A stratified fuzzy decision-making approach for engineering procurement construction project quality evaluation with triangular fuzzy neutrosophic information

Abstract

EPC (Engineering, Procedure, Construction) refers to the implementation of “procurement, construction, and design” for a certain project, which has a very similar meaning to general engineering contracting. The general contracting mode of EPC is that the construction enterprise, as the owner, contracts the construction project to the general contracting enterprise in a direct form. The engineering procurement construction project quality evaluation is looked as the multi-attribute decision-making (MADM). The triangular fuzzy neutrosophic sets (TFNSs) is more suitable for expressing uncertain information during the engineering procurement construction project quality evaluation. Grey relational analysis (GRA) method is a very active branch of grey system theory, whose basic idea is to determine whether the connections between different sequences are close based on the similarity of the geometric shapes of sequence curves. In this paper, the triangular fuzzy neutrosophic number GRA (TFNN-GRA) method is put up under triangular fuzzy neutrosophic sets (TFNSs) with completely unknown weight information. The information entropy is employed to obtain the weight values under TFNSs. Then, commenting GRA method with TFNSs, the TFNN-GRA is designed and the decision steps for MADM are constructed. Finally, a numerical example for engineering procurement construction project quality evaluation was given and some comparative analysis is employed to verify the advantages of TFNN-GRA method.

1. Introduction

Decision making has been an important activity of humanity since ancient times [1, 2, 3, 4]. With the rapid development of the economy in recent years, decision-making has gradually penetrated into practical issues in multiple fields, such as employment selection and stock market analysis [5, 6, 7, 8]. Whether decision-makers can make the right decisions quickly and accurately not only affects individuals but also relates to the development of the enterprise [4, 9, 10]. For enterprises, how to allocate their resources and make the most suitable choice for their development path affects their future. The study of Multiple attribute decision making (MADM) has important practical significance and high practical value [11, 12, 13, 14, 15, 16]. In the past ten years, the research on MADM has become more and more the society and researchers are interested, the more relevant research results are produced [17, 18, 19]. The fuzzy sets (FSs) [20] were put up to solve the MADM problems [21, 22]. The intuitionistic FSs (IFS) [23] were also put up to solve the MADM. Smarandache [24] put up the neutrosophic sets (NSs). In order to practical application, the single-value neutrosophic sets (SVNSs) [25] and interval-value neutrosophic sets (INSs) [26] were put forward as subclasses of NSs. According to the existing literatures. NSs is a generalization of FS and IFS. Sahin and Kucuk [27] gave the introduction of entropy measure of SVNSs. Li et al. [28] put forward the Heronian mean under SVNSs. Ye [29, 30, 31, 32, 33] gave some similarity measures for SVNSs. Liu et al. [34] put forward the Hamacher operatos into NSs. Biswas et al. [35] put forward the triangular fuzzy neutrosophic numbers (TFNNs). Chakraborty et al. [36] put forward the de-neutrosophication concept for neutrosophic number for TFNNs. Meng et al. [37] put forward the triangular fuzzy neutrosophic preference for ERP software selection. Wang et al. [38] put forward the TFNN-VIKOR for MADM Irvanizam et al. [39] put forward the TFNNs-MABAC for MADM.

There are three shortcomings in disposing of the MAGDM problems under TFNSs environment that to form our incentives in the following:

(1)

In complex situations, the triangular fuzzy neutrosophic sets (TFNSs) is more suitable for expressing more answers of DMs which is depicted with triangular fuzzy numbers (TFNs), which is beyond the capabilities of NSs and INSs, thus, TFNSs is selected to depict decision-making information in the paper.

