Research on social ecological evaluation of the spatial form of old urban blocks based on dual probabilistic linguistic term sets

Abstract

Under the rapid process of urbanization, many early renovated urban villages have also encountered many problems. Due to the rapid development of urban construction and the continuous changes in spatial functions, early renovated urban villages have already encountered problems such as unreasonable commercial distribution, lack of parking spaces, reduced commercial vitality, and commercial activities crowding out affecting the normal lives of villagers. There is a serious contradiction between the need for development and the quality of life of villagers. Due to the fixed nature of architectural space, only by fully understanding the essential morphological characteristics of the space can we find the optimal solutions for different space usage functions, and obtain the matching of the optimal solutions in the existing space requirements. The social ecological evaluation of the spatial form of old urban blocks is a multi-attribute group decision making (MAGDM). Recently, the grey relational analysis (GRA) and CRITIC method has been used to cope with MAGDM issues. The dual probabilistic linguistic term sets (DPLTSs) are used as a tool for characterizing uncertain information during the social ecological evaluation of the spatial form of old urban blocks. In this manuscript, the dual probabilistic linguistic GRA (DPL-GRA) method is built to solve the MAGDM under DPLTSs. The CRITIC method is used to obtain the attributes weights. In the end, a numerical case study for social ecological evaluation of the spatial form of old urban blocks is given to validate the proposed method.

Keywords

Multi-attribute group decision making (MAGDM)dual probabilistic linguistic term sets GRA CRITIC social ecological evaluation

1 Introduction

Multiple attribute decision making (MADM) is a problem of sorting alternatives when each alternative has many attribute values. In traditional MADM, the attribute values of alternatives are just numbers [1 –5]. At that time, data are mainly numbers. But with the development of digital technology, now, the data, which named Big Data, contains various information. The types of data are no longer just numbers [6 –10]. The traditional multiple attribute decision making methods need to improved according to the time. Torra [11] proposed hesitant fuzzy sets (HFSs). Hesitant fuzzy set is a function that contains a set of membership values for each element in the domain and makes it possible to consider all the possible values [12 –16]. It is considered to be a better aggregate calculation function, but the structure of this set only concludes the membership values. In 2012, Zhu, Xu and Xia [17] introduced a new extension of fuzzy set, called dual hesitate fuzzy sets (DHFSs), which contains two parts: the membership hesitancy function and the non-membership hesitancy function. It provides a way to allow decision makers and experts to think about the pros and cons of an alternative under a certain attribute from two opposite sides [18 –22]. In actual situation, the weight of each membership and non-membership is different. But either in HFSs or DHFSs, it assumes that the weight of each membership and non-membership is equal to each other. So, Zhu and Xu [23] proposed the probability-hesitant fuzzy sets (P-HFS) and Hao, Xu, Zhao and Su [24] proposed the probability dual hesitant fuzzy sets (P-DHFS). These two functions give probabilities to each degree of membership and non-membership [25 –31]. In order not to lose the information in the variables during the decision-making process, Zadeh [32] first introduced the linguistic variables. Scholars have proposed more term sets on this basis, such as hesitant fuzzy linguistic term sets (HFLTSs) [33] and Probabilistic linguistic term sets (PLTS) [34]. In 2017, the dual probabilistic linguistic term set (DPLTS) was proposed by Xie, Xu and Ren [35], which contains linguistic term sets’ membership degree and non-membership degree and weights for each degree. In real life, this function is most suitable for multi-criteria group decision making problems, which were often qualitative analysis in the past. In this paper, DPLTS is the background for making decisions, and we will study the decision-making method under the dual probabilistic linguistic term set (DPLTS) condition.

Gray Relation Analysis (GRA) is a multi-factor statistical analysis method, whose purpose is to analyze the relative strength of a project affected by other factors in a gray system [36]. Specifically, assuming and knowing that a certain indicator may be related to several other factors, then analyze which factor is more related to this indicator and which factor is relatively weaker, and so on [37]. By sorting the factors and getting an analysis result, it is easy to know which of the factors is more related to the indicator decision-makers are concerned about [38]. GRA is widely used in MADM research field [39 –44]. Wei [45] established an optimization model based on the basic ideal of traditional GRA method to deal with intuitionistic fuzzy MADM with incompletely known weight information. In failure mode and effect analysis, Hua, Jing and Martínez [46] used GRA combined with decision-making trial and evaluation laboratory (DEMATEL) method to objectively determine the objective weight of attribute. Li, Xu, Liu and Wei [47] extended grey relation analysis to probabilistic linguistic term sets and proposed a new consensus judgment mechanism in the field of social network group decision-making. In order to increased safety of warehouse facilities, Hsu, Hwang and Tsou [48] calculated the weights critical indicators by BWM method, and ranked the critical risks of picking and material-handling accidents in a warehouse facility through GRA method. Another difference in my method is the way of determining the weight. The Criteria Importance Through Intercriteria Correlation (CRITIC) method [49]. The essence of this method is to determine the objective weight which are calculated with the contrast intensity and conflict contained in the elements of the decision matrices [50]. It was proposed for firms ranking at the beginning, and now it has been extended for many situations where objective weight is required and objective ordering of solutions is required. In this paper, we will use the core of the CRITIC: the method of determining weight.

Rapid urbanization has made urban blocks an important area of planning practice, reshaping spatial forms in four aspects: commercialization, centralization, motorization, and internal and external spatial relationships. This has led to rapid changes in social ecology, so it is necessary to analyze it from a social ecological perspective. Aiming to proposed a new method and perspective to solve this problem, this article conducts a new evaluating model for appraising the spatial form of old urban blocks. Firstly, the grey relational analysis (GRA) and CRITIC method has been used to cope with MAGDM issues. Then,the dual probabilistic linguistic term sets (DPLTSs) are used as a tool for characterizing uncertain information during the social ecological evaluation of the spatial form of old urban blocks. In this manuscript, the dual probabilistic linguistic GRA (DPL-GRA) method is built to solve the MAGDM under DPLTSs. The CRITIC method is used to obtain the attributes weights [51].

The main contribution of this paper is: 1. DPLTS have never been used in the social ecological evaluation of the spatial form of old urban blocks, this research applied this set to practical evaluation of old urban blocks. 2. A new model is proposed, this is the first time to use GRA method to deal with DPLTS. 3. The practical significance of this model is to evaluate the Social Ecological of Old Urban Blocks, which helps to achieve urban ecology, form coordinated development of society, economy, and nature and overall ecology, and achieve integrated development. It is a sustainable development model.

