LogTODIM framework for MAGDM with neutrosophic sets: Energy conservation and emission reduction case

Abstract

The significant acceleration of China’s urbanization process has greatly promoted economic development. At the same time, the massive construction of urban buildings has also caused many problems such as environmental pollution and increased energy consumption. Therefore, in architectural design, it is particularly important to pay attention to the sustainable development of the environment, handle the relationship between good people and nature under the guidance of the concept of green development, and focus on the recycling of resources. Focusing on resource utilization in architectural design and based on certain planning can better control the construction process of urban buildings, save energy consumption, reduce construction costs, and contribute to the green development of cities. The comprehensive evaluation of energy conservation and emission reduction of construction projects is a classical MAGDM problems. Recently, the Logarithmic TODIM (LogTODIM) method has been employed to cope with MAGDM issues. The single-valued neutrosophic sets (SVNSs) are used as a tool for characterizing uncertain information during the comprehensive evaluation of energy conservation and emission reduction of construction projects. In this paper, the single-valued neutrosophic number Logarithmic TODIM (SVNN-LogTODIM) method is built to solve the MAGDM under SVNSs. In the end, a numerical case study for comprehensive evaluation of energy conservation and emission reduction of construction projects is given to validate the proposed method.

1. Introduction

The central and local governments at all levels should increase capital investment, provide policy support, support the development of building energy-saving technology, and accelerate the pace of building energy-saving technology innovation [1, 2]. Organize and promote major technical research and research, focus on developing a number of core technologies required for low-carbon buildings, such as solar energy technology, small-scale wind power generation technology, water resource recycling and water ecological restoration technology, and efficient cooling technology, and continuously improve new technologies for building energy conservation [3, 4]. Strengthen international cooperation and exchange, timely track the international trend of building energy conservation technology innovation, actively introduce and absorb new theories, standards, technologies, materials, and processes of building energy conservation, so as to rapidly improve the technical level of building energy conservation in China [5, 6, 7]. Vigorously promote efficient, high-quality and inexpensive products, improve the reuse rate of construction resources, and reduce construction and renovation costs. Strengthen supervision over all aspects of building energy conservation and emission reduction. It is recommended that the government detect and publicize the energy consumption level of buildings. A special adjustment and review system for design quality review should be established in the architectural design process, and a special acceptance system for building energy conservation and emission reduction should be established in the completion acceptance process [8, 9, 10]. A regulatory mechanism covering all aspects from design, construction drawing review, construction, completion acceptance filing, to sales and use should be established as soon as possible. At the same time, if the building does not meet the requirements for energy conservation and emission reduction, it shall be resolutely rejected for acceptance; For buildings that meet the requirements for energy conservation and emission reduction, their energy conservation and emission reduction levels should be evaluated, indicating the amount or proportion of energy conservation and emission reduction [11, 12]. We will effectively strengthen supervision, implement shutdown and rectification for illegal projects, circulate a notice of criticism for illegal enterprises, and punish illegal acts in accordance with the law. Building energy conservation and emission reduction is an important content of energy conservation and emission reduction in China. In recent years, China has made significant progress in building energy conservation and emission reduction. However, at present, building energy conservation and emission reduction are facing four major difficulties: high cost of energy conservation and emission reduction, great difficulty in promotion, unclear regulatory system, unreasonable design of assessment index system, low enthusiasm of local governments, multiple building users, and difficulty in forming joint forces, which directly hinder the further development of building energy conservation and emission reduction work in China [13, 14, 15]. In order to accelerate the promotion of building energy conservation, it is necessary to strengthen the research and development of building energy conservation technology, rationalize the departmental management system, adopt economic incentives, fully play the role of government regulation, accelerate the use of renewable energy in buildings, and form a consensus on building energy conservation and emission reduction [16, 17, 18].

Multi-attribute decision making (MADM) is to sort or select the best alternatives under the decision information [19, 20, 21, 22, 23, 24]. It is widely used in energy, politics, environment, commerce and other fields [25, 26, 27, 28, 29, 30]. With the increasing complexity of the MADM problems [31, 32, 33, 34], it is of great necessity to consider the DMs’ psychological factors [35]. Tversky and Kahneman [36] built the prospect theory (PT) under risk. Gomes and Lima [37] built the TODIM for MADM issues under risk. The comprehensive evaluation of energy conservation and emission reduction of construction projects is classical MAGDM. Recently, Logarithmic TODIM (LogTODIM) method [38] has been used to cope with MAGDM issues. The SVNSs [39, 40, 41, 42, 43, 44, 45, 46, 47] are used as a tool for characterizing uncertain information during the comprehensive evaluation of energy conservation and emission reduction of construction projects. In this paper, the single-valued neutrosophic number Logarithmic TODIM (SVNN-LogTODIM) method is built to solve the MAGDM under SVNSs. In the end, a numerical case study for comprehensive evaluation of energy conservation and emission reduction of construction projects is given to validate the proposed method. The main contribution research motivation and aim of this paper is constructed as follows: (1) The Entropy method is utilized to obtain the weight information under SVNSs; (2) the single-valued neutrosophic number Logarithmic TODIM (SVNN-LogTODIM) method is built to solve the MAGDM under SVNSs; (3) Finally, a numerical case study for comprehensive evaluation of energy conservation and emission reduction of construction projects is constructed and (4) some decision comparisons are made to show some advantages of SVNN-LogTODIM approach.

The structure of this paper is listed below. In Section 2, the SVNSs is introduced. In Section 3, SVNN-LogTODIM method is designed under SVNSs with entropy. Section 4 gives an illustrative case for comprehensive evaluation of energy conservation and emission reduction of construction projects and some comparative analysis. Some remarks are given in Section 5.

