‖·‖}_2$ represents the number of binomials of the vector, and element ${\overline{v}}^i=\frac{1}{n}{\displaystyle \sum _{l=1}^n{v}_l^i}$ in ${\overline{\mathrm{V}}}^i$, element ${\overline{v}}^j=\frac{1}{n}{\displaystyle \sum _{l=1}^n{v}_l^j}$ in ${\overline{\mathrm{V}}}^j$.

Theorem 1. 

$0\le r\left({\mathrm{V}}^i,{\mathrm{V}}^j\right)\le 1.$

Definition 2. 

Given a set $S$ containing $S$ n-dimensional vectors $V^i=({\nu }_1^i,{\nu }_2^i,\cdots ,{\nu }_n^i)$, then the average preference of the set $S$ is as follows:

$$Y=\frac{1}{s}{\displaystyle \sum _{i=1}^s{V}^i}$$

(4)

Based on the above theorems and definitions, the DEA–DA cyclic model of a partial binary tree is formed, as shown in Figure 3. First, decision-making members were categorized on the basis of subjective characterization into $S_1,S_2,\cdots ,S_h$. The corresponding sample sizes in each category are $l_1,l_2,\cdots ,l_h$. The first stage of classification is as follows: denote $S_1$ by $G_1$, merge $S_2,\cdots ,S_h$, and denote the new set formed by the merger by $G_2$. The MIP DEA–DA model was applied to perform an initial discriminative categorization of $G_1$ and $G_2$, resulting in two categories $G_1$ and $G_2$. Compare whether the elements in $G_1^{(1)}$, $G_2^{(1)}$ and $G_1$, $G_2$ are the same. If they are the same, the model is stable, and there is no need to classify them again, ending the loop. If they are not the same, discriminate and classify $G_1^{(1)}$ and $G_2^{(1)}$ again according to the MIP DEA–DA model until the loop stops when the result after n classifications is the same as the previous one, i.e., $G_1^{(n)}=G_1^{(n-1)}$, $G_2^{(n)}=G_2^{(n-1)}$. The final $G_1^{(n-1)}$ then ends up in the first category, with the final result denoted as $G_1^{\prime }$. The number of elements in $S_1,S_2,\cdots ,S_h$ is denoted by $n_1,n_2,\cdots ,n_k$ in $G_1^{\prime }$. In the formed $G_1^{\prime }$, if $n_i=\max (n_1,n_2,\cdots n_h)$, it means that $S_i$ represents $G_1^{\prime }$ and reflects the main properties of $G_1^{\prime }$. Accordingly, let $G_2^{(n)}={(S^{\prime }}_1,{S^{\prime }}_2,\cdots ,{(S^{\prime }}_h)$ and ${S^{\prime }}_i$ be part in $S_i$, i.e., the part separated from $G_1$, then the degree of aggregation of the elements with the categories is calculated, and the separated elements are assigned to ${S^{\prime }}_1,{S^{\prime }}_2,\cdots ,{S^{\prime }}_{i-1},{S^{\prime }}_{i+1},{S^{\prime }}_h$, and thus a new classification is formed, which leads to a further second stage of discrimination and categorization.

A new classification was derived from the first stage of discrimination and categorization, which served as the initial classification for the second stage. Denote the elements in part ${S^{\prime }}_i$ by $V^j(j=1,2,\cdots ,k)$ and $k=l_i-n_i$ by the number of elements, thus computing the preference vector $Y^p$ of $S_p^{\prime }(p=1,2,\cdots ,h)$ according to Equations (4) and (5) and the degree of aggregation $r(V^j,Y^p)$ of $V^j$ and $Y^p$ by Equation (3).

Suppose $r(V^j,Y^q)=\max \{r(V^j,Y^p),p=1,2,\cdots ,h\}(q=1,2,\cdots ,h)$, then $Y^p$ is most similar to $V^j$, so $V^j$ can be assigned to $S_q$. Based on the above process, $h-1$ new sets are obtained, and the new set is denoted by $S_1,S_2,\cdots S_{h-1}$ with the corresponding element $l_1,l_2,\cdots l_{h-1}$, respectively.

By cycling through the above steps, $G_{h-1}^{\prime }=G_2^{(n)}$ is finally obtained after $h-1$ classification stages, and $h$ categories can be obtained, as shown in Figure 4.

3.2. Aggregate Group Conflict Metrics for Decision-Making in Old Neighborhoods Renovations

3.2.1. Degree of Conflict in Decision-Making Aggregation Groups

The methods for determining the conflict degree indicators are constantly developing and innovating, mainly the TOPSIS method, the entropy weight method, and the gray correlation method, etc. Different methods have their own advantages and disadvantages and scope of application, and the selection of the appropriate method needs to take into account a number of factors, such as the object of the study, the characteristics of the data, and the purpose of the study. This chapter carries out the construction of the degree of conflict indicators for the aggregation group of decision-makers in the transformation of old neighborhoods on the basis of the vector clustering structure of decision-making members’ preferences.

Definition 3. 

Aggregate group ${\mathrm{\Omega }}^{\ast }$ conflict level indicator $\phi$ is defined as

$$\phi =1-\frac{1}{{\displaystyle \sum _{i<j}^K(n_i+n_j)}}{\displaystyle \sum _{i<j}^K(n_i+n_j)\cdot r(G^i,G^j)}$$

(5)

In the above equation, $r\left(G^i,G^j\right)=\frac{\left(\left|G^i-{\overline{G}}^i\right|\right)\cdot {\left(\left|G^j-{\overline{G}}^j\right|\right)}^T}{{‖G^i-{\overline{G}}^i‖}_2\cdot {‖G^j-{\overline{G}}^j‖}_2}$, is the degree of cohesion between the preference vectors $S^i$ and $S^j$ of two aggregates $S^i$ and $S^j$, and $G^k={{\displaystyle \sum _{V^i\in S^k}{V}^i}/‖{\displaystyle \sum _{V^i\in S^k}{V}^i}‖}_2$ is the aggregated preference vector, $1\le i,j\le K$.