This new method has following advantages: 1. This is a total objective method. Blocks are closely related to people’s lives. In the evaluation process, there are too many factors that are factors in experts’ daily lives. Therefore, experts can easily bring their own preferences and biases into the decision-making process and incorrectly determine the weight of attributes. Using CRITIC method can avoid the preference of experts. 2. DPLTS can contain lots of fuzzy information, and this is beneficia for appraising the spatial form of old urban blocks.

The structure of this paper is listed below. In Section 2, the DPLTSs is introduced. In Section 3, DPL-GRA method is designed for MAGDM under DPLTSs with CRITIC weight method. Section 4 gives an illustrative case for social ecological evaluation of the spatial form of old urban blocks and some comparative analysis. Some remarks are given in Section 5.

2 Basic knowledges

Some basic knowledge and operations about DPLTS.

Definition 1 [52]. Hypothetically, ℵ = {ℵ _z|z = - Ξ, ⋯ , -1, 0, 1, ⋯ Ξ} is a finite and discontinuous collection of languages, subsequently, DPLTS can be set out: $M_{\partial} = {< z, G (\partial), NG (\partial) > | z \in Z}$ $G (\partial) = {G^{(t)} (\partial^{(t)}) | G^{(t)} \in ℵ, \partial^{(t)} \geq 0, \sum_{t = 1}^{Δ G (\partial)} \partial^{(t)} \leq 1}$ $NG (\partial) = {N G^{(t)} (\partial^{(t)}) | N G^{(t)} \in ℵ, \partial^{(t)} \geq 0, \sum_{t = 1}^{Δ NG (\partial)} \partial^{(t)} \leq 1}$ the the grade of membership and the grade of non-membership of z ∈ Z are denoted as G (∂) , NG (∂) respectively. ΔG (∂) is the magnitude of elements in G (∂), G^(t) (∂^(t)) is the t-th element of the G (∂), the proportion of the t-th G (∂) is ∂^(t).

Analogously, ΔNG (∂) is the magnitude of elements in NG (∂), NG^(t) (∂^(t)) is the t-th element of the NG (∂), the proportion of the t-th NG (∂) is ∂^(t). $Here, ℵ_{- Ξ} \leq max {G (\partial)} \oplus max {G (\partial)} \leq ℵ_{Ξ}$ $ℵ_{- Ξ} \leq min {NG (\partial)} \oplus min {NG (\partial)} \leq ℵ_{Ξ}$ Some operations for language terms are listed below [53]: $ℵ_{z_{1}} \oplus ℵ_{z_{2}} = ℵ_{z_{1} + z_{2}}$ (1) $ℵ_{z_{1}} \otimes ℵ_{z_{2}} = ℵ_{z_{1} \otimes z_{2}}$ (2) $ℓ ℵ_{z} = ℵ_{ℓ z}$ (3)

$M_{1} \oplus M_{2} = ⋃_{\binom{G_{1} (\partial) \in M_{1}, N G_{1} (\partial) \in M_{1},}{G_{2} (\partial) \in M_{2}, N G_{2} (\partial) \in M_{2}}} \begin{matrix} {(Γ^{- 1} (Γ ({G_{1}}^{(t)}) + Γ ({G_{2}}^{(t)}) - Γ ({G_{1}}^{(t)}) Γ ({G_{2}}^{(t)}))) ({\partial_{1}}^{(t)} {\partial_{2}}^{(t)}), \\ (N G_{1} \otimes N G_{2}) ({\partial_{1}}^{(t)} {\partial_{2}}^{(t)})} \end{matrix}$ (4)

$M_{1} \otimes M_{2} = ⋃_{\binom{G_{1} (\partial) \in M_{1}, N G_{1} (\partial) \in M_{1},}{G_{2} (\partial) \in M_{2}, N G_{2} (\partial) \in M_{2}}} \begin{matrix} {(G_{1} \otimes G_{2})) ({\partial_{1}}^{(t)} {\partial_{2}}^{(t)}), \\ (Γ^{- 1} (Γ (N {G_{1}}^{(t)}) + Γ (N {G_{2}}^{(t)}) - Γ (N {G_{1}}^{(t)}) Γ (N {G_{2}}^{(t)})) ({\partial_{1}}^{(t)} {\partial_{2}}^{(t)})} \end{matrix}$ (5)

$ℓ M = {Γ^{- 1} (1 - (1 - Γ (G^{(t)})))^{ℓ} (\partial^{(t)}), (NG (\partial)^{(t)})^{ℓ} (\partial^{(t)})}$ (6)Γ (G^(t)) and Γ (NG^(t)) are numeric variable, they belong to hesitation fuzzy set. The equivalent conversion function from the hesitation fuzzy linguistic term set to the hesitation fuzzy set can refer to the literature of Gou, Xu and Liao [54]: $Γ (G^{(t)}) = \frac{z + Ξ}{2 Ξ} = ƛ$ (7) $Γ^{- 1} (Γ (G^{(t)})) = ℵ_{(2 ƛ - 1) Ξ}$ (8)

Γ^-1 and Γ are reversible transformation functions: Γ : [- Ξ, Ξ] → [0, 1] and Γ^-1 : [0, 1] → [- Ξ, Ξ].

Definition 2 [52]. The score function of the DPLTS is: $DS (M_{\partial}) = ℵ_{O_{1} - O_{2}}$ (9) where $o_{1} = \sum_{t = 1}^{Δ G (\partial)} Γ (G (\partial)) \partial^{(t)} / \sum_{t = 1}^{Δ G (\partial)} \partial^{(t)}$ , $o_{2} = \sum_{t = 1}^{Δ NG (\partial)} Γ (NG (\partial)) \partial^{(t)} / \sum_{t = 1}^{Δ NG (\partial)} \partial^{(t)}$ .

Definition 3 [52]. The precision mapping of the DPLTS is: $\begin{matrix} DPF (M_{\partial}) = (\sum_{t = 1}^{Δ G} (\partial^{(t)} (Γ (G (\partial^{(t)})) - o_{1}))^{2})^{\frac{1}{2}} / \sum_{t = 1}^{Δ G} \partial^{(t)} + \\ (\sum_{t = 1}^{Δ NG} (\partial^{(t)} (Γ (NG (\partial^{(t)})) - o_{1}))^{2})^{\frac{1}{2}} / \sum_{t = 1}^{Δ NG} \partial^{(t)} \end{matrix}$ (10)

According to definitions (2) and (3), the size comparison rules of any two dual probability languages are as follows:

If DS (M_1∂) > DS (M_2∂), then M_1∂ > M_2∂; If DS (M_1∂) < DS (M_2∂), then M_1∂ < M_2∂;

If DS (M_1∂) = DS (M_2∂), then continue to compare the exact values of M_1∂ and M_2∂,

If DPF (M_1∂) = DPF (M_2∂), then M_1∂ = M_2∂;

If DPF (M_1∂) < DPF (M_2∂), then M_1∂ < M_2∂;

If DPF (M_1∂) > DPF (M_2∂), then M_1∂ > M_2∂.