(2)

EPC (Engineering Procurement Construction) refers to the general contracting of the entire process or several stages of the design, procurement, construction, and other aspects of an engineering construction project by the contractor, commissioned by the owner, in accordance with the contract [40, 41, 42]. And be responsible for the quality, safety, cost, and progress of the contracted projects. Under the policy documents and relevant guidance opinions jointly issued by the National Development and Reform Commission and the Ministry of Housing and Urban Rural Development to promote the “Management Measures for General Contracting of Housing Construction and Municipal Infrastructure Projects” and “Guiding Opinions on Promoting the Development of Full Process Engineering Consulting Services”, In principle, state-owned enterprises and government investment projects need to be equipped with an EPC engineering general contracting management team led by an EPC project manager and an EPC project manager to carry out the construction and implementation of the engineering general contracting project [43, 44, 45, 46]. In the EPC mode, Engineering not only includes specific design work, but may also include the overall planning of the entire construction project content, as well as the planning and specific work of the entire construction project implementation organization management; Procurement is not a general procurement of construction equipment and materials, but needs to further include the procurement of professional equipment and materials; Construction should be translated as “construction”, which includes construction, installation, testing, technical training, etc. [47, 48, 49, 50]. The engineering procurement construction project quality evaluation is looked as MADM with indeterminate information. The TFNSs is an appropriate form to express the indeterminate information in the engineering procurement construction project quality evaluation. Therefore, in this paper, the triangular fuzzy neutrosophic numbers GRA (TFNN-GRA) is put up based on GRA method and used to engineering procurement construction project quality evaluation.

(3)

In MADM problem, to fuse information is based on different methods. And GRA method [51] is one of the most commonly used methods, and its core idea is to select the optimal alternative based on the shape similarity degree between potential alternatives and positive ideal and negative ideal solutions. So far, GRA method has been used in few uncertain environments, thus, it is an interesting topic to study the GRA under TFNSs. Xie [52] constructed the TFNN-GRA method under TFNSs with completely known weight information. However, The method of Xie [52] can’t cope with the TFNN-GRA method under TFNSs with completely unknown weight information. Thus, how to cope with the TFNN-GRA method under TFNSs with completely unknown weight information is an interesting research topic.

Inspired by these decision factors, our contributions include the following: this study aims to expand the engineering procurement construction project quality evaluation in TFNSs environment and provide theoretical and methodological support for related evaluation decisions. Thus, the purpose of this paper is to build the TFNN-GRA according to the traditional GRA method [51] and TFNSs to study MADM problems with completely unknown weight information. Through in-depth research analysis, the research contribution of the paper is summarized: (1) the TFNN-GRA is proposed for MADM problem with TFNSs with completely unknown weight information; (2) The information entropy is employed to obtain the weight values under TFNSs; (3) the TFNN-GRA in constructed to cope with the engineering procurement construction project quality evaluation proposed; (4) Through the comparative analysis, it is found that TFNN-GRA for engineering procurement construction project quality evaluation proposed in the study are effective.

In order to do so, the research structure of this work is: the definition of TFNNSs is given in Section 2. The TFNNGRA method is put up for MADM with completely unknown weight information in Section 3. A numerical example for engineering procurement construction project quality evaluation is used to show the TFNNGRA in Section 4. Section 5 gives the conclusions.

2. Preliminaries

Biswas et al. [35] designed the TFNSs.

Definition 1 [35]. Let $\mathrm{\Theta }$ be a fix set the TFNSs $PP$ is:

$PP=\{(\theta ,PA(\theta ),PB(\theta ),PC(\theta ))|\theta \in \mathrm{\Theta }\}$(1)

where $PA(\theta ),PB(\theta ),PC(\theta )\in [0,1]$ is truthmembership, indeterminacymembership and falsitymembership by TFNs.

$PA(\theta )=(P{A}^L(\theta ),P{A}^M(\theta ),P{A}^U(\theta )),0⩽P{A}^L(\theta )⩽P{A}^M(\theta )⩽P{A}^U(\theta )⩽1$(2) $PB(\theta )=(P{B}^L(\theta ),P{B}^M(\theta ),P{B}^U(\theta )),0⩽P{B}^L(\theta )⩽P{B}^M(\theta )⩽P{B}^U(\theta )⩽1$(3) $PB(\theta )=(P{B}^L(\theta ),P{B}^M(\theta ),P{B}^U(\theta )),0⩽P{B}^L(\theta )⩽P{B}^M(\theta )⩽P{B}^U(\theta )⩽1$(4)

For convenience, $PP=\{(P{A}^L,P{A}^M,P{A}^U),(P{B}^L,P{B}^M,P{B}^U),(P{C}^L,P{C}^M,P{C}^U)\}$ is named as a TFNN and meet $0⩽P{A}^U+P{B}^U+P{C}^U⩽3$ .