The conflict level indicator of the aggregation group ${\mathrm{\Omega }}^{\ast }$ combines the weight of the number of members, which reflects the average of the degree of convergence of the aggregation preference vectors in the whole decision-making group so that the aggregation with a larger number of members has a greater impact on the conflict level of the group. When there is only one aggregation in the aggregation group ${\mathrm{\Omega }}^{\ast }$, then $\phi =0$; accordingly, when the number of aggregations increases, the group is in conflict, and the level of group conflict is directly proportional to the index of group conflict degree, that is, the larger the value of $\phi$, the larger the level of conflict, and the easier it is to form conflicts.

When the number of aggregates is more than 1, conflicts are bound to exist, and when the conflict level indicator is small, conflicts can be reduced, and decision-making can be optimized through coordination methods. However, when the conflict level indicator is large, the conflict cannot be coordinated, and then it is necessary to reduce the conflict level by introducing a third-party coordinator, adding voting methods, etc. Conflict thresholds $\delta$ are often used to define the degree of conflict, that is, the critical level of conflict in the decision-making process, which, when exceeded, indicates that the decision-making process needs to be interrupted and external methods used to reduce the decision-making conflict; whereas, within the threshold, decision-making can continue, but it is also necessary to avoid conflict as much as possible through organizational consultation to make the decision-making more rational. The conflict threshold $\delta$ is usually taken as $\xi /M$, the ratio of the number of aggregations to the total number of members.

3.2.2. Decision-Making Aggregated Group Conflict Vector

The conflict level indicator indicates the level of conflict between aggregates and aggregates, and to determine the perspective from which to reduce the level of conflict, it is also necessary to quantify the conflict between the aggregates and the entire decision-making community and, therefore, it is necessary to calculate the aggregated group conflict vector, which can be calculated to identify aggregates that have a high level of conflict with the group, and thus to coordinate the conflict.

Definition 4. 

The aggregated group conflict vector ${\mathrm{\Omega }}^{\ast }$ is defined as $\mathrm{\Psi }=({\psi }_1,{\psi }_2,\dots ,{\psi }_k)$ and the components ${\psi }_k(k=1,2,\dots ,K)$ are defined as

$${\psi }_k=1-r_k(S^k,E)=1-\frac{(S^k-{\overline{S}}^k)\cdot {(\left|E-\overline{E}\right|)}^T}{{‖S^k-{\overline{S}}^k‖}_2\cdot {‖E-\overline{E}‖}_2}$$

(6)

where $E$ is the large group preference vector,

$$E={{\displaystyle \sum _{k=1}^K\frac{n_k}{M}}{S}^k/‖{\displaystyle \sum _{k=1}^K\frac{n_k}{M}}{S}^k‖}_2$$

(7)

${\psi }_k$ indicates the degree of conflict between aggregation $S^k$ and the group as a whole, with larger values of ${\psi }_k$ indicating that there is a large discrepancy between aggregation $S^k$ and the group as a whole in the decision-making group and that there is a need for consultation with the members of $S^k$.

4. Case Study

4.1. Background of the Project

One city identified 832 neighborhoods built before 2000 without property management rooms in 683 neighborhoods, the number of buildings with potential housing safety hazards was 719, there were leaky houses in 645 neighborhoods, the need to renovate electricity in 498 neighborhoods, a fire escape was not open in 749 neighborhoods, there was no rainwater harvesting system in 694 neighborhoods, 723 neighborhoods required facade renovation, and there were 145 neighborhoods where fire water was not separated from domestic water. There were 679 neighborhoods without fitness equipment, 587 neighborhoods without kindergartens, 588 neighborhoods without community medical institutions, and 700 neighborhoods without elderly care institutions.

As these districts were built early, the supporting infrastructure and building bodies have aged badly and were basically built in the 1970s and 1990s. They face problems such as deterioration of the housing stock and ancillary facilities, uneconomical operation and maintenance of the neighborhoods, deterioration of the quality of living, and potential safety hazards. The neighborhoods were built at the time of China’s economic development, with less public space, higher building density, and poor supporting facilities. In terms of social composition, the aging rate is extremely high, and the income of the residents is generally low, which makes it difficult to implement renovation and for older people to move up and down the stairs.

4.2. Decision-Making Situations

These decision-making situations are the relevant guidelines to be followed for the renovation, as determined by the authors and the expert group following a joint consultation on this project.

(1)

Overall principle

Principle of fairness: In the decision-making process, it is necessary to ensure justice and fairness, take care of the interests of all residents, and try to avoid damage to the interests of a few people.

Principle of sustainable development: Long-term development should be considered in the process of remodeling to ensure that the remodeled buildings and environment can be maintained in good condition for a long time.

Principle of environmental friendliness: In the process of renovation, damage to the environment should be minimized, the ecological environment should be protected, and a livable environment should be built.

Benefit maximization principle: The renovation plan should be evaluated scientifically to maximize the effect of the renovation and improve the social and economic benefits of the renovation and the satisfaction of the residents.

Principle of health and safety: In the process of renovation, construction and residence safety should be guaranteed, especially focusing on the health and safety of the elderly, children, and other vulnerable groups.