Definition 4 [52]. The dual probabilistic linguistic weighted aggregated (DPLWA) operator: $DPLWA (M_{1}, M_{2} \dots M_{n}) = \oplus_{α = 1}^{k} ζ_{α} M_{α}$ (11)

ζ_α is the corresponding proportion of M_α.

3 DPL-GRA method for MAGDM under DPLTSs

In this section, we will propose GRA method based on dual probability language, and utilize information entropy to solve the proportion of each attribute. the specific work is as follows.

Assume there are H schemes and K properties, M_hk is the expert’s evaluation of plan h under attribute k.

Step 1. Switch the expert evaluation information of cost attributes into benefit data.

Step 2. We use CRITIC method to determine the weights of attributes.

First, we use the score value to calculate the correlation coefficient. $ρ_{kl} = \frac{\sum_{h = 1}^{H} (DS (M_{hk}) - D \bar{S} (M_{hk})) (DS (M_{hl}) - D \bar{S} (M_{hl}))}{\sqrt{\sum_{h = 1}^{H} {(DS (M_{hk}) - D \bar{S} (M_{hk}))}^{2} \sum_{h = 1}^{H} {(DS (M_{hl}) - D \bar{S} (M_{hl}))}^{2}}}$ (12) $D {\bar{S}}_{(M_{hk})} = \frac{1}{H} \sum_{h = 1}^{H} D S_{(M_{hk})}; h = 1, \dots \dots, H$

Then, the standard deviation of each attribute: $σ_{k} = \sqrt{\frac{1}{H - 1} \sum_{h = 1}^{H} {(DS (M_{hk}) - D \bar{S} (M_{hk}))}^{2}}; h = 1, \dots \dots, H$ (13) The index (C) is calculated using the following equation: $α_{k} = σ_{k} \sum_{l = 1}^{K} (1 - ρ_{kl}); l = 1, \dots \dots, K$ (14) The weight of attribute is: $β_{k} = \frac{α_{k}}{\sum_{k = 1}^{K} α_{k}}; k = 1, \dots \dots, K$ (15)

Step 4. Define the dual probability language terms PIS (DPLTS-PIS) $U_{k}^{+}$ and the dual probability language terms NIS (DPLTS-NIS) $U_{k}^{-}$ as: $U_{k}^{+} = max_{h} (M_{hk}) = (max_{h} (G_{hk} (\partial)), min_{h} ({NG}_{hk} (\partial)))$ (16) $U_{k}^{-} = max_{h} (M_{hk}) = (min_{h} (G_{hk} (\partial)), max_{h} ({NG}_{hk} (\partial)))$ (17)

Step 5. Compute the grey relational coefficient (GRC) of every alternative between each alternative and DPLTS-PIS and the GRC between each alternative and DPLTS-NIS as:

$\begin{matrix} DPLTS - PIS (π_{hk}) = \frac{min_{1 \leq h \leq H} min_{1 \leq k \leq K} DPLTSED (M_{hk}, U_{k}^{+}) + ℓ max_{1 \leq h \leq H} max_{1 \leq k \leq K} DPLTSED (M_{hk}, U_{k}^{+})}{DPLTSED (M_{hk}, U_{k}^{+}) + ℓ max_{1 \leq h \leq H} max_{1 \leq k \leq K} DPLTSED (M_{hk}, U_{k}^{+})} \\ DPLTS - NIS (π_{hk}) = \frac{min_{1 \leq h \leq H} min_{1 \leq k \leq K} DPLTSED (M_{hk}, U_{k}^{-}) + ℓ max_{1 \leq h \leq H} max_{1 \leq k \leq K} DPLTSED (M_{hk}, U_{k}^{-})}{DPLTSED (M_{hk}, U_{k}^{-}) + ℓ max_{1 \leq h \leq H} max_{1 \leq k \leq K} DPLTSED (M_{hk}, U_{k}^{-})} \\ h = 1, 2, \dots, H, k = 1, 2, \dots, K, \end{matrix}$ (18)

Step 6. Deriving the degree of GRC of all alternatives from DPLTS-PIS and DPLTS-NIS:

$DPLTS - PIS (π_{h}) = \sum_{k = 1}^{K} β_{k} DPLTS - PIS (π_{hk}), h = 1, 2, \dots, H,$ (19)

$DPLTS - NIS (π_{h}) = \sum_{k = 1}^{K} β_{k} DPLTS - NIS (π_{hk}), h = 1, 2, \dots, H,$ (20)

Step 7. Compute each alternative’s relative relational degree (DPLTS-RRD) of all alternatives from DPLTS-PIS:

$DPLTS - RR D_{h} = \frac{DPLTS - NIS (π_{h})}{DPLTS - PIS (π_{h}) + DPLTS - NIS (π_{h})}, h = 1, 2, \dots, H$ (21)

Step 8. According to the DPLTS - RRD_h (h = 1, 2, …, H). The highest value of DPLTS - RRD_h (h = 1, 2, …, H) is, the optimal alternative is.

4 Case study and comparative analysis

4.1 Case study

The practice of urban blocks is bound to change the spatial form. Some scholars advocate the planning idea of “uncontrollable urban growth and plastic Urban morphology”, and propose a design method based on Urban morphology, which is in agreement with Daniel G Park and others who regard Urban morphology design criteria as an alternative method of land development management. As the focus of the above-mentioned block planning practice shifts from land to form, the core lies in how to design the spatial form, and there is little involvement in how to evaluate it, especially in the social ecological evaluation of the block spatial form. In fact, spatial form and social ecology are one and two sides of a block, and the process of shaping spatial form will inevitably change the social ecology. After all, spatial form is only a material carrier, and social ecology is the true essence of the block. Therefore, evaluating the spatial form of blocks from the perspective of social ecology will be a new way to reflect on the effectiveness of block planning practice, which will help to re understand the essence, methods, and implementation mechanisms of block planning. The social ecological evaluation of the spatial form of old urban blocks is a MAGDM. Therefore, the social ecological evaluation of the spatial form of old urban blocks is presented to demonstrate the approach developed in this essay. There is a panel with five potential old urban blocks H_i (i = 1, 2, 3, 4, 5) to choose. The experts select four attributes to assess the five old urban blocks:

➀K₁ is function selection. Functional selection is the main content of shaping spatial form and the foundation of cognitive block development, which includes comprehensive behaviors of value judgment, selection, comparison, and guidance. It has characteristics of dynamism, complexity, and comprehensiveness, and is often dominated by one function accompanied by multiple auxiliary functions, which will directly affect spatial form and its social ecology. It mainly includes two aspects: functional positioning and functional selection. The functional positioning of a block depends on the commercial or community orientation, development motivation, and design scheme determined by the value of the plot, and ultimately determines the social and ecological outcomes of the block’s functions. The selection of block functions implies a trade-off between functional complexity and is an important indicator for measuring social ecology. Its two ends are highly singular and complex, which can be measured using the Simpson Index in the field of ecology. The positioning and selection of block functions need to be comprehensively judged from both internal and external dimensions. The internal characteristics determine the innovation and adaptability of the neighborhood, while the external characteristics mainly refer to the impact of external population and policies on the functionality of the neighborhood, thereby changing the social ecology of the neighborhood. Functional selection has material significance for the social ecology of neighborhoods. Under the market mechanism, it manifests as the commercialization of the historical and cultural value of neighborhoods for the purpose of economic growth, inevitably leading to the marginalization of social ecology.

➁K₂ is material properties. Material attributes are the material carriers of social and ecological organization activities, as well as the results of social and ecological activities. They have characteristics such as substantiality, stability, extensibility, and naturalness, which will affect residents’ use of the built environment, such as communication, transportation, and economic and social life. The material attributes are mainly manifested in three aspects: spatial location primacy, physical scale, and natural characteristics.

➂K₃ is spatial behavior. Spatial behavior is a manifestation of the specific activities of social ecological organizations, a concrete manifestation of urban vitality, manifested as social interaction activities, and presents the social ecology of street space. In this regard, the “urban theater” proposed by Louis Mumford, the “street ballet” proposed by Jane Jacobs and the “communication space” proposed by Yang Gail all well illustrate the richness of street space behavior. There is a close relationship between spatial behavior and spatial form. A good spatial form is not only the result of people’s spatial behavior, but also plays a guiding or restricting role in spatial behavior. A clear and clear spatial form is more likely to have a positive impact on people’s behavior, such as a sense of stability, security, and solemnity. On the contrary, negative spatial behavior will occur. Spatial behavior also depends on the behavioral goals and beliefs of different groups of people, forming the foundation for the diversity, interactivity, and sociality of spatial behavior. Among them, diversity is the sum of various behavioral possibilities in the neighborhood, and these behavioral choices are hidden within individuals; Interactivity is the mutual connection and influence between spatial behaviors based on multiple possibilities, which is the root cause of block space as a social interaction activity; Sociology forms the spatial foundation of civil society.

➃K₄ is place spirit. The spirit of place is the ultimate criterion for measuring the social ecology of block space, emphasizing the significance and importance of block space. It has collective and local characteristics and is the bond of social connection. The spirit of place originated from the ancient Roman faith, believing that people and places have place significance due to the protection of gods. Block space, as a field with its own structure, should be classified and analyzed based on “space” and “characteristics”. “Space” implies the elements that make up a place, which is a three-dimensional organization; “characteristics” refer to “atmosphere”, which is the most abundant characteristic in the place, “which constitutes one of the important indicators of social ecology. The spirit of place is first manifested as the integrity of the existence of the neighborhood and the community residents’ awareness of neighborhood identity. The existence of a block includes two aspects: the type of building that serves as the main filling material of the block and the iconic space that serves as a cultural symbol of the block, which are key factors in understanding the spatial form. Street identity consciousness can be divided into cultural belief and emotional experience, including space integrity, life convenience, religious belief and cultural identity. The spirit of place has a significant impact on the social ecology of the neighborhood. A good place spirit brings an orderly social ecological environment, which can drive urban vitality, increase social interaction, and enrich social heterogeneity. On the contrary, bad and alienated place spirit will destroy the social ecology. From this, it can be seen that the spirit of place is the core element that reflects the social ecology of the neighborhood, and it is the collective unconscious cognition of the community towards the neighborhood as a place, manifested as a symbol and metaphor of spatialform.

All attributes are beneficial one. The five potential old urban blocks H_i (i = 1, 2, 3, 4, 5) are to be evaluated with linguistic scale with the four defined criteria by three experts E_t (t = 1, 2, 3). The discrete linguistic term set was S ={ s_t|t = -3, - 2, - 1, 0, 1, 2, 3 }. The decision information is listed in Tables 1–3.

Table 1
The decision matrix (DM₁) from the 1st DM

Altrenatives K ₁ K ₂ K ₃ K ₄

H ₁ $\begin{matrix} {G_{- 1} (0.7), N G_{2} (0.2)}, \\ {G_{- 2} (0.1), N G_{- 3} (0.7)} \end{matrix}$ $\begin{matrix} {G_{- 2} (0.3), N G_{1} (0.3)}, \\ {G_{- 1} (0.2), N G_{0} (0.4)} \end{matrix}$ $\begin{matrix} {G_{- 3} (0.5), N G_{1} (0.2)}, \\ {G_{- 2} (0.2), N G_{- 1} (0.3)} \end{matrix}$ $\begin{matrix} {G_{- 2} (0.2), N G_{- 1} (0.1)}, \\ {G_{- 1} (0.3), N G_{3} (0.2)} \end{matrix}$

H ₂ $\begin{matrix} {G_{- 2} (0.3), N G_{2} (0.4)}, \\ {G_{- 1} (0.2), N G_{0} (0.5)} \end{matrix}$ $\begin{matrix} {G_{- 3} (0.4), N G_{0} (0.2)}, \\ {G_{- 2} (0.1), N G_{3} (0.3)} \end{matrix}$ $\begin{matrix} {G_{- 1} (0.6), N G_{1} (0.1)}, \\ {G_{- 2} (0.1), N G_{- 1} (0.2)} \end{matrix}$ $\begin{matrix} {G_{1} (0.3), N G_{3} (0.2)}, \\ {G_{- 2} (0.1), N G_{- 3} (0.2)} \end{matrix}$

H ₃ $\begin{matrix} {G_{1} (0.4), N G_{3} (0.2)}, \\ {G_{- 1} (0.1), N G_{1} (0.5)} \end{matrix}$ $\begin{matrix} {G_{- 3} (0.8), N G_{2} (0.2)}, \\ {G_{- 2} (0.1), N G_{- 1} (0.5)} \end{matrix}$ $\begin{matrix} {G_{0} (0.3), N G_{2} (0.2)}, \\ {G_{1} (0.1), N G_{3} (0.5)} \end{matrix}$ $\begin{matrix} {G_{- 2} (0.5), N G_{2} (0.4)}, \\ {G_{0} (0.2), N G_{3} (0.6)} \end{matrix}$