2. Preliminaries

Wang et al. [39] built the SVNSs.

Definition 1 [39]. The SVNSs $A$ in $\mathrm{\Phi }$ is characterized:

$PA=\{(\varphi ,P{T}_A(\varphi ),P{I}_A(\varphi ),P{F}_A(\varphi ))|\varphi \in \mathrm{\Phi }\}$(1)

where the $P{T}_A(\varphi ),P{I}_A(\varphi ),P{F}_A(\varphi )$ depicts the truth-membership, indeterminacy-membership and falsity-membership, $P{T}_A(\varphi ),P{I}_A(\varphi ),P{F}_A(\varphi )\in [0,1]$ and satisfies $0⩽P{T}_A(\varphi )+P{I}_A(\varphi )+P{F}_A(\varphi )⩽3$ .

The single-valued neutrosophic number (SVNN) is names as $PA=(P{T}_A,P{I}_A,P{F}_A)$ , where $P{T}_A,P{I}_A,P{F}_A\in [0,1]$ , and $0⩽P{T}_A+P{I}_A+P{F}_A⩽3$ .

Definition 2 [48]. Let $PA=(P{T}_A,P{I}_A,P{F}_A)$ be a SVNN, a score value is defined:

$SV(PA)={\displaystyle \frac{(2+P{T}_A-P{I}_A-P{F}_A)}{3}},S(PA)\in [0,1].$(2)

Definition 3 [48]. Let $PA=(P{T}_A,P{I}_A,P{F}_A)$ be a SVNN, an accuracy value is defined:

$AV(PA)=P{T}_A-P{F}_A,AV(PA)\in [-1,1].$(3)

Peng et al. [48] defined the order relation between two SVNNs.

Definition 4 [48]. Let $PA=(P{T}_A,P{I}_A,P{F}_A)$ and $PB=(P{T}_B,P{I}_B,P{F}_B)$ be two given SVNNs, $SV(PA)=\frac{(2+P{T}_A-P{I}_A-P{F}_A)}{3}$ and $SV(PB)=\frac{(2+P{T}_B-P{I}_B-P{F}_B)}{3}$ , and $AV(A)=P{T}_A-P{F}_A$ and $AV(XB)=P{T}_B-P{F}_B$ , then if $SV(PA)<SV(PB)$ , then $PA<PB$ ; if $SV(PA)=SV(PB)$ , then (1) if $AV(PA)=AV(PB)$ , then $PA=PB$ ; (2) if $AV(PA)<AV(PB)$ , then $PA<PB$ .

Definition 5 [39]. Let $PA=(P{T}_A,P{I}_A,P{F}_A)$ and $PB=(P{T}_B,P{I}_B,P{F}_B)$ be two SVNNs, the given operations are presented:

(1) $PA\oplus PB=(P{T}_A+P{T}_B-P{T}_{A}P{T}_B,P{I}_{A}P{I}_B,P{F}_{A}P{F}_B)$ ; (2) $PA\otimes PB=(P{T}_{A}P{T}_B,P{I}_A+P{I}_B-P{I}_{A}P{I}_B,P{F}_A+P{F}_B-P{F}_{A}P{F}_B)$ ; (3) $\lambda PA=(1-{(1-P{T}_A)}^{\lambda },{(P{I}_A)}^{\lambda },{(P{F}_A)}^{\lambda }),\lambda >0$ ; (4) ${(PA)}^{\lambda }=({(P{T}_A)}^{\lambda },{(P{I}_A)}^{\lambda },1-{(1-P{F}_A)}^{\lambda }),\lambda >0$ .

Definition 6 [49]. Let $PA=(P{T}_A,P{I}_A,P{F}_A)$ and $PB=(P{T}_B,P{I}_B,P{F}_B)$ , then the Hamming distance is defined:

$HD(PA,PB)={\displaystyle \frac{|P{T}_A-P{T}_B|+|P{I}_A-P{I}_B|+|P{F}_A-P{F}_B|}{3}}$(4)

The SVNNWA and SVNNWG operator is defined as follow:

Definition 7 [48]. Let $P{A}_j=(P{T}_j,P{I}_j,P{F}_j)$ be a group of SVNNs, the SVNNWA is presented:

$\text{𝑆𝑉𝑁𝑁𝑊𝐴}(P{A}_1,P{A}_2,\mathrm{\dots },P{A}_n)$ $=p{w}_{1}P{A}_1\oplus p{w}_{2}P{A}_2\mathrm{\dots }\oplus p{w}_{n}P{A}_n=\underset{j=1}{\overset{n}{\oplus }}p{w}_{j}cP{A}_j$(5) $=(1-{\displaystyle \prod _{j=1}^n}{(1-P{T}_{ij})}^{p{w}_j},{\displaystyle \prod _{k=1}^l}{(P{F}_{ij}^k)}^{p{w}_j},{\displaystyle \prod _{k=1}^l}{(P{T}_{ij}^k)}^{p{w}_j})$

where $pw={(p{w}_1,p{w}_2,\mathrm{\dots },p{w}_n)}^T$ be weight of $P{A}_j$ , $p{w}_j>0$ , $\sum _{j=1}^{n}p{w}_j=1$ .