Principle of in situ remodeling: In the process of remodeling, it is necessary to maintain the historical and cultural value of the building as much as possible and avoid excessive destruction, and in-situ remodeling and other methods can be used for remodeling;

(2)

Planning Objectives

Taking the improvement of residents’ quality of life as the starting and ending point, and by means of urban organic renewal, actively promote the functional improvement, space tapping, and service upgrading of old districts, connecting scattered old districts to form a safe, convenient, green, clean and orderly community for good living, so as to make the public’s sense of acquisition, happiness, and safety significantly enhanced, while stimulating the city’s potential for development;

(3)

Transformation Strategy

The old neighborhoods within the city’s central urban area are connected to a piece of the spatial distribution of relatively concentrated, scattered distribution of old neighborhoods integrated into a piece of the community (10-min living circle) as a unit to create a piece of good living community, forming a more concentrated demonstration effect of good living. This level of the central city is outside the scope of the old neighborhoods, according to the residents of the transformation of the strong will, step by step according to the needs of the residents of its “menu” transformation;

(4)

Financing

National and provincial subsidies on the part of the old neighborhood transformation of the state, province, city, and district financial subsidies at the four levels of the total sum of funds is not less than the standard of 7600 yuan per household in 2019. City and district side by adjusting and optimizing the structure of local government general bond expenditure, transferring part of the funds for the renovation of old neighborhoods, exploring the issuance of special bonds for the renovation of old neighborhoods, the people’s government (management committee) of each city district (park) should be used annually for the construction and maintenance of the backstreets and lanes of the construction and maintenance of the funds and other urban construction funds to the transformation of the old neighborhoods to tilt the local inputs. State-owned franchised enterprises or franchised pipeline units outside of the household are supported to renovate and upgrade professional business facilities and equipment, introduce support policies that attract social capital participation, and renovate and upgrade household water, power, gas, and communications. Allow the use of unused and inefficient space in the community to expand or revitalize the new construction, the construction of community supporting public services and commercial facilities. Owners, in accordance with the “who benefits, who contributes” principle, encourage residents to participate in the renovation of old neighborhoods, distinguish between different transformation content, and determine the responsibility of the residents to contribute and share the proportion;

(5)

Related guarantees

Set up the city’s leading group for the renovation of old neighborhoods; follow the principle of “municipal coordination, district-level implementation and market-oriented operation”, and boldly explore and innovate; under the premise of not adding new hidden government debt, and in accordance with the principle of maximizing the economic and social benefits of the district, the project is allowed to adjust the planning, change the use of the land, transfer the land under an agreement, simplify the approval procedures, and give financial tax and property operation incentives and other policy support for the transformation of old neighborhoods to provide strong support; firmly grasp the opportunity of the Ministry of Housing and Urban–Rural Development pilot cities for the transformation of old neighborhoods, build a mechanism covering the nine aspects of coordination, project generation, the government residents share, market-oriented operation, financial support, the masses of people to build, project management, resource integration, and long-term management; strict supervision and assessment. Carry out regular research, on-site observation, briefing, and evaluation activities, identify problems, solve problems, summarize the experience, exchange results, and enhance the effectiveness of the work; open to the community information related to the renovation of old neighborhoods, timely announcement of the progress of work and other matters of personal interest to residents, and widely accepted by the public supervision.

4.3. Data Processing

4.3.1. Subjective Qualitative Classification of Survey Respondents

This paper divides the conflict factors into four categories: interest subjects, public participation process, policy and social environment, and project decision-making mode. The interest subjects include public participation intention, expected benefits of participants, inter-subject negotiation degree, and direct cost. The public participation process includes the integrity of the participation process, the degree of community publicity, the effectiveness of the participation process, and the transparency of the participation process. The policy and social environment include the degree of openness of government information, the degree of impact on the environment, the degree of perfection of laws and regulations, the fairness and justice of bidding, the intensity of government supervision, and the preferential policies adopted by the government; the project decision-making model includes the interaction between experts and the public, the degree of public participation, the settlement of contract disputes, the number of professionals, the clarity of participation objectives, and the quality of construction. This paper invited 77 experts who are very familiar with the renovation of old residential areas as research objects by means of a questionnaire survey based on their knowledge of or experience in participating in decision-making conflicts in the renovation of old neighborhoods. Their preference information is given for the four main categories of conflict factors mentioned in the article by scoring the level of recognition that can respond to the effective reduction of decision-making conflicts. According to the nature of the work of the survey subjects, the information given by experts is divided into the following five categories:

$$S_1^1=\left\{d_k\left|d_k\right.\in G,\mathit{The}\mathit{nature}\mathit{of}\mathit{the}\mathit{decision}\mathit{maker}’\mathit{s}\mathit{work}=\mathit{Community}\mathit{staff}\right\}=\left\{d_1,d_2\cdots d_{20}\right\}$$

$$\begin{array}{l}{S}_2^1=\left\{d_k\left|d_k\right.\in G,5\succ \mathit{The}\mathit{nature}\mathit{of}\mathit{the}\mathit{decision}\mathit{maker}’\mathit{s}\mathit{work}=\mathit{Researchers}\left(\mathit{universities}\mathit{and}\mathit{research}\mathit{institutions}\right)\right\}\\ =\left\{d_{21},d_{22}\cdots d_{32}\right\}\end{array}$$

$$S_3^1=\left\{d_k\left|d_k\right.\in G,10\succ \mathit{The}\mathit{nature}\mathit{of}\mathit{the}\mathit{decision}\mathit{maker}’\mathit{s}\mathit{work}=\mathit{Building}/\mathit{construction}\mathit{unit}\right\}=\left\{d_{33},d_{34}\cdots d_{50}\right\}$$

$$\begin{array}{l}{S}_4^1=\left\{d_k\left|d_k\right.\in G,15\succ \mathit{The}\mathit{nature}\mathit{of}\mathit{the}\mathit{decision}\mathit{maker}’\mathit{s}\mathit{work}=\mathit{Government}\mathit{and}\mathit{public}\mathit{institutions}\right\}\\ =\left\{d_{51},d_{52}\cdots d_{69}\right\}\end{array}$$

$$S_5^1=\left\{d_k\left|d_k\right.\in G,\mathit{The}\mathit{nature}\mathit{of}\mathit{the}\mathit{decision}\mathit{maker}’\mathit{s}\mathit{work}=others\right\}=\left\{d_{70},d_{71}\cdots d_{77}\right\}$$