H ₄ $\begin{matrix} {G_{0} (0.3), N G_{1} (0.5)}, \\ {G - 2 (0.5), N G_{1} (0.4)} \end{matrix}$ $\begin{matrix} {G_{- 3} (0.2), N G_{- 2} (0.2)}, \\ {G_{- 2} (0.1), N G_{- 1} (0.9)} \end{matrix}$ $\begin{matrix} {G_{- 3} (0.3), N G_{3} (0.7)}, \\ {G_{- 2} (0.2), N G_{0} (0.8)} \end{matrix}$ $\begin{matrix} {G_{- 1} (0.9), N G_{2} (0.1)}, \\ {G_{2} (0.4), N G_{3} (0.4)} \end{matrix}$

H ₅ $\begin{matrix} {G_{- 3} (0.2), N G_{1} (0.2)}, \\ {G_{- 1} (0.1), N G_{0} (0.1)} \end{matrix}$ $\begin{matrix} {G_{- 1} (0.3), N G_{2} (0.3)}, \\ {G_{- 1} (0.2), N G_{1} (0.3)} \end{matrix}$ $\begin{matrix} {G_{1} (0.5), N G_{2} (0.5)}, \\ {G_{- 1} (0.2), N G_{2} (0.5)} \end{matrix}$ $\begin{matrix} {G_{1} (0.1), N G_{2} (0.2)}, \\ {G_{- 2} (0.1), N G_{3} (0.3)} \end{matrix}$

Altrenatives	K ₁	K ₂	K ₃	K ₄
H ₁	$\begin{matrix} {G_{- 1} (0.7), N G_{2} (0.2)}, \\ {G_{- 2} (0.1), N G_{- 3} (0.7)} \end{matrix}$	$\begin{matrix} {G_{- 2} (0.3), N G_{1} (0.3)}, \\ {G_{- 1} (0.2), N G_{0} (0.4)} \end{matrix}$	$\begin{matrix} {G_{- 3} (0.5), N G_{1} (0.2)}, \\ {G_{- 2} (0.2), N G_{- 1} (0.3)} \end{matrix}$	$\begin{matrix} {G_{- 2} (0.2), N G_{- 1} (0.1)}, \\ {G_{- 1} (0.3), N G_{3} (0.2)} \end{matrix}$
H ₂	$\begin{matrix} {G_{- 2} (0.3), N G_{2} (0.4)}, \\ {G_{- 1} (0.2), N G_{0} (0.5)} \end{matrix}$	$\begin{matrix} {G_{- 3} (0.4), N G_{0} (0.2)}, \\ {G_{- 2} (0.1), N G_{3} (0.3)} \end{matrix}$	$\begin{matrix} {G_{- 1} (0.6), N G_{1} (0.1)}, \\ {G_{- 2} (0.1), N G_{- 1} (0.2)} \end{matrix}$	$\begin{matrix} {G_{1} (0.3), N G_{3} (0.2)}, \\ {G_{- 2} (0.1), N G_{- 3} (0.2)} \end{matrix}$
H ₃	$\begin{matrix} {G_{1} (0.4), N G_{3} (0.2)}, \\ {G_{- 1} (0.1), N G_{1} (0.5)} \end{matrix}$	$\begin{matrix} {G_{- 3} (0.8), N G_{2} (0.2)}, \\ {G_{- 2} (0.1), N G_{- 1} (0.5)} \end{matrix}$	$\begin{matrix} {G_{0} (0.3), N G_{2} (0.2)}, \\ {G_{1} (0.1), N G_{3} (0.5)} \end{matrix}$	$\begin{matrix} {G_{- 2} (0.5), N G_{2} (0.4)}, \\ {G_{0} (0.2), N G_{3} (0.6)} \end{matrix}$
H ₄	$\begin{matrix} {G_{0} (0.3), N G_{1} (0.5)}, \\ {G - 2 (0.5), N G_{1} (0.4)} \end{matrix}$	$\begin{matrix} {G_{- 3} (0.2), N G_{- 2} (0.2)}, \\ {G_{- 2} (0.1), N G_{- 1} (0.9)} \end{matrix}$	$\begin{matrix} {G_{- 3} (0.3), N G_{3} (0.7)}, \\ {G_{- 2} (0.2), N G_{0} (0.8)} \end{matrix}$	$\begin{matrix} {G_{- 1} (0.9), N G_{2} (0.1)}, \\ {G_{2} (0.4), N G_{3} (0.4)} \end{matrix}$
H ₅	$\begin{matrix} {G_{- 3} (0.2), N G_{1} (0.2)}, \\ {G_{- 1} (0.1), N G_{0} (0.1)} \end{matrix}$	$\begin{matrix} {G_{- 1} (0.3), N G_{2} (0.3)}, \\ {G_{- 1} (0.2), N G_{1} (0.3)} \end{matrix}$	$\begin{matrix} {G_{1} (0.5), N G_{2} (0.5)}, \\ {G_{- 1} (0.2), N G_{2} (0.5)} \end{matrix}$	$\begin{matrix} {G_{1} (0.1), N G_{2} (0.2)}, \\ {G_{- 2} (0.1), N G_{3} (0.3)} \end{matrix}$