Definition 8 [48]. Let $P{A}_j=(P{T}_j,P{I}_j,P{F}_j)$ be a group of SVNNs, the SVNNWG operator is:

$\text{𝑆𝑉𝑁𝑁𝑊𝐺}(P{A}_1,P{A}_2,\mathrm{\dots },P{A}_n)$ $={(P{A}_1)}^{p{w}_1}\otimes {(P{A}_2)}^{p{w}_2},\mathrm{\dots }\otimes {(P{A}_n)}^{p{w}_n}=\underset{j=1}{\overset{n}{\otimes }}{(P{A}_j)}^{p{w}_j}$(6) $=({\displaystyle \prod _{j=1}^n}{(P{T}_{ij})}^{p{w}_j},1-{\displaystyle \prod _{k=1}^l}{(1-P{F}_{ij}^k)}^{p{w}_j},1-{\displaystyle \prod _{k=1}^l}{(1-P{T}_{ij}^k)}^{p{w}_j})$

where $pw={(p{w}_1,p{w}_2,\mathrm{\dots },p{w}_n)}^T$ be weight of $P{A}_j$ , $p{w}_j>0$ , $\sum _{j=1}^{n}p{w}_j=1$ .

3. SVNN-LogTODIM method for MAGDM with entropy weight

3.1 SVNN-MAGDM issues description

In this section, SVNN-LogTODIM method is built for MAGDM. Let $PA=\{P{A}_1,P{A}_2,\mathrm{\dots },P{A}_m\}$ be alternatives, and the attributes set $PG=\{P{G}_1,P{G}_2,\mathrm{\dots },P{G}_n\}$ with weight $x\omega$ , where $p{\omega }_j\in [0,1]$ , $\sum _{j=1}^{n}p{\omega }_j=1$ and a set of invited experts $PE=\{P{E}_1,P{E}_2,\mathrm{\dots },P{E}_q\}$ , let expert’s weight be $\{p{w}_1,p{w}_2,\mathrm{\dots },p{w}_t\}$ .

Then, SVNN-LogTODIM method is built for MAGDM. The calculating steps are depicted:

(1). Build the SVNN-matrix $P{R}^t={[p{\varphi }_{ij}^t]}_{m\times n}={(P{T}_{ij}^t,P{I}_{ij}^t,P{F}_{ij}^t)}_{m\times n}$ and calculate the average matrix $PR={[p{\varphi }_{ij}]}_{m\times n}$ :

$\begin{array}{c}\begin{array}{cccccccccccccccccccc}\hfill P{G}_1\hfill & \hfill P{G}_2\hfill & \hfill \mathrm{\dots }\hfill & \hfill P{G}_n\hfill & & & & & & & & & & & & & & & & \end{array}\hfill \\ PR={[p{\varphi }_{ij}^t]}_{m\times n}=\begin{array}{cccccccccccccccccccc}\hfill P{A}_1\hfill & & & & & & & & & & & & & & & & & & & \\ \hfill P{A}_2\hfill & & & & & & & & & & & & & & & & & & & \\ \hfill \mathrm{⋮}\hfill & & & & & & & & & & & & & & & & & & & \\ \hfill P{A}_m\hfill & & & & & & & & & & & & & & & & & & & \end{array}\left[\begin{array}{cccccccccccccccccccc}\hfill p{\varphi }_{11}^t\hfill & \hfill p{\varphi }_{12}^t\hfill & \hfill \mathrm{\dots }\hfill & \hfill p{\varphi }_{1n}^t\hfill & & & & & & & & & & & & & & & & \\ \hfill p{\varphi }_{21}^t\hfill & \hfill p{\varphi }_{22}^t\hfill & \hfill \mathrm{\dots }\hfill & \hfill p{\varphi }_{2n}^t\hfill & & & & & & & & & & & & & & & & \\ \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & & & & & & & & & & & & & & & & \\ \hfill p{\varphi }_{m1}^t\hfill & \hfill p{\varphi }_{m2}^t\hfill & \hfill \mathrm{\dots }\hfill & \hfill p{\varphi }_{mn}^t\hfill & & & & & & & & & & & & & & & & \end{array}\right]\hfill \end{array}$(7) $\begin{array}{c}\begin{array}{cccccccccccccccccccc}\hfill P{G}_1\hfill & \hfill P{G}_2\hfill & \hfill \mathrm{\dots }\hfill & \hfill P{G}_n\hfill & & & & & & & & & & & & & & & & \end{array}\hfill \\ PR={[p{\varphi }_{ij}]}_{m\times n}=\begin{array}{cccccccccccccccccccc}\hfill P{A}_1\hfill & & & & & & & & & & & & & & & & & & & \\ \hfill P{A}_2\hfill & & & & & & & & & & & & & & & & & & & \\ \hfill \mathrm{⋮}\hfill & & & & & & & & & & & & & & & & & & & \\ \hfill P{A}_m\hfill & & & & & & & & & & & & & & & & & & & \end{array}\left[\begin{array}{cccccccccccccccccccc}\hfill p{\varphi }_{11}\hfill & \hfill p{\varphi }_{12}\hfill & \hfill \mathrm{\dots }\hfill & \hfill p{\varphi }_{1n}\hfill & & & & & & & & & & & & & & & & \\ \hfill p{\varphi }_{21}\hfill & \hfill p{\varphi }_{22}\hfill & \hfill \mathrm{\dots }\hfill & \hfill p{\varphi }_{2n}\hfill & & & & & & & & & & & & & & & & \\ \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & & & & & & & & & & & & & & & & \\ \hfill p{\varphi }_{m1}\hfill & \hfill p{\varphi }_{m2}\hfill & \hfill \mathrm{\dots }\hfill & \hfill p{\varphi }_{mn}\hfill & & & & & & & & & & & & & & & & \end{array}\right]\hfill \end{array}$(8)