4.3.2. Partial Binary Tree-Based DEA–DA Cyclic Modeling for Survey Respondent Reclassification

Let $G_1=S_1,G_2=S_2\cup S_3\cup S_4\cup S_5$. Using Linguo 18.0 software and substituting it into the MIP DEA–DA model for the COI and HO two-stage classification cycle, the results are shown in

Table 1. The weights of parameters and indexes in the first stage of the MIP DEA–DA model.

	First Classification n = 1		Second Classification n = 2		Third Classification n = 3		Fourth Classification n = 4
	COI	HO	COI	HO	COI	HO	COI
1	−0.1298	0.1914	−0.1021	0.1456	−0.0707	0.0980	−0.0546
2	0.0230	−0.0596	0.0117	0.0635	0.0417	−0.0242 × 10−7	0.0483
3	0.0698	−0.1012	0.0432	−0.0763	0.0157	−0.0659	0.0251
4	0.0492	0.0242 × 10−7	0.0584	−0.1144	0.0373	−0.0410	0.0245
5	−0.1190	0.0889	−0.1017	0.1261	−0.0530	−0.0248 × 10−2	−0.0820
6	0.0263	−0.0270	0.0358	−0.0408	0.0530	−0.0607	0.0501
7	0.0361	−0.0516	−0.0017	0.0417	−0.0494	0.1513	−0.0066
8	0.0452	−0.0446	0.0603	−0.1286	0.0346	−0.0729	0.0070
9	−0.0463	−0.0242 × 10−7	−0.1148	−0.0242 × 10−7	−0.1644	0.1030	−0.1676
10	0.0405	−0.0243	0.0603	−0.0744	0.0477	−0.1059	0.0440
11	0.0261	0.0013	0.0472	−0.0506	0.0722	−0.0262	0.0734
12	0.0241	−0.0242 × 10−7	0.0153	0.0108	0.0089	−0.0242 × 10−7	0.0152
13	−0.0130	0.0242 × 10−7	−0.0113	0.0409	0.0133	−0.0242 × 10−7	−0.0047
14	0.0023	0.0242 × 10−7	0.0385	0.0165	0.0606	−0.0269	0.0713
15	−0.1314	0.2055	0.1224	−0.0321	0.0659	−0.1011	0.0797
16	0.0470	−0.0357	−0.0201	−0.0242 × 10−7	−0.0146	0.1086	0.0242 × 10−5
17	0.0724	−0.1524	−0.0367	0.0074	−0.0257	−0.0108	−0.0375
18	0.0570	−0.0117	0.0185	−0.0242 × 10−7	0.0578	−0.0277	0.0608
19	−0.0242	−0.0047	−0.0613	0.0303	−0.0851	−0.0242 × 10−7	−0.1141
20	−0.0175	−0.0242 × 10−7	−0.0386	−0.0242 × 10−7	−0.0286	−0.0242 × 10−7	−0.0334
	S	0.0475	S	0.0223	S	0.0072	S	−0.0701
	E	0.0242 × 10−5	E	0.0242 × 10−5	E	0.0242 × 10−5	E	0.0242 × 10−5
	D	0.2184	D	0.1425	D	0.1134	D	−0.0012
	C	−0.1014	C	−0.1565	C	−0.4426

S represents the distance of the discriminant function from the discriminant d.

and

Table 2. The results of the first stage of the circular classification of survey subjects.