Definition 2 [35]. Let $P{P}_1=\{(P{A}_1^L,P{A}_1^M,P{A}_1^U),(P{B}_1^L,P{B}_1^M,P{B}_1^U),(P{C}_1^L,P{C}_1^M,P{C}_1^U)\}$ , $P{P}_2=\{(P{A}_2^L,P{A}_2^M,P{A}_2^U),(P{B}_2^L,P{B}_2^M,P{B}_2^U),(P{C}_2^L,P{C}_2^M,P{C}_2^U)\}$ and $PP=\{(P{A}^L,P{A}^M,P{A}^U),(P{B}^L,P{B}^M,P{B}^U),(P{C}^L,P{C}^M,P{C}^U)\}$ , the operational laws are designed:

$(1)P{P}_1\oplus P{P}_2=\left\{\begin{array}{c}\left(\begin{array}{c}P{A}_1^L+P{A}_2^L-P{A}_1^{L}P{A}_2^L,\hfill \\ P{A}_1^M+P{A}_2^M-P{A}_1^{M}P{A}_2^M,P{A}_1^U+P{A}_2^U-P{A}_1^{U}P{A}_2^U\hfill \end{array}\right),\hfill \\ (P{B}_1^{L}P{B}_2^L,P{B}_1^{M}P{B}_2^M,P{B}_1^{U}P{B}_2^U),\hfill \\ (P{C}_1^{L}P{C}_2^L,P{C}_1^{M}P{C}_2^M,P{C}_1^{U}P{C}_2^U)\hfill \end{array}\right\};$ $(2)P{P}_1\otimes P{P}_2=\left\{\begin{array}{c}(P{A}_1^{L}P{A}_2^L,P{A}_1^{M}P{A}_2^M,P{A}_1^{U}P{A}_2^U),\hfill \\ \left(\begin{array}{c}P{B}_1^L+P{B}_2^L-P{B}_1^{L}P{B}_2^L,\hfill \\ P{B}_1^M+P{B}_2^M-P{B}_1^{M}P{B}_2^M,P{B}_1^U+P{B}_2^U-P{B}_1^{U}P{B}_2^U\hfill \end{array}\right),\hfill \\ \left(\begin{array}{c}P{C}_1^L+P{C}_2^L-P{C}_1^{L}P{C}_2^L,\hfill \\ P{C}_1^M+P{C}_2^M-P{C}_1^{M}P{C}_2^M,P{C}_1^U+P{C}_2^U-P{C}_1^{U}P{C}_2^U\hfill \end{array}\right)\hfill \end{array}\right\};$ $(3)\lambda PP=\left\{\begin{array}{c}(1-{(1-P{A}^L)}^{\lambda },1-{(1-P{A}^M)}^{\lambda },1-{(1-P{A}^U)}^{\lambda }),\hfill \\ ({(P{B}^L)}^{\lambda },{(P{B}^M)}^{\lambda },{(P{B}^U)}^{\lambda }),\hfill \\ ({(P{C}^L)}^{\lambda },{(P{C}^M)}^{\lambda },{(P{C}^U)}^{\lambda })\hfill \end{array}\right\},\lambda >0;$ $(4)P{P}^{\lambda }=\left\{\begin{array}{c}({(P{A}^L)}^{\lambda },{(P{A}^M)}^{\lambda },{(P{A}^U)}^{\lambda }),\hfill \\ (1-{(1-P{B}^L)}^{\lambda },1-{(1-P{B}^M)}^{\lambda },1-{(1-P{B}^U)}^{\lambda }),\hfill \\ (1-{(1-P{C}^L)}^{\lambda },1-{(1-P{C}^M)}^{\lambda },1-{(1-P{C}^U)}^{\lambda })\hfill \end{array}\right\},\lambda >0.$

From Definition 2, the operation laws have following properties.