Based on SVNNWA, the $PR={[p{\varphi }_{ij}]}_{m\times n}={(P{T}_{ij},P{I}_{ij},P{F}_{ij})}_{m\times n}$ is:

$p{\varphi }_{ij}=p{w}_{1}p{\varphi }_{ij}^1\oplus p{w}_{2}p{\varphi }_{ij}^2\oplus \mathrm{\dots }\oplus p{w}_{t}p{\varphi }_{ij}^t$ (9) $=(1-{\displaystyle \prod _{k=1}^t}{(P{T}_{ij}^t)}^{p{w}_j},{\displaystyle \prod _{k=1}^t}{(P{I}_{ij}^t)}^{p{w}_j},{\displaystyle \prod _{k=1}^t}{(P{F}_{ij}^t)}^{p{w}_j})$

(2). Normalize the $PR={[p{\varphi }_{ij}]}_{m\times n}$ into $PN={[P{N}_{ij}]}_{m\times n}$ .

For benefit attributes:

$P{N}_{ij}=({\text{𝑁𝑃𝑇}}_{ij},{\text{𝑁𝑃𝐼}}_{ij},{\text{𝑁𝑃𝐹}}_{ij})=p{\varphi }_{ij}=(P{T}_{ij},P{I}_{ij},P{F}_{ij})$(10)

For cost attributes:

$P{N}_{ij}=({\text{𝑁𝑃𝑇}}_{ij},{\text{𝑁𝑃𝐼}}_{ij},{\text{𝑁𝑃𝐹}}_{ij})=(P{F}_{ij},P{I}_{ij},P{T}_{ij})$(11)

3.2 Compute the attributes weight by using information entropy

Entropy [50] is a conventional theory to derive weight. Firstly, the normalized SVNN-matrix ${\text{𝑆𝑉𝑁𝑁𝑀}}_{ij}$ is obtained:

${\text{𝑆𝑉𝑁𝑁𝑀}}_{ij}={\displaystyle \frac{\frac{\frac{AV({\text{𝑁𝑃𝑇}}_{ij},{\text{𝑁𝑃𝐼}}_{ij},{\text{𝑁𝑃𝐹}}_{ij})+1}{2}}{SV({\text{𝑁𝑃𝑇}}_{ij},{\text{𝑁𝑃𝐼}}_{ij},{\text{𝑁𝑃𝐹}}_{ij})}}{\sum _{i=1}^m\frac{\frac{AV({\text{𝑁𝑃𝑇}}_{ij},{\text{𝑁𝑃𝐼}}_{ij},{\text{𝑁𝑃𝐹}}_{ij})+1}{2}}{SV({\text{𝑁𝑃𝑇}}_{ij},{\text{𝑁𝑃𝐼}}_{ij},{\text{𝑁𝑃𝐹}}_{ij})}}},$(12)

Then, the Shannon information entropy $\text{𝑆𝐼𝐸}=({\text{𝑆𝐼𝐸}}_1,{\text{𝑆𝐼𝐸}}_2,\mathrm{\dots },{\text{𝑆𝐼𝐸}}_n)$ is defined:

${\text{𝑆𝐼𝐸}}_j=-{\displaystyle \frac{1}{\ln m}}{\displaystyle \sum _{i=1}^m}{\text{𝑆𝑉𝑁𝑁𝑀}}_{ij}\ln {\text{𝑆𝑉𝑁𝑁𝑀}}_{ij}$(13)

and ${\text{𝑆𝑉𝑁𝑁𝑀}}_{ij}\ln {\text{𝑆𝑉𝑁𝑁𝑀}}_{ij}=0$ if ${\text{𝑆𝑉𝑁𝑁𝑀}}_{ij}=0$ .

Then, the weights $p\omega =(p{\omega }_1,p{\omega }_2,\mathrm{\dots },p{\omega }_n)$ is defined:

$p{\omega }_j={\displaystyle \frac{1-{\text{𝑆𝐼𝐸}}_j}{\sum _{j=1}^n(1-{\text{𝑆𝐼𝐸}}_j)}},j=1,2,\mathrm{\dots },n.$(14)

3.3 SVNN-LogTODIM method for MAGDM

In such section, the SVNN-LogTODIM method is built for MAGDM.

(1) Define relative weight of $P{G}_j$ :

$rp{\omega }_j=p{\omega }_j/p{\omega }_j^{*},$(15)

$p{\omega }_j^{*}$ is the maximum weight of the attributes, and $0⩽p{\omega }_j⩽1$ .

(2) The dominance degree ${\mathrm{\Pi }}_j(P{A}_i,P{A}_t)$ of $P{A}_i$ over $P{A}_t$ for $P{G}_j$ is obtained:

${\mathrm{\Pi }}_j(P{A}_i,P{A}_t)=\{\begin{array}{cccccccccccccccccccc}\hfill {\displaystyle \frac{rp{\omega }_j\times \log (1+10\rho HD(P{N}_{ij},P{N}_{tj}))}{{\sum }_{j=1}^{n}rp{\omega }_j}}\hfill & \hfill \text{if }SV(P{N}_{ij})>SV(P{N}_{tj})\hfill & & & & & & & & & & & & & & & & & & \\ \hfill 0\hfill & \hfill \text{if }SV(P{N}_{ij})=SV(P{N}_{tj})\hfill & & & & & & & & & & & & & & & & & & \\ \hfill -{\displaystyle \frac{rp{\omega }_j\times \lambda \log (1+10\rho HD(P{N}_{ij},P{N}_{tj}))}{{\sum }_{j=1}^{n}rp{\omega }_j}}\hfill & \hfill \text{if }SV(P{N}_{ij})<SV(P{N}_{tj})\hfill & & & & & & & & & & & & & & & & & & \end{array}$(16)

where $\lambda$ is the amplification parameter and $\lambda \in [1,5]$ and $\rho \in N^{+}$ indicates how significant the decision is according to the agent’s perception [38].