		First Classification n = 1		Second Classification n = 2		Third Classification n = 3		Fourth Classification n = 4
Decision-Maker	Initial Class	COI	HO	COI	HO	COI	HO	COI
1	$G1$	$G1$	—	$G1$	—	$G1$	—	$G1$
2	$G1$	$G1$	—	$G1$	—	$G1$	—	$G1$
3	$G1$	$G1$	—	$G1$	—	*	$G1$	$G1$
4	$G1$	$G1$	—	*	$G1$	*	$G1$	$G1$
5	$G1$	$G1$	—	$G1$	—	$G1$	—	$G1$
6	$G1$	*	$G1$	*	$G1$	*	$G1$	$G1$
7	$G1$	*	$G1$	*	$G1$	*	$G1$	$G1$
8	$G1$	$G1$	—	$G1$	—	$G1$	—	$G1$
9	$G1$	*	$G1$	$G1$	—	$G1$	—	$G1$
10	$G1$	*	$G1$	*	$G1$	$G1$	—	$G1$
11	$G1$	$G1$	—	*	$G1$	$G1$	—	$G1$
12	$G1$	*	$G1$	*	$G1$	*	$G1$	$G1$
13	$G1$	*	$G1$	*	$G1$	*	$G1$	$G1$
14	$G1$	*	$G1$	$G1$	—	$G1$	—	$G1$
15	$G1$	*	$G1$	*	$G1$	$G1$	—	$G1$
16	$G1$	*	$G1$	$G1$	—	$G1$	—	$G1$
17	$G1$	*	$G1$	*	$G1$	*	$G1$	$G1$
18	$G1$	*	$G1$	$G1$	—	$G1$	—	$G1$
19	$G1$	$G1$	—	$G1$	—	*	$G1$	$G1$
20	$G1$	*	$G1$	*	$G1$	*	$G1$	$G1$
21	$G2$	$G2$	—	*	$G1$	*	$G1$	$G1$
22	$G2$	$G2$	—	$G2$	—	$G2$	—	$G2$
23	$G2$	*	$G2$	*	$G1$	*	$G1$	$G1$
24	$G2$	$G2$	—	$G2$	—	*	$G2$	$G2$
25	$G2$	$G2$	—	$G2$	—	$G2$	—	$G2$
26	$G2$	$G2$	—	$G2$	—	$G2$	—	$G2$
27	$G2$	$G2$	—	$G2$	—	$G2$	—	$G2$
28	$G2$	*	$G2$	*	$G1$	*	$G1$	$G1$
29	$G2$	$G2$	—	$G2$	—	$G2$	—	$G2$
30	$G2$	$G2$	—	$G2$	—	*	$G1$	$G1$
31	$G2$	$G2$	—	$G2$	—	$G2$	—	$G2$
32	$G2$	*	$G2$	*	$G2$	*	$G1$	$G1$
33	$G2$	*	$G2$	$G2$	—	$G2$	—	$G2$
34	$G2$	$G2$	—	*	$G1$	$G1$	—	$G1$
35	$G2$	*	$G2$	*	$G2$	$G2$	—	$G2$
36	$G2$	$G2$	—	*	$G1$	$G1$	—	$G1$
37	$G2$	$G2$	—	$G2$	—	$G2$	—	$G2$
38	$G2$	*	$G1$	*	$G1$	*	$G1$	$G1$
39	$G2$	*	$G2$	*	$G2$	$G2$	—	$G2$
40	$G2$	$G2$	—	$G2$	—	*	$G1$	$G1$
41	$G2$	*	$G2$	*	$G2$	$G1$	—	$G1$
42	$G2$	$G2$	—	*	$G1$	*	$G1$	$G1$
43	$G2$	*	$G2$	$G2$	—	$G2$	—	$G2$
44	$G2$	$G2$	—	$G2$	—	$G2$	—	$G2$
45	$G2$	*	$G1$	*	$G1$	*	$G1$	$G1$
46	$G2$	*	$G2$	*	$G2$	$G1$	—	$G1$
47	$G2$	*	$G2$	*	$G2$	$G2$	—	$G2$
48	$G2$	*	$G1$	*	$G1$	*	$G1$	$G1$
49	$G2$	*	$G2$	*	$G2$	$G2$	—	$G2$
50	$G2$	$G2$	—	$G2$	—	$G2$	—	$G2$
51	$G2$	$G2$	—	$G2$	—	$G2$	—	$G2$
52	$G2$	$G2$	—	*	$G1$	*	$G1$	$G1$
53	$G2$	*	$G1$	*		$G1$		$G1$
54	$G2$	*	$G2$	*	$G2$	*	$G1$	$G1$
55	$G2$	*	$G2$	$G2$	—	$G2$	—	$G2$
56	$G2$	*	$G2$	$G2$	—	$G2$	—	$G2$
57	$G2$	*	$G1$	$G1$	—	$G1$	—	$G1$
58	$G2$	*	$G2$	*	$G2$	$G1$	—	$G1$
59	$G2$	$G2$	—	$G2$	—	$G2$	—	$G2$
60	$G2$	$G2$	—	$G2$	—	$G2$	—	$G2$
61	$G2$	*	$G2$	*	$G2$	*	$G1$	$G1$
62	$G2$	*	$G1$	$G1$	—	$G1$	—	$G1$
63	$G2$	$G2$	—	*	$G1$	$G2$	—	$G2$
64	$G2$	$G2$	—	$G2$	—	$G2$	—	$G2$
65	$G2$	*	$G2$	*	$G2$	$G1$	—	$G1$
66	$G2$	$G2$	—	$G2$	—	$G2$	—	$G2$
67	$G2$	$G2$	—	$G2$	—	$G1$	—	$G1$
68	$G2$	$G2$	—	$G2$	—	$G1$	—	$G1$
69	$G2$	$G2$	—	$G2$	—	$G2$	—	$G2$
70	$G2$	$G2$	—	$G2$	—	$G1$	—	$G1$
71	$G2$	*	$G2$	$G2$	—	$G2$	—	$G2$
72	$G2$	*	$G2$	*	$G2$	$G1$	—	$G1$
73	$G2$	*	$G1$	$G1$	—	*	$G1$	$G1$
74	$G2$	*	$G1$	$G1$	—	$G1$	—	$G1$
75	$G2$	*	$G2$	*	$G2$	*	$G1$	$G1$
76	$G2$	*	$G2$	*	$G2$	*	$G1$	$G1$
77	$G2$	*	$G2$	*	$G2$	*	$G1$	$G1$

* represents the cross-misclassification part.

.

Through four cycles, the classification reaches stability, and the first class of samples is obtained as follows:

$$\begin{array}{c}{G}_1^{\prime }=\{d_1,d_2,\dots ,d_{21},d_{23},d_{28},d_{30},d_{32},d_{34},d_{36},d_{38},d_{40},d_{41},d_{42},d_{45},d_{46},d_{48},d_{52},d_{53},d_{54},\\ d_{57},d_{58},d_{61},d_{62},d_{65},d_{67},d_{68},d_{70},d_{72},d_{73},d_{74},d_{75},d_{76},d_{77}\}\end{array}$$

$G_1^{\prime }$ contains 51 elements, respectively, from $S_1$, $S_2$, $S_3$ and $S_4$, denoted as $S_1^{1\prime }$, while denoting $S_2^1$ that divides the elements as $S_2^{1\prime }$, $S_3^1$ as $S_3^{1\prime }$, $S_4^1$ as $S_4^{1\prime }$, $S_5^1$ as $S_5^{1\prime }$. Let $S_1^2=S_2^{1\prime }$; $S_2^2=S_3^{1\prime }$; $S_3^2=S_4^{1\prime }$; $S_4^2=S_5^{1\prime }$ be the second stage of initial classification for another cycle.