Table 2

The decision matrix (DM₂) from the 2nd DM

Altrenatives	K ₁	K ₂	K ₃	K ₄
H ₁	$\begin{matrix} {G_{1} (0.5), N G_{2} (0.2)}, \\ {G_{2} (0.3), N G_{3} (0.6)} \end{matrix}$	$\begin{matrix} {G_{0} (0.2), N G_{1} (0.3)}, \\ {G_{- 1} (0.2), N G_{0} (0.7)} \end{matrix}$	$\begin{matrix} {G_{1} (0.2), N G_{2} (0.1)}, \\ {G_{0} (0.4), N G_{2} (0.6)} \end{matrix}$	$\begin{matrix} {G_{- 1} (0.4), N G_{3} (0.2)}, \\ {G_{- 2} (0.5), N G_{- 1} (0.3)} \end{matrix}$
H ₂	$\begin{matrix} {G_{1} (0.3), N G_{2} (0.5)}, \\ {G_{- 1} (0.3), N G_{1} (0.7)} \end{matrix}$	$\begin{matrix} {G_{- 1} (0.3), N G_{2} (0.4)}, \\ {G_{0} (0.4), N G_{3} (0.5)} \end{matrix}$	$\begin{matrix} {G_{- 2} (0.4), N G_{2} (0.3)}, \\ {G_{- 2} (0.1), N G_{- 3} (0.7)} \end{matrix}$	$\begin{matrix} {G_{- 3} (0.5), N G_{2} (0.3)}, \\ {G_{- 2} (0.1), N G_{- 1} (0.1)} \end{matrix}$
H ₃	$\begin{matrix} {G_{- 3} (0.4), N G_{- 2} (0.2)}, \\ {G_{- 2} (0.3), N G_{3} (0.4)} \end{matrix}$	$\begin{matrix} {G_{- 2} (0.4), N G_{0} (0.5)}, \\ {G_{- 2} (0.8), N G_{3} (0.2)} \end{matrix}$	$\begin{matrix} {G_{- 2} (0.5), N G_{2} (0.1)}, \\ {G_{0} (0.1), N G_{1} (0.3)} \end{matrix}$	$\begin{matrix} {G_{- 1} (0.7), N G_{2} (0.2)}, \\ {G_{- 1} (0.5), N G_{3} (0.4)} \end{matrix}$
H ₄	$\begin{matrix} {G_{- 3} (0.3), N G_{2} (0.5)}, \\ {G_{- 2} (0.2), N G_{0} (0.1)} \end{matrix}$	$\begin{matrix} {G_{- 1} (0.4), N G_{3} (0.3)}, \\ {G_{2} (0.3), N G_{3} (0.6)} \end{matrix}$	$\begin{matrix} {G_{- 3} (0.6), N G_{1} (0.2)}, \\ {G_{- 2} (0.6), N G_{- 1} (0.1)} \end{matrix}$	$\begin{matrix} {G_{1} (0.6), N G_{2} (0.3)}, \\ {G_{- 1} (0.1), N G_{1} (0.4)} \end{matrix}$
H ₅	$\begin{matrix} {G_{1} (0.5), N G_{2} (0.2)}, \\ {G_{2} (0.6), N G_{3} (0.3)} \end{matrix}$	$\begin{matrix} {G_{- 3} (0.4), N G_{- 2} (0.2)}, \\ {G_{- 2} (0.3), N G_{- 1} (0.1)} \end{matrix}$	$\begin{matrix} {G_{- 1} (0.1), N G_{2} (0.2)}, \\ {G_{0} (0.3), N G_{1} (0.5)} \end{matrix}$	$\begin{matrix} {G_{- 1} (0.3), N G_{1} (0.2)}, \\ {G_{- 2} (0.2), N G_{- 1} (0.4)} \end{matrix}$

Table 3

The decision matrix (DM₃) from the 3rd DM

Altrenatives	K ₁	K ₂	K ₃	K ₄
H ₁	$\begin{matrix} {G_{1} (0.6), N G_{3} (0.2)}, \\ {G_{- 2} (0.4), N G_{0} (0.6)} \end{matrix}$	$\begin{matrix} {G_{2} (0.7), N G_{3} (0.3)}, \\ {G_{- 1} (0.5), N G_{0} (0.4)} \end{matrix}$	$\begin{matrix} {G_{1} (0.4), N G_{3} (0.2)}, \\ {G_{- 2} (0.5), N G_{0} (0.1)} \end{matrix}$	$\begin{matrix} {G_{0} (0.5), N G_{2} (0.2)}, \\ {G_{- 2} (0.4), N G_{3} (0.6)} \end{matrix}$
H ₂	$\begin{matrix} {G_{- 1} (0.4), N G_{1} (0.2)}, \\ {G_{0} (0.3), N G_{3} (0.4)} \end{matrix}$	$\begin{matrix} {G_{- 1} (0.8), N G_{0} (0.2)}, \\ {G_{2} (0.7), N G_{3} (0.3)} \end{matrix}$	$\begin{matrix} {G_{1} (0.3), N G_{2} (0.2)}, \\ {G_{- 2} (0.3), N G_{- 1} (0.7)} \end{matrix}$	$\begin{matrix} {G_{1} (0.5), N G_{3} (0.4)}, \\ {G_{- 2} (0.3), N G_{- 1} (0.7)} \end{matrix}$
H ₃	$\begin{matrix} {G_{- 3} (0.6), N G_{2} (0.3)}, \\ {G_{0} (0.4), N G_{3} (0.4)} \end{matrix}$	$\begin{matrix} {G_{- 1} (0.2), N G_{1} (0.5)}, \\ {G_{- 2} (0.3), N G_{3} (0.4)} \end{matrix}$	$\begin{matrix} {G_{- 2} (0.4), N G_{2} (0.6)}, \\ {G_{0} (0.2), N G_{2} (0.7)} \end{matrix}$	$\begin{matrix} {G_{- 3} (0.3), N G_{2} (0.2)}, \\ {G_{- 2} (0.5), N G_{0} (0.2)} \end{matrix}$
H ₄	$\begin{matrix} {G_{- 2} (0.3), N G_{2} (0.2)}, \\ {G_{- 2} (0.4), N G_{- 1} (0.4)} \end{matrix}$	$\begin{matrix} {G_{- 1} (0.1), N G_{3} (0.4)}, \\ {G_{- 2} (0.2), N G_{2} (0.5)} \end{matrix}$	$\begin{matrix} {G_{- 2} (0.3), N G_{1} (0.6)}, \\ {G_{- 2} (0.2), N G_{3} (0.4)} \end{matrix}$	$\begin{matrix} {G_{- 1} (0.8), N G_{2} (0.2)}, \\ {G_{- 2} (0.1), N G_{1} (0.8)} \end{matrix}$
H ₅	$\begin{matrix} {G_{- 1} (0.3), N G_{0} (0.2)}, \\ {G_{2} (0.1), N G_{- 3} (0.9)} \end{matrix}$	$\begin{matrix} {G_{1} (0.4), N G_{2} (0.2)}, \\ {G_{- 2} (0.1), N G_{3} (0.6)} \end{matrix}$	$\begin{matrix} {G_{- 2} (0.5), N G_{2} (0.2)}, \\ {G_{- 2} (0.1), N G_{3} (0.3)} \end{matrix}$	$\begin{matrix} {G_{0} (0.6), N G_{1} (0.3)}, \\ {G_{- 1} (0.2), N G_{2} (0.4)} \end{matrix}$