The dominance degree matrix ${\mathrm{\Pi }}_j(P{A}_i)(j=1,2,\mathrm{\dots },n)$ with respect to $P{G}_j$ is defined:

${\mathrm{\Pi }}_j(P{A}_i)={[{\mathrm{\Pi }}_j(P{A}_i,P{A}_t)]}_{m\times m}$ $\begin{array}{cccccccccccccccccccc}\hfill P{A}_1\hfill & \hfill P{A}_2\hfill & \hfill \mathrm{\dots }\hfill & \hfill P{A}_m\hfill & & & & & & & & & & & & & & & & \end{array}$ $=\begin{array}{cccccccccccccccccccc}\hfill P{A}_1\hfill & & & & & & & & & & & & & & & & & & & \\ \hfill P{A}_2\hfill & & & & & & & & & & & & & & & & & & & \\ \hfill \mathrm{⋮}\hfill & & & & & & & & & & & & & & & & & & & \\ \hfill P{A}_m\hfill & & & & & & & & & & & & & & & & & & & \end{array}\left[\begin{array}{cccccccccccccccccccc}\hfill 0\hfill & \hfill {\mathrm{\Pi }}_j(P{A}_1,P{A}_2)\hfill & \hfill \mathrm{\dots }\hfill & \hfill {\mathrm{\Pi }}_j(P{A}_1,P{A}_m)\hfill & & & & & & & & & & & & & & & & \\ \hfill {\mathrm{\Pi }}_j(P{A}_2,P{A}_1)\hfill & \hfill 0\hfill & \hfill \mathrm{\dots }\hfill & \hfill {\mathrm{\Pi }}_j(P{A}_2,P{A}_m)\hfill & & & & & & & & & & & & & & & & \\ \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{\dots }\hfill & \hfill \mathrm{⋮}\hfill & & & & & & & & & & & & & & & & \\ \hfill {\mathrm{\Pi }}_j(P{A}_m,P{A}_1)\hfill & \hfill {\mathrm{\Pi }}_j(P{A}_m,P{A}_2)\hfill & \hfill \mathrm{\dots }\hfill & \hfill 0\hfill & & & & & & & & & & & & & & & & \end{array}\right]$

(3) Produce the $\mathrm{\Pi }(P{A}_i,P{A}_t)$ of $P{A}_i$ over other alternatives for $P{A}_t$ :

$\mathrm{\Pi }(P{A}_i,P{A}_t)={\displaystyle \sum _{j=1}^n}{\mathrm{\Pi }}_j(P{A}_i,P{A}_t)$(17)

The $\mathrm{\Pi }=\mathrm{\Pi }{(P{A}_i,P{A}_t)}_{m\times m}$ is defined:

$\mathrm{\Pi }=\mathrm{\Pi }{(P{A}_i,P{A}_t)}_{m\times m}$ $=\left[\begin{array}{cccccccccccccccccccc}& \hfill P{A}_1\hfill & \hfill P{A}_2\hfill & \hfill \mathrm{\dots }\hfill & \hfill P{A}_m\hfill & & & & & & & & & & & & & & & \\ \hfill P{A}_1\hfill & \hfill {\displaystyle \sum _{j=1}^n}{\mathrm{\Pi }}_j(P{A}_1,P{A}_1)\hfill & \hfill {\displaystyle \sum _{j=1}^n}{\mathrm{\Pi }}_j(P{A}_1,P{A}_2)\hfill & \hfill \mathrm{\dots }\hfill & \hfill {\displaystyle \sum _{j=1}^n}{\mathrm{\Pi }}_j(P{A}_1,P{A}_m)\hfill & & & & & & & & & & & & & & & \\ \hfill P{A}_2\hfill & \hfill {\displaystyle \sum _{j=1}^n}{\mathrm{\Pi }}_j(P{A}_2,P{A}_1)\hfill & \hfill {\displaystyle \sum _{j=1}^n}{\mathrm{\Pi }}_j(P{A}_2,P{A}_2)\hfill & \hfill \mathrm{\dots }\hfill & \hfill {\displaystyle \sum _{j=1}^n}{\mathrm{\Pi }}_j(P{A}_2,P{A}_m)\hfill & & & & & & & & & & & & & & & \\ \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & \hfill \mathrm{⋮}\hfill & & & & & & & & & & & & & & & \\ \hfill P{A}_m\hfill & \hfill {\displaystyle \sum _{j=1}^n}{\mathrm{\Pi }}_j(P{A}_m,P{A}_1)\hfill & \hfill {\displaystyle \sum _{j=1}^n}{\mathrm{\Pi }}_j(P{A}_m,P{A}_2)\hfill & \hfill \mathrm{\dots }\hfill & \hfill {\displaystyle \sum _{j=1}^n}{\mathrm{\Pi }}_j(P{A}_m,P{A}_m)\hfill & & & & & & & & & & & & & & & \end{array}\right]$

(4) Finally, the SVNN overall dominance degree $\text{𝑆𝑉𝑁𝑁𝑂𝐷𝐷}(P{A}_i)$ of alternative $P{A}_i$ is obtained:

$\text{𝑆𝑉𝑁𝑁𝑂𝐷𝐷}(P{A}_i)={\displaystyle \frac{\sum _{t=1}^m\mathrm{\Pi }(P{A}_i,P{A}_t)-\underset{i}{min}\left\{\sum _{t=1}^m\mathrm{\Pi }(P{A}_i,P{A}_t)\right\}}{\underset{i}{max}\left\{\sum _{t=1}^m\mathrm{\Pi }(P{A}_i,P{A}_t)\right\}-\underset{i}{min}\left\{\sum _{t=1}^m\mathrm{\Pi }(P{A}_i,P{A}_t)\right\}}}.$(18)

(5) The order could be ranked in accordance with $\text{𝑆𝑉𝑁𝑁𝑂𝐷𝐷}(P{A}_i)(i=1,2,\mathrm{\dots },m)$ , that is, the greater $\text{𝑆𝑉𝑁𝑁𝑂𝐷𝐷}(P{A}_i)(i=1,2,\mathrm{\dots },m)$ , the better the alternative $P{A}_i$ is.

4. An empirical example and comparative analysis

4.1 An empirical example

China currently has a building area of 40 billion square meters, but its energy efficiency in building is relatively low, and the energy consumption per unit building is 2–3 times higher than that of countries under the same climatic conditions. Domestic building energy consumption accounts for about 1/3 of the total energy consumption of the whole society, and has become the three “major energy users” in China along with industrial energy consumption and transportation energy consumption, which has to some extent exacerbated the increasingly serious energy crisis in China. At present, China is in a period of rapid urbanization, and the construction of new rural areas is also constantly advancing. The construction industry is developing at the right time, and the building area is rapidly increasing. Each year, 200 million square meters of new housing area is added, exceeding the total annual building area of all developed countries. It can be seen that doing a good job in building energy conservation and emission reduction has important practical significance. It can not only alleviate China’s energy supply, but also reduce environmental pollution, which is conducive to sustainable economic and social development. China has made significant progress in actively developing green buildings and vigorously reducing building energy consumption, but there are still some difficulties that cannot be ignored, restricting the implementation of building energy conservation and emission reduction work. With policy support and guidance, low-carbon buildings have been promoted in most provinces, regions, and cities in China, and low-carbon building communities have also emerged in some cities. Jiangsu Province will build a national low-carbon ecological demonstration zone that integrates low-carbon manufacturing industry cluster, low-carbon modern service industry cluster, and low-carbon demonstration application cluster. It is also China’s first low-carbon demonstration zone and an international low-carbon new technology exchange and promotion base. Hangzhou has designed and built a low-carbon science and technology museum in accordance with the standard of “National Green Building Three Star”, mainly using natural and recyclable materials in building materials. It has pioneered the perfect combination of solar hot water supply, heating, refrigeration, and photovoltaic grid connected power generation technologies with buildings. The overall energy efficiency of the building has reached 88%, making it a truly low-carbon building. The continuous completion of these low-carbon buildings has played a good exemplary role in leading and driving new trends in China’s architectural development. The comprehensive evaluation of energy conservation and emission reduction of construction projects is a classical MAGDM issue. Therefore, the comprehensive evaluation of energy conservation and emission reduction of construction projects is presented to demonstrate the approach developed in this essay. There is a panel with five potential construction projects $P{A}_i(i=1,2,3,4,5,6)$ to choose. The experts select six attributes to assess the five construction projects: ⟀ PG₁ is material saving for construction enterprises; ⟁ PG₂ is saving energy for construction enterprises; ⟂ PG₃ is environmental management ability of construction enterprises; ⟃ PG₄ is management ability of construction enterprises; ⟄ PG₅ is social benefit of construction enterprises; ⟅ PG₆ is innovation ability of construction enterprises. All are beneficial one. The five possible construction projects $P{A}_i(i=1,2,3,4,5,6)$ are to be evaluated with SVNNs with the four criteria by three experts $P{E}_t(t=1,2,3)$ (Suppose expert’s weight is (0.3406, 0.3706, 0.2878).

Table 1

Linguistic scale and SVNNs

Linguistic terms	SVNNs
Exceedingly Terrible-PET	(0.0000, 1.0000, 1.0000)
Very Terrible-PVT	(0.1000, 0.9000, 0.9000)
Terrible-PT	(0.3000, 0.7000, 0.7000)
Medium-PM	(0.5000, 0.5000, 0.5000)
Well-PW	(0.7000, 0.3000, 0.3000)
Very Well-PVW	(0.9000, 0.1000, 0.1000)
Exceedingly Well-PEW	(1.0000, 0.0000, 0.0000)

The SVNN-LogTODIM method is used to solve the comprehensive evaluation of energy conservation and emission reduction of construction projects.

Step 1. Construct the SVNN matrix $P{R}^t={[p{\varphi }_{ij}^t]}_{6\times 6}={(P{T}_{ij}^t,P{I}_{ij}^t,P{F}_{ij}^t)}_{6\times 6}$ (see Tables 2–4).