The following three subsequent cycles are the same as in the first stage, yielding the following classification results:

$$G_2^{\prime }=\left\{d_{22},d_{24},d_{25},d_{26},d_{27},d_{29},d_{31}\right\}$$

$$G_3^{\prime }=\left\{d_{33},d_{35},d_{37},d_{39},d_{43},d_{44},d_{47},d_{49},d_{50}\right\}$$

$$G_4^{\prime }=\left\{d_{51},d_{55},d_{56},d_{59},d_{60},d_{63},d_{64},d_{66},d_{69}\right\}$$

$$G_5^{\prime }=\left\{d_{71}\right\}$$

$G_2^{\prime }$ contains seven elements, $G_3^{\prime }$ and $G_4^{\prime }$ each contains nine elements, $G_5^{\prime }$ contains only one element, and $G_5^{\prime }$ is a minority opinion because the number of members in it is less than two. In a large group decision-making process, there is a need to protect minority views but not to jeopardize the entire large group decision-making process by individual views. For this reason, this paper reclassifies $d_{71}$, a single element, by dividing it into $G_1^{\prime },G_2^{\prime },G_3^{\prime },G_4^{\prime }$. From Equations (4) and (5), the average preference of $G_1^{\prime }$, $G_2^{\prime }$, $G_3^{\prime }$, $G_4^{\prime }$ is as follows:

$$\begin{array}{l}{Y}_1=\frac{1}{51}{\displaystyle \sum _{k=1}^{51}{d}_k=}=5.555.555.515.435.245.065.065.255.495.535.615.375.475.55\\ 5.295.185.295.454.925.16)\end{array}$$

$$\begin{array}{l}{Y}_2=\frac{1}{7}{\displaystyle \sum _{k=1}^7{d}_k=}=4.714.294.573.864.714.294.574.434.1443.434.294.574.434.71\\ 4.574.574.864.865)\end{array}$$

$$\begin{array}{l}{Y}_3=\frac{1}{9}{\displaystyle \sum _{k=1}^9{d}_k=}=5.1155.445.2254.225.335.225.225.225.225.2254.565.335.67\\ 55.115.785.67)\end{array}$$

$$\begin{array}{l}{Y}_4=\frac{1}{9}{\displaystyle \sum _{k=1}^9{d}_k=}=5.5655.445.445.565.335.6754.7854.674.894.894.225.114.78\\ 5.335.225.565.56)\end{array}$$

Calculate the degree of polymerization between $d_{71}$ and $G_1^{\prime },G_2^{\prime },G_3^{\prime },G_4^{\prime }$ according to Equation (3) for 0.899, 0.926, 0.915, and 0.92, respectively. Therefore, dividing $d_{71}$ into $G_2^{\prime }$ yields a new classification result and the final classification result for

$$\begin{array}{l}{S_1}^{\prime }=\{d_1,d_2,\dots ,d_{21},d_{23},d_{28},d_{30},d_{32},d_{34},d_{36},d_{38},d_{40},d_{41},d_{42},d_{45},d_{46},d_{48},d_{52},d_{53},d_{54},\\ d_{57},d_{58},d_{61},d_{62},d_{65},d_{67},d_{68},d_{70},d_{72},d_{73},d_{74},d_{75},d_{76},d_{77}\}\end{array}$$

$$S_2^{\prime }=\left\{d_{22},d_{24},d_{25},d_{26},d_{27},d_{29},d_{31},{\mathrm{d}}_{71}\right\}$$

$$S_3^{\prime }=\left\{d_{33},d_{35},d_{37},d_{39},d_{43},d_{44},d_{47},d_{49},d_{50}\right\}$$

$$S_4^{\prime }=\left\{d_{51},d_{55},d_{56},d_{59},d_{60},d_{63},d_{64},d_{66},d_{69}\right\}$$

4.3.3. Cluster Group Conflict Quantification

The aggregation preference vectors for the above four aggregations are first calculated, as shown in

Table 3. Expert Aggregation and Aggregation Preference Vector.

Aggregation Sk′	Expert Members nk	Aggregation Preference Vector Gk
${S_1}^{\prime }$	51	(0.232, 0.233, 0.23, 0.227, 0.219, 0.211, 0.211, 0.22, 0.23, 0.231, 0.234, 0.224, 0.229, 0.232, 0.221, 0.216, 0.221, 0.228, 0.201, 0.215)
${S_2}^{\prime }$	8	(0.22, 0.207, 0.22, 0.176, 0.245, 0.226, 0.239, 0.233, 0.196, 0.201, 0.163, 0.214, 0.22, 0.201, 0.245, 0.245, 0.239, 0.251, 0.251, 0.251)
${S_3}^{\prime }$	9	(0.220, 0.215, 0.35, 0.225, 0.215, 0.182, 0.23, 0.225, 0.225, 0.225, 0.225, 0.225, 0.215, 0.196, 0.23, 0.244, 0.215, 0.22, 0.249, 0.244)
${S_4}^{\prime }$	9	(0.241, 0.217, 0.236, 0.236, 0.241, 0.231, 0.245, 0.217, 0.207, 0.217, 0.202, 0.212, 0.212, 0.183, 0.221, 0.207, 0.231, 0.226, 0.241, 0.241)

.