Table 4

The global decision matrix

Altrenatives	K ₁	K ₂
H ₁	$\begin{matrix} {G_{0.4802} (0.6000), N G_{2.2894} (0.2000)}, \\ {G_{0.0760} (0.2667), N G_{0.0000} (0.6333)} \end{matrix}$	$\begin{matrix} {G_{0.5338} (0.4000), N G_{1.4422} (0.3000)}, \\ {G_{- 1.0000} (0.3000), N G_{0.0000} (0.5000)} \end{matrix}$
H ₂	$\begin{matrix} {G_{- 0.4200} (0.3333), N G_{1.5874} (0.3667)}, \\ {G_{- 0.6342} (0.2667), N G_{0.0000} (0.5333)} \end{matrix}$	$\begin{matrix} {G_{- 1.5789} (0.5000), N G_{0.0000} (0.2667)}, \\ {G_{- 0.6000} (0.4000), N G_{3.0000} (0.3667)} \end{matrix}$
H ₃	$\begin{matrix} {G_{- 1.1602} (0.4667), N G_{- 2.2894} (0.2333)}, \\ {G_{- 0.9149} (0.2667), N G_{2.0801} (0.4333)} \end{matrix}$	$\begin{matrix} {G_{- 1.9324} (0.4667), N G_{0.0000} (0.4000)}, \\ {G_{- 2.0000} (0.4000), N G_{- 2.0801} (0.3667)} \end{matrix}$
H ₄	$\begin{matrix} {G_{- 1.4814} (0.3000), N G_{1.5874} (0.4000)}, \\ {G_{- 2.0000} (0.3667), N G_{0.0000} (0.3000)} \end{matrix}$	$\begin{matrix} {G_{- 1.5789} (0.2333), N G_{- 2.6207} (0.3000)}, \\ {G_{- 0.6667} (0.2000), N G_{- 1.4422} (0.3333)} \end{matrix}$
H ₅	$\begin{matrix} {G_{- 0.6342} (0.3333), N G_{0.0000} (0.2000)}, \\ {G_{1.4126} (0.2667), N G_{0.0000} (0.4333)} \end{matrix}$	$\begin{matrix} {G_{- 0.6342} (0.3667), N G_{- 2.0000} (0.2333)}, \\ {G_{- 1.6667} (0.2000), N G_{- 1.4422} (0.3333)} \end{matrix}$
	K ₃	K ₄
H ₁	$\begin{matrix} {G_{0.1155} (0.3667), N G_{1.8171} (0.1667)}, \\ {G_{- 1.2172} (0.3667), N G_{0.0000} (0.3333)} \end{matrix}$	$\begin{matrix} {G_{- 0.9149} (0.3667), N G_{- 1.8171} (0.1667)}, \\ {G_{- 1.6416} (0.4000), N G_{- 2.0801} (0.3667)} \end{matrix}$
H ₂	$\begin{matrix} {G_{- 0.4200} (0.4333), N G_{1.5874} (0.2000)}, \\ {G_{- 2.0000} (0.1667), N G_{- 1.4422} (0.5333)} \end{matrix}$	$\begin{matrix} {G_{0.1155} (0.4333), N G_{2.6207} (0.3000)}, \\ {G_{- 2.0000} (0.1667), N G_{- 1.4422} (0.3333)} \end{matrix}$
H ₃	$\begin{matrix} {G_{- 1.2172} (0.4000), N G_{2.0000} (0.3000)}, \\ {G_{0.3793} (0.1333), N G_{1.8171} (0.5000)} \end{matrix}$	$\begin{matrix} {G_{- 1.9324} (0.5000), N G_{2.0000} (0.2667)}, \\ {G_{- 0.9149} (0.4000), N G_{0.0000} (0.4000)} \end{matrix}$
H ₄	$\begin{matrix} {G_{- 2.6462} (0.4000), N G_{1.4422} (0.5000)}, \\ {G_{- 2.0000} (0.3333), N G_{0.0000} (0.4333)} \end{matrix}$	$\begin{matrix} {G_{- 0.1748} (0.7667), N G_{2.0000} (0.2000)}, \\ {G_{0.2856} (0.2000), N G_{1.4422} (0.5333)} \end{matrix}$
H ₅	$\begin{matrix} {G_{- 0.4200} (0.3667), N G_{2.0000} (0.3000)}, \\ {G_{- 0.9149} (0.2000), N G_{1.8171} (0.4333)} \end{matrix}$	$\begin{matrix} {G_{0.1155} (0.3333), N G_{1.2599} (0.2333)}, \\ {G_{- 1.6416} (0.1667), N G_{- 1.8171} (0.3667)} \end{matrix}$

Table 5

Positive ideal point

K ₁	K ₂
$\begin{matrix} {G_{0.4802} (0.6000), N G_{- 2.2894} (0.2333)}, \\ {G_{1.4126} (0.2667), N G_{0.0000} (0.3000)} \end{matrix}$	$\begin{matrix} {G_{0.5338} (0.4000), N G_{- 2.6207} (0.3000)}, \\ {G_{- 0.6000} (0.4000), N G_{- 2.0801} (0.3667)} \end{matrix}$
K ₃	K ₄
$\begin{matrix} {G_{0.1155} (0.3667), N G_{1.8171} (0.1667)}, \\ {G_{- 1.2172} (0.3667), G_{- 1.4422} (0.5333)} \end{matrix}$	$\begin{matrix} {G_{- 0.1748} (0.7667), N G_{- 1.8171} (0.1667)}, \\ {G_{- 0.9149} (0.4000), N G_{2.0801} (0.3667)} \end{matrix}$

Table 6

Negative ideal point

K ₁	K ₂
$\begin{matrix} {G_{- 1.4814} (0.3000), N G_{1.5874} (0.4000)}, \\ {G_{- 2.0000} (0.3667), N G_{2.0801} (0.4333)} \end{matrix}$	$\begin{matrix} {G_{- 1.5789} (0.2333), N G_{1.4422} (0.3000)}, \\ {G_{- 2.0000} (0.4000), N G_{3.0000} (0.3667)} \end{matrix}$
K ₃	K ₄
$\begin{matrix} {G_{- 2.6462} (0.4000), N G_{1.4422} (0.5000)}, \\ {G_{- 2.0000} (0.1667), N G_{1.8171} (0.5000)} \end{matrix}$	$\begin{matrix} {G_{- 1.9324} (0.5000), N G_{2.6207} (0.3000)}, \\ {G_{- 2.0000} (0.1667), N G_{1.4422} (0.5333)} \end{matrix}$

The DPL-GRA method is used to solve the social ecological evaluation of the spatial form of old urban blocks.

Step 1. The global decision matrix can be obtained by weighted average operator, and the weight of each expert is 1/3.