Table 2

Evaluation information by $P{E}_1$

	PG1	PG2	PG3	PG4	PG5	PG6
PA1	PVW	PW	PM	PT	PM	PVW
PA2	PVW	PT	PVT	PM	PVT	PM
PA3	PM	PVW	PT	PW	PVT	PM
PA4	PVT	PM	PVW	PW	PT	PW
PA5	PVT	PVW	PM	PW	PVW	PW
PA6	PW	PVT	PM	PVW	PW	PT

Table 3

Evaluation information by $P{E}_2$

	PG1	PG2	PG3	PG4	PG5	PG6
PA1	PM	PT	PM	PT	PW	PVW
PA2	PVT	PVW	PW	PVW	PVT	PVT
PA3	PVW	PW	PM	PW	PVT	PVW
PA4	PVT	PM	PW	PVW	PVW	PW
PA5	PM	PVT	PVT	PVT	PVT	PM
PA6	PW	PVW	PVT	PM	PW	PVW

Table 4

Evaluation information by $P{E}_3$

	PG1	PG2	PG3	PG4	PG5	PG6
PA1	PVW	PVT	PVW	PW	PW	PVW
PA2	PW	PM	PVT	PT	PVT	PVW
PA3	PM	PT	PT	PM	PVT	PT
PA4	PW	PVW	PT	PM	PT	PM
PA5	PVT	PVW	PM	PW	PT	PM
PA6	PM	PVT	PT	PVW	PT	PM

Then according to SVNNWA, the $PR={[p{\varphi }_{ij}]}_{6\times 6}$ is obtained (see Table 5).

Table 5

The $PR={[p{\varphi }_{ij}]}_{6\times 6}$

	PG1	PG2
PA1	(0.4392, 0.5112, 0.2306)	(0.6771, 0.8970, 0.1142)
PA2	(0.7437, 0.4809, 0.3973)	(0.4692, 0.2620, 0.6021)
PA3	(0.4823, 0.5038, 0.4739)	(0.2703, 0.2635, 0.4764)
PA4	(0.2376, 0.7978, 0.6845)	(0.4591, 0.5072, 0.2967)
PA5	(0.3494, 0.1468, 0.1902)	(0.6642, 0.7392, 0.6705)
PA6	(0.2672, 0.2413, 0.4097)	(0.3126, 0.3021, 0.2632)
	PG4	PG3
PA1	(0.3118, 0.1237, 0.1762)	(0.3424, 0.5059, 0.6351)
PA2	(0.8652, 0.5238, 0.4857)	(0.3058, 0.3709, 0.3485)
PA3	(0.4825, 0.8216, 0.2439)	(0.6563, 0.8293, 0.5142)
PA4	(0.8683, 0.6708, 0.5943)	(0.4728, 0.7301, 0.7293)
PA5	(0.6234, 0.5673, 0.7984)	(0.1901, 0.4760, 0.4167)
PA6	(0.2306, 0.3465, 0.4265)	(0.3089, 0.1457, 0.1692)
	PG5	PG6
PA1	(0.5452, 0.6163, 0.4031)	(0.3703, 0.6147, 0.2214)
PA2	(0.4502, 0.6103, 0.5071)	(0.6461, 0.4507, 0.3721)
PA3	(0.1783, 0.4542, 0.4006)	(0.4012, 0.3451, 0.5762)
PA4	(0.3201, 0.4047, 0.5239)	(0.2908, 0.1015, 0.1541)
PA5	(0.3025, 0.3206, 0.3008)	(0.4269, 0.3016, 0.2635)
PA6	(0.4012, 0.3451, 0.5762)	(0.1671, 0.4437, 0.4116)

Step 2. Normalize the $PR={[p{\varphi }_{ij}]}_{6\times 6}$ into $PN={[P{N}_{ij}]}_{6\times 6}$ (see Table 6).

Table 6

The $PN={[P{N}_{ij}]}_{6\times 6}$

	PG1	PG2
PA1	(0.4392, 0.5112, 0.2306)	(0.6771, 0.8970, 0.1142)
PA2	(0.7437, 0.4809, 0.3973)	(0.4692, 0.2620, 0.6021)
PA3	(0.4823, 0.5038, 0.4739)	(0.2703, 0.2635, 0.4764)
PA4	(0.2376, 0.7978, 0.6845)	(0.4591, 0.5072, 0.2967)
PA5	(0.3494, 0.1468, 0.1902)	(0.6642, 0.7392, 0.6705)
PA6	(0.2672, 0.2413, 0.4097)	(0.3126, 0.3021, 0.2632)
	PG4	PG3
PA1	(0.3118, 0.1237, 0.1762)	(0.3424, 0.5059, 0.6351)
PA2	(0.8652, 0.5238, 0.4857)	(0.3058, 0.3709, 0.3485)
PA3	(0.4825, 0.8216, 0.2439)	(0.6563, 0.8293, 0.5142)
PA4	(0.8683, 0.6708, 0.5943)	(0.4728, 0.7301, 0.7293)
PA5	(0.6234, 0.5673, 0.7984)	(0.1901, 0.4760, 0.4167)
PA6	(0.2306, 0.3465, 0.4265)	(0.3089, 0.1457, 0.1692)
	PG5	PG6
PA1	(0.5452, 0.6163, 0.4031)	(0.3703, 0.6147, 0.2214)
PA2	(0.4502, 0.6103, 0.5071)	(0.6461, 0.4507, 0.3721)
PA3	(0.1783, 0.4542, 0.4006)	(0.4012, 0.3451, 0.5762)
PA4	(0.3201, 0.4047, 0.5239)	(0.2908, 0.1015, 0.1541)
PA5	(0.3025, 0.3206, 0.3008)	(0.4269, 0.3016, 0.2635)
PA6	(0.4012, 0.3451, 0.5762)	(0.1671, 0.4437, 0.4116)

Step 3. Obtain the weight values (see Table 7).

Table 7

The attributes weight

	PG1	PG2	PG3	PG4	PG5	PG6
$p\omega$	0.1909	0.1730	0.1896	0.1559	0.1653	0.1253

Step 4. Obtain the relative weight (see Table 8).