There is no minority opinion in the clustering results, and an indicator of the degree of conflict of the decision makers’ aggregated groups $\phi =0.244$ is calculated using Equation (5). Depending on the conflict degree threshold $\delta$, let $\xi /M=51/77=0.66$, then $\phi <\delta$, which indicates a low level of conflict between aggregates in that decision-making group. Then, the decision-making group conflict vector is calculated according to Equation (6), resulting in $\mathrm{\Psi }=(0.425,0.513,0.851,0.493)$, which shows the degree of conflict between each aggregation with ${S_3}^{\prime }>{S_2}^{\prime }>{S_4}^{\prime }>{S_1}^{\prime }$. The aggregation with the highest level of conflict is ${S_3}^{\prime }$, indicates that it has a high level of conflict with the entire decision-making group. The overall evaluation score for each aggregation is $\begin{array}{l}\{5.15,4.92,5.19,4.9,5.17,4.78,5.2,5.03,4.84,4.94,4.69,4.93,4.93,4.58,5.15,5.12,\\ 5.09,5.2,5.31,5.34\}.\end{array}$

4.4. Analysis of Results

Based on the calculation results, the conflict level indicator between the four aggregates is 0.244 less than the conflict threshold, and the conflict between the aggregates is small. However, among the four aggregations, ${S_3}^{\prime }$ has the largest conflict vector with the whole group of decision-makers, indicating the need to coordinate the relationship between ${S_3}^{\prime }$ and the whole group.

In ${S_3}^{\prime }$, the factor “interaction between experts and the public” scored the highest at 5.78, followed by the number of professionals (5.67) and contractual dispute resolution (5.67), all of which are derived from the project’s decision-making model. It indicates that the clustering focuses on the professionalism of the transformation. An important part of the renovation process for old neighborhoods involves different levels of different industries of interest, a need to bring together experts in all aspects of the planning, design, and implementation of the entire process, and the need for professionals to provide guidance throughout. It is also necessary to summarize the public’s opinions, strengthen communication and interaction with the public during the renovation process, and improve or adjust the contents of the renovation through the public’s opinions. “Expected benefits for the participant”, “Effectiveness of the public participation process”, and “Degree of public participation” also have high scores of 5.44, 5.33, and 5.33, respectively. Old urban district renovation projects, through the improvement of the city’s basic functions and other aspects of the overall development of the city, lead to the realization of a well-off society. Meanwhile, urban development can lead to economic upgrading, improve the living standards of the residents, improve the quality of life, and lead to the development of the construction industry, and therefore, the need to balance the interests of all parties involved in the interests of the main body, and to achieve the coordination of the public benefits and personal interests, to avoid conflicts and contradictions caused by the interests of the incoherence. Throughout the transformation process of the old neighborhoods, different levels of the public have different opinions on the transformation preferences, the implementation of the transformation plan, the quality of construction, etc. The ability of the decision-makers to read, understand, and adopt the public’s opinions directly affects the effectiveness of the public’s participation and also affects the success of the transformation project in realizing the interests of the public. In addition, the degree and depth of public participation are hampered by insufficient public participation channels and an unclear participation system. There are two ways for Chinese citizens to participate in public policy: institutional and non-institutional. Institutional policy participation is the main channel, but there are too few opportunities for ordinary citizens to participate directly. The cost of public participation in the public policy process through non-institutional channels is often high.

From the perspective of the ratings of the whole group of decision-makers, the ratings of “the expected benefits of the participant” and “the public’s willingness to participate” are higher in terms of the participant’s main body. It has long been a contradiction that all parties have divergent interests when it comes to renovating old urban neighborhoods. Renovating old neighborhoods involves a wide range of stakeholders with different interests, and the effective coordination of public benefits and personal interests can not only drive the urban economy but also realize the overall development of the city. In the transformation process, the public’s “unintentional participation” in public affairs limits the level of governance in the renovation of old neighborhoods. Enhancing the public’s willingness to participate not only summarizes the public’s opinions but also promotes the smooth progress of the renovation process. Regarding the public participation process, the ratings of “effectiveness of the participation process” and “completeness of the participation process” are relatively high. The whole process of the old neighborhood renovation project, from pre-planning to renovation design to project construction and acceptance as well as later neighborhood management, requires listening to and reading the public’s opinions so as to truly realize the effective and complete participation of the public. In terms of policy and social environment, the overall scores did not reach 5 but were all around 4.9, indicating that the old neighborhood renovation projects are mainly affected by factors inherent in the projects themselves, but they also need the overall constraints of national laws and regulations. At the same time, the whole renovation process needs to be open and transparent, maintaining the principle of fairness and impartiality and ensuring that the renovation of old neighborhoods is carried out in a legal and compliant manner. In terms of the project decision-making model process, the overall score is higher than 5, indicating that in the process of renovation of old neighborhoods, it is essential to ensure the professionalism of the renovation and the accuracy of the decision-making process, as well as the degree and depth of the public participation, and to guarantee the quality of the construction, so that old neighborhoods can be renovated of high-quality and high-level.

5. Discussion

Renovations of old urban neighborhoods are essential measures of urban renewal; however, due to the inaccurate prejudgment of the impact of the consequences of the decision-making program, coupled with the existence of differences in the values and understanding of the decision-makers, often leads to conflict in the process of decision-making and implementation, resulting in a tremendous waste of manpower, material resources, and financial resources and losses, which cause an irreversible situation. This paper selects four types of conflict factors as the evaluation attributes of the old neighborhood renovation project, collects the evaluation matrix of experts on the renovation work of a city’s old neighborhood with the help of examples, clusters the experts using the MIP DEA–DA model and measures the degree of conflict between the experts, and searches for where the conflict lies in the decision-making of the renovation of the city’s old neighborhood. As a result of the study, the imbalance of interests among participants is a long-standing contradiction since the renovation process of old neighborhoods involves participants from different levels and industries, and their interests vary, creating an imbalance of interests. In addition, the renovation process also needs to gather experts in all aspects, from planning design to implementation, to provide professional guidance, and at the same time, strengthen communication and interaction with the public, sum up the public’s views based on the public’s perception of the renovation of the content of the modification or adjustment. In general, the urban renovation of old districts’ decision-making conflict lies in the balance between the interests of the main benefits and decision-making professionalism, not only to ensure that the renovation of professionalism and decision-making are accurate but also to ensure that the degree and depth of public participation in the development of public benefits and economic benefits while ensuring the professionalism of the decision-making.