Step 2. Attribute weight is solved using CRITIC method, according to equations (12), (13), (14) and (15), the weights can be obtained. $β_{1} = 0.1976, β_{2} = 0.2307, β_{3} = 0.4211, β_{4} = 0.1506$

Step 3. Identify positive and negative ideal points. Using equation (16) to determine PIS and (17) to determine NIS. The results are shown below.

Step 4. Compute the grey relational coefficient (GRC) of every alternative between each alternative and DPLTS-PIS and the GRC between each alternative and DPLTS-NIS. Tables 7 and 8 show the results.

Table 7

The positive ideal point grey relational coefficient

	K ₁	K ₂	K ₃	K ₄
H ₁	0.5357	0.4749	1.0000	0.5268
H ₂	0.4157	0.4204	0.8090	0.4577
H ₃	0.4488	0.5085	0.4645	0.4128
H ₄	0.3790	0.5603	0.4588	0.4099
H ₅	0.5275	0.6550	0.5193	0.5114

Table 8

The negative ideal point grey relational coefficient

	K ₁	K ₂	K ₃	K ₄
H ₁	0.6521	0.7996	0.5790	0.5383
H ₂	1.0000	0.9652	0.5823	0.6257
H ₃	0.6869	0.6549	0.9212	0.7733
H ₄	0.8004	0.6567	0.8606	0.6628
H ₅	0.6792	0.6143	0.8487	0.6081

Step 5. Deriving the degree of GRC of all alternatives from DPLTS-PIS and DPLTS-NIS.(See Table 9):

Table 9

The degree of GRC of all alternatives from DPLTS-PIS and DPLTS-NIS

	H ₁	H ₂	H ₃	H ₄	H ₅
GRC-PIS	0.7159	0.5888	0.4637	0.4591	0.5511
GRC-NISx	0.6382	0.7597	0.7912	0.7719	0.7249

Step 6. Compute each alternative’s relative relational degree (DPLTS-RRD) of all alternatives from DPLTS-PIS and the result is demonstrated in Table 10.

Table 10

The relative relational degree (DPLTS-RRD) of all alternatives from DPLTS-PIS

H ₁	H ₂	H ₃	H ₄	H ₅
0.5287	0.4366	0.3695	0.3729	0.4319

Step 7. Sort (ζ_i) in descending order, we can get H₁ > H₂ > H₅ > H₄ > H₃. So, the best alternative is H₁.

4.2 Comparative analysis

Then, the DPL-GRA method is compared with DPLWA operator [52], DPL-closeness coefficient method [55], DPL-weighted correlation coefficient [56] and DPL-TODIM [57]. The comparative decision results are shown in Table 11.

Table 11
Order of the different methods

Order

DPLWA operator [52] H₁ > H₂ > H₅ > H₄ > H₃

DPL-closeness coefficient method [55] H₁ > H₂ > H₅ > H₄ > H₃

DPL-weighted correlation coefficient [56] H₁ > H₂ > H₅ > H₄ > H₃

DPL-TODIM [57] H₁ > H₂ > H₄ > H₅ > H₃

DPL-GRA H₁ > H₂ > H₅ > H₄ > H₃

	Order
DPLWA operator [52]	H₁ > H₂ > H₅ > H₄ > H₃
DPL-closeness coefficient method [55]	H₁ > H₂ > H₅ > H₄ > H₃
DPL-weighted correlation coefficient [56]	H₁ > H₂ > H₅ > H₄ > H₃
DPL-TODIM [57]	H₁ > H₂ > H₄ > H₅ > H₃
DPL-GRA	H₁ > H₂ > H₅ > H₄ > H₃

From the above detailed analysis, it could be seen that these five models have the same optimal choice and these seven methods’ order are slightly different. H₁ is always the best alternative, and H₃ is always the worst one. The ranking of H₄ and H₅ are slightly different because that TODIM method is an outranking method, but the other methods only consider the distance between alternatives with PIS and NIS. So the results obtained are slightly different. This verifies the DPL-GRA method is reasonable and effective.

5 Conclusions

Urbanization is a major issue in today’s era. With the development of urban modernization, the paid use of land, the rise of real estate, and the operation of market economy mechanisms, urban construction is unfolding at an unprecedented speed, and urban development has entered a new stage. Urban construction is mainly reflected in the development of new areas and the renewal of old cities. The development of new areas has to some extent alleviated the pressure on the old city, adjusted the population, transportation, and industrial structure of the old city, and the old city is generally located in the core area of the city. The differential rent effect of land makes it a hot development topic for real estate developers. The social ecological evaluation of the spatial form of old urban blocks is a MAGDM. In this paper, we proposed a new method to deal with MAGDM problem under DPLTSs. We use CRITIC method to determine the objective weights of attributes, use DPLWA function to integrate decision matrices into a total decision matrix, and use DPL-GRA method to sort alternatives. This method can deal with the objective sorting issues, such as choose supplier, select medical treatment and so on, and can be used to subjective sorting issues through changing the parameter values and replace objective weights with subjective weights. In the end, a numerical case study for social ecological evaluation of the spatial form of old urban blocks is given to validate the proposed method. Through the comparison of DPL-GRA method with other two methods, closeness coefficient method and correlation coefficient method, it is easy to know the superiority of this new method. It can make better use of the information in the decision matrix and select the best-performing program through the performance of each alternative in the case of important attributes or preference attributes. Its calculation process is more specific, DMs can know each alternative’s dominance degree under each attribute over other alternatives through pairwise comparison. If the situation changes or the DMs have their own preference, they can quickly choose the best alternative.

There are some shortcomings in this study. 1. The amount of data in this experiment is small. The number of options is too small to fully reflect the differences between different decision-making models. 2. All the attributes are beneficial attributes; no cost attributes are taken into consideration.

The future research directions are mentioned as follows. 1. The model can be improved to contain subjective preferences of decision makers. 2. The distance measure and operators can be modified to deal with big data which has much more fuzzy information [58 –62].

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Research on social ecological evaluation of the spatial form of old urban blocks based on dual probabilistic linguistic term sets

Abstract

Keywords

1 Introduction

2 Basic knowledges

4.1 Case study

Table 11 Order of the different methods Order DPLWA operator [52] H1 > H2 > H5 > H4 > H3 DPL-closeness coefficient method [55] H1 > H2 > H5 > H4 > H3 DPL-weighted correlation coefficient [56] H1 > H2 > H5 > H4 > H3 DPL-TODIM [57] H1 > H2 > H4 > H5 > H3 DPL-GRA H1 > H2 > H5 > H4 > H3

References