Table 8

The relative attributes weight

	PG1	PG2	PG3	PG4	PG5	PG6
$rp\omega$	1.0000	0.9062	0.9932	0.8167	0.8659	0.6564

Step 5. Obtain the $\mathrm{\Pi }=\mathrm{\Pi }{(P{A}_i,P{A}_t)}_{5\times 5}$ matrix (see Table 9).

Table 9

The $\mathrm{\Pi }=\mathrm{\Pi }{(P{A}_i,P{A}_t)}_{5\times 5}$

Alternatives	PA1	PA2	PA3	PA4	PA5	PA6
PA1	0.0000	$-$ 1.7628	2.1633	2.8379	$-$ 1.0103	0.4928
PA2	0.2147	0.0000	2.0947	0.2268	2.1187	0.2593
PA3	$-$ 1.5659	$-$ 2.3537	0.0000	$-$ 3.4884	1.4654	1.5914
PA4	$-$ 0.7076	$-$ 0.9730	2.5549	0.0000	0.6834	0.2888
PA5	1.3794	$-$ 2.4927	$-$ 0.1613	$-$ 3.1836	0.0000	1.2191
PA6	$-$ 0.1011	$-$ 0.1811	$-$ 0.4153	$-$ 2.0162	$-$ 2.5706	0.0000

Step 6. Calculate the $\text{𝑆𝑉𝑁𝑁𝑂𝐷𝐷}(P{A}_i)(i=1,2,\mathrm{\dots },5)$ (Table 10).

Table 10

The $\text{𝑆𝑉𝑁𝑁𝑂𝐷𝐷}(P{A}_i)$

Alternatives	PA1	PA2	PA3	PA4	PA5	PA6
SVNNODD	0.7849	1.0000	0.0915	0.6992	0.2005	0.0000

Step 7. Finally, the order could be obtained: $P{A}_2\succ P{A}_1\succ P{A}_4\succ P{A}_5\succ P{A}_3\succ P{A}_6$ , and thus the optimal construction project is $P{A}_2$ .

4.2 Comparative analysis

Then, the SVNN-LogTODIM method is compared with SVNNWA operator [48], SVNNWG operator [48], SVNN-CODAS method [51], SVNN-EDAS method [52] and SVNN-TODIM method [53]. The comparative decision results are shown in Table 11.

Table 11

Ordering of the different methods

	Ordering
SVNNWA operator [48]	$P{A}_2\succ P{A}_1\succ P{A}_4\succ P{A}_5\succ P{A}_3\succ P{A}_6$
SVNNWG operator [48]	$P{A}_2\succ P{A}_1\succ P{A}_4\succ P{A}_3\succ P{A}_5\succ P{A}_6$
SVNN-CODAS method [51]	$P{A}_2\succ P{A}_1\succ P{A}_4\succ P{A}_5\succ P{A}_3\succ P{A}_6$
SVNN-EDAS method [52]	$P{A}_2\succ P{A}_1\succ P{A}_4\succ P{A}_5\succ P{A}_3\succ P{A}_6$
SVNN-TODIM method [53]	$P{A}_2\succ P{A}_1\succ P{A}_4\succ P{A}_5\succ P{A}_3\succ P{A}_6$

In accordance with WS coefficients [54, 55], the WS coefficient calculation between the SVNNWA operator [48], SVNNWG operator [48], SVNN-CODAS method [51], SVNN-EDAS method [52], SVNN-TODIM method [53] and the proposed SVNN-LogTODIM method is 1.0000, 0.9010, 1.0000, 1.0000, 1.0000, respectively. The WS coefficient shows the ranking results of the proposed SVNN-LogTODIM method are same to the ranking results of the SVNNWA operator [48], SVNN-CODAS method [51], SVNN-EDAS method [52], SVNN-TODIM method [53]; the WS coefficient shows the ranking results of the proposed SVNN-LogTODIM method are slightly different to the ranking results of the SVNNWG operator [48]. Furthermore, the reason for this subtle difference is that SVNNWG operator [48] emphasize the individual’s influence. This verifies the SVNN-LogTODIM method is reasonable and effective.

5. Conclusion

Compared with developed countries, China’s new construction materials industry is still relatively backward, and there are gaps in equipment and application of new construction technologies. Therefore, it is necessary to further increase investment in scientific research and talent, which will directly increase the cost of using new construction materials and technologies in China. According to the report “Cost and Economic Benefits of Green Buildings” published by the Task Force on Sustainable Buildings of over 40 government agencies around the world, the cost of a building with green environmental characteristics is on average 2% higher than that of ordinary buildings, while China estimates that the construction cost will increase by 1000 yuan/square meter. Most of these construction costs will be passed on to consumers, greatly increasing their costs. Some people have conducted a survey and found that if the price of low-carbon housing is not much higher than that of ordinary housing, most home buyers can still accept it. However, if it is higher than 10%, there are few people willing to purchase low-carbon housing. High construction costs have brought great difficulties to the promotion of green buildings. The comprehensive evaluation of energy conservation and emission reduction of construction projects is classical MAGDM. Recently, the Logarithmic TODIM (LogTODIM) method has been employed to cope with MAGDM issues. The single-valued neutrosophic sets (SVNSs) are used as a tool for characterizing uncertain information during the comprehensive evaluation of energy conservation and emission reduction of construction projects. In this paper, the single-valued neutrosophic number Logarithmic TODIM (SVNN-LogTODIM) method is built to solve the MAGDM under SVNSs. In the end, a numerical case study for comprehensive evaluation of energy conservation and emission reduction of construction projects is given to validate the proposed method.

Looking forward to the future works, the combination of different decision support approaches can provide a direction for the development of scientific decision models [56, 57, 58, 59, 60, 61, 62, 63, 64].