The advantage of applying the conflict degree calculation method in group decision-making to public participation in old neighborhood renovation decision-making and constructing a partial binary tree MIP DEA–DA cyclic clustering model based on the preference information of decision-making experts is that it can effectively avoid the subjectivity and arbitrariness of clustering decision-makers according to certain quantitative information of individuals, and then measure the conflict between clusters and large groups, which can provide a new idea for quantifying the conflict in the decision-making of old neighborhood renovation. This study also further refines the decision-making process for urban retrofitting of older neighborhoods based on group decision-making theory. The practical implications of this paper are that it helps to build a set of multi-faceted participation mechanisms with residents as the main body, integrates the public’s point of view into the decision-making process of the renovation of old neighborhoods, and scientifically measures the preferences and needs of the main body of the decision-making process, so as to reach a consensus of the main body of the decision-making process. To a certain extent, it helps to improve the effect of public participation, build a good civil society, fundamentally eliminate conflicts in the decision-making process of old neighborhood renovation, promote scientific and democratic decision-making of old neighborhood renovation, and then improve the scientific and democratic level of urban renewal planning.

Due to the wide range of aspects involved in the renovation of old neighborhoods and the complexity of the actual situation, in practice, we can only try to consider but also cannot avoid the omission of a certain aspect. This paper can put forward relevant policy recommendations for the decision-making conflicts arising in the process of renovation of old neighborhoods. Firstly, since the public’s “unintentional participation” in public affairs during the renovation process has limited the level of governance of the renovation of old neighborhoods, we need to enhance public participation willingness and strengthen consultation. During the renovation process, a multi-channel feedback mechanism should be provided to collect public feedback in a timely manner so as to solve problems and improve the work in a timely manner.

Secondly, standard procedures for multi-actor participation in the renovation of old neighborhoods should be developed and improved, and a synergistic system of multiple cooperation should be put into place. As different stakeholders have different interests, establishing a relevant organizational department is essential in order to understand and coordinate the interests of multiple stakeholders so as to develop a set of standardized participation procedures accepted by all parties. At the same time, a mechanism for improving and revising the procedure should be established, with the objective of ensuring professionalism of decision-making to a certain extent through effective and reasonable amendments and improvements. In addition, the market mechanism should be used in the implementation of the renovation of old neighborhoods in the establishment of the main body due to the flexibility of the market mechanism so that the multi-interested subject’s demands can be satisfied.

Thirdly, the results in terms of policy and social environment suggest the need for open and transparent decision-making processes in old neighborhood improvement projects. The integrity of the participatory process is the most significant of the public participation process factors, so improving the integrity of the participatory process is particularly important for reducing conflict. Decision-making processes need to be open and transparent so that all residents can understand the details and process of decision-making, thus increasing their trust and acceptance of the decisions.

Fourthly, the findings of the study show that the lack of public participation channels also affects the success of the renovation projects, so there is a need to provide diversified ways of participation so as to allow more members of the public to participate in the decision-making process. For example, the government can collect opinions and suggestions online, interact with the public on social media, etc., to involve more people. Through diversified participation channels, it will be easier for the public to participate in decision-making, thus increasing the depth and breadth of public participation.

Fifthly, in response to the plurality of decision-making bodies and the possible conflict of interests, government departments should play a key role in coordination. The Government should communicate fully with various stakeholders to understand the needs and requirements of all parties and balance the interests of all parties involved. For example, the Government can hold meetings in the form of hearings and symposiums to consult residents and property companies, listen to their suggestions and opinions, and respond to residents’ concerns and problems in a timely manner. In addition, the Government can also collect residents’ views and feedback through questionnaires and consultation hotlines, so as to balance the interests of all parties. It can also facilitate the implementation of the transformation of old neighborhoods through the introduction of a market mechanism.

There are still several limitations in this paper. Although the conflict quantification method of old neighborhood renovation decision-making proposed in this paper can be used to determine the degree of conflict within the decision-making group involved in renovating old neighborhoods, it needs to be combined with the actual situation, specific analysis of specific problems, and it still needs to fully investigate the local environment before decision-making, investigate and analyze the main body of the decision-making process, and promote the communication and exchange of the decision-makers in the renovation decision-making. In the future, further research is needed to study the renovation decision-making of old districts under public participation, and after measuring the conflict, it is necessary to introduce a conflict coordination mechanism to resolve the conflict and improve the dynamic interaction mechanism in the transformation decision-making of old districts under public participation. In addition, it is also necessary to establish a post-evaluation model of the decision-making effect to post-evaluate the effect of the whole decision-making in order to increase the rationality and scientificity of the decision-making.

6. Conclusions

This paper combines group decision-making theory and practice to explore the conflicts in the decision-making of old neighborhood renovation and uses the MIP-DEA–DA model to quantify the conflicts in the old neighborhood renovation in the actual situation and puts forward some informative opinions on the decision-making of old neighborhood renovation under public participation. The results of the study show that the conflict in old neighborhood transformation decision-making exists between the balance of benefits of interest subjects and decision-making professionalism. Therefore, in practice, decision-makers need to be more concerned with the balance of interests between the participating subjects to avoid conflicts and contradictions caused by the imbalance of interests, as well as the development of both economic and public benefits simultaneously. While doing so, they should also ensure the professionalism of their decision-making so that old neighborhoods are transformed at a high level and with high quality.