$(1)P{P}_1\oplus P{P}_2=P{P}_2\oplus P{P}_1,P{P}_1\otimes P{P}_2=P{P}_2\otimes P{P}_1,{({(P{P}_1)}^{{\lambda }_1})}^{{\lambda }_2}={(P{P}_1)}^{{\lambda }_1{\lambda }_2};$(5) $(2)\lambda (P{P}_1\oplus P{P}_2)=\lambda P{P}_1\oplus \lambda P{P}_2,{(P{P}_1\otimes P{P}_2)}^{\lambda }={(P{P}_1)}^{\lambda }\otimes {(P{P}_2)}^{\lambda };$(6) $(3){\lambda }_{1}P{P}_1\oplus {\lambda }_{2}P{P}_1=({\lambda }_1+{\lambda }_2)P{P}_1,{(P{P}_1)}^{{\lambda }_1}\otimes {(P{P}_1)}^{{\lambda }_2}={(P{P}_1)}^{({\lambda }_1+{\lambda }_2)}.$(7)

Definition 3 [35]. Let $PP=\{(P{A}^L,P{A}^M,P{A}^U),(P{B}^L,P{B}^M,P{B}^U),(P{C}^L,P{C}^M,P{C}^U)\}$ , the score and accuracy functions are designed:

$SF(PP)={\displaystyle \frac{1}{12}}\left[\begin{array}{c}8+(P{A}^L+2P{A}^M+P{A}^U)\hfill \\ -(P{B}^L+2P{B}^M+P{B}^U)\hfill \\ -(P{C}^L+2P{C}^M+P{C}^U)\hfill \end{array}\right],SF(PP)\in [0,1]$(8) $AF(PP)={\displaystyle \frac{1}{4}}\left[\begin{array}{c}(P{A}^L+2P{A}^M+P{A}^U)\hfill \\ -(P{B}^L+2P{B}^M+P{B}^U)\hfill \end{array}\right],AF(PP)\in [-1,1]$(9)

For two TFNNs $P{P}_1$ and $P{P}_2$ , then

$(1)\text{if }SF(P{P}_1)\prec SF(P{P}_2),P{P}_1\prec P{P}_2;$ $(2)\text{if }SF(P{P}_1)=SF(P{P}_2),AF(P{P}_1)\prec AF(P{P}_2),P{P}_1\prec P{P}_2;$ $(3)\text{if }SF(P{P}_1)=SF(P{P}_2),AF(P{P}_1)=AF(P{P}_2),P{P}_1=P{P}_2.$

Definition 4 [39]. Let $P{P}_1=\{(P{A}_1^L,P{A}_1^M,P{A}_1^U),(P{B}_1^L,P{B}_1^M,P{B}_1^U),(P{C}_1^L,P{C}_1^M,P{C}_1^U)\}$ , $P{P}_2=\{(P{A}_2^L,P{A}_2^M,P{A}_2^U),(P{B}_2^L,P{B}_2^M,P{B}_2^U),(P{C}_2^L,P{C}_2^M,P{C}_2^U)\}$ , the Euclidean distance is designed:

$ED(P{P}_1,P{P}_2)=\sqrt{{\displaystyle \frac{1}{9}}\left(\begin{array}{c}{|P{A}_1^L-P{A}_2^L|}^2+{|P{A}_1^M-P{A}_2^M|}^2+{|P{A}_1^U-P{A}_2^U|}^2\hfill \\ \\ +{|P{B}_1^L-P{B}_2^L|}^2+{|P{B}_1^M-P{B}_2^M|}^2+{|P{B}_1^U-P{B}_2^U|}^2\hfill \\ \\ +{|P{C}_1^L-P{C}_2^L|}^2+{|P{C}_1^M-P{C}_2^M|}^2+{|P{C}_1^U-P{C}_2^U|}^2\hfill \end{array}\right)}$(10)

3. GRA method for MADM issue with TFNNs

Then, the TFNNGRA is designed for MADM. Suppose there are $m$ alternatives $\{P{X}_1,P{X}_2,\mathrm{\dots },P{X}_m\}$ , $n$ devised attributes $\{P{G}_1,P{G}_2,\mathrm{\dots },P{G}_n\}$ and the steps of TFNNGRA method for MADM are devised (Fig. 1).

Figure 1.

The decision framework of the TFNN-GRA for MADM.

Step 1. Build the TFNNmatrix $PR={[P{R}_{ij}]}_{m\times n}$ :

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

where

$P{R}_{ij}=\left\{\begin{array}{c}((P{A}_{ij}^L),(P{A}_{ij}^M),(P{A}_{ij}^U)),\hfill \\ \\ ((P{B}_{ij}^L),(P{B}_{ij}^M),(P{B}_{ij}^U)),\hfill \\ \\ ((P{C}_{ij}^L),(P{C}_{ij}^M),(P{C}_{ij}^U))\hfill \end{array}\right\}.$

Step 2. Normalize the $PR={[P{R}_{ij}]}_{m\times n}$ to $NR={[{\text{𝑁𝑃𝑅}}_{ij}]}_{m\times n}$ .

For benefit attributes:

${\text{𝑁𝑃𝑅}}_{ij}=\left\{\begin{array}{c}(({\text{𝑁𝑃𝐴}}_{ij}^L),({\text{𝑁𝑃𝐴}}_{ij}^M),({\text{𝑁𝑃𝐴}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐵}}_{ij}^L),({\text{𝑁𝑃𝐵}}_{ij}^M),({\text{𝑁𝑃𝐵}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐶}}_{ij}^L),({\text{𝑁𝑃𝐶}}_{ij}^M),({\text{𝑁𝑃𝐶}}_{ij}^U))\hfill \end{array}\right\}=\left\{\begin{array}{c}((P{A}_{ij}^L),(P{A}_{ij}^M),(P{A}_{ij}^U)),\hfill \\ \\ ((P{B}_{ij}^L),(P{B}_{ij}^M),(P{B}_{ij}^U)),\hfill \\ \\ ((P{C}_{ij}^L),(P{C}_{ij}^M),(P{C}_{ij}^U))\hfill \end{array}\right\}$(12)

For cost attributes:

${\text{𝑁𝑃𝑅}}_{ij}=\left\{\begin{array}{c}(({\text{𝑁𝑃𝐴}}_{ij}^L),({\text{𝑁𝑃𝐴}}_{ij}^M),({\text{𝑁𝑃𝐴}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐵}}_{ij}^L),({\text{𝑁𝑃𝐵}}_{ij}^M),({\text{𝑁𝑃𝐵}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐶}}_{ij}^L),({\text{𝑁𝑃𝐶}}_{ij}^M),({\text{𝑁𝑃𝐶}}_{ij}^U))\hfill \end{array}\right\}=\left\{\begin{array}{c}((P{C}_{ij}^L),(P{C}_{ij}^M),(P{C}_{ij}^U)),\hfill \\ \\ ((P{B}_{ij}^L),(P{B}_{ij}^M),(P{B}_{ij}^U)),\hfill \\ \\ ((P{A}_{ij}^L),(P{A}_{ij}^M),(P{A}_{ij}^U))\hfill \end{array}\right\}$(13)

Step 3. Obtain the attributes weight by entropy method.

Many scholars focused on to obtain the weight under different environment [53, 54, 55, 56, 57]. Entropy [58] is a conventional method to obtain weight. Firstly, the normalized TFNN-matrix ${\text{𝑁𝑁𝑅}}_{ij}$ is obtained:

${\text{𝑁𝑁𝑅}}_{ij}={\displaystyle \frac{SF\left\{\begin{array}{c}(({\text{𝑁𝑃𝐴}}_{ij}^L),({\text{𝑁𝑃𝐴}}_{ij}^M),({\text{𝑁𝑃𝐴}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐵}}_{ij}^L),({\text{𝑁𝑃𝐵}}_{ij}^M),({\text{𝑁𝑃𝐵}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐶}}_{ij}^L),({\text{𝑁𝑃𝐶}}_{ij}^M),({\text{𝑁𝑃𝐶}}_{ij}^U))\hfill \end{array}\right\}}{\sum _{i=1}^{m}SF\left\{\begin{array}{c}(({\text{𝑁𝑃𝐴}}_{ij}^L),({\text{𝑁𝑃𝐴}}_{ij}^M),({\text{𝑁𝑃𝐴}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐵}}_{ij}^L),({\text{𝑁𝑃𝐵}}_{ij}^M),({\text{𝑁𝑃𝐵}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐶}}_{ij}^L),({\text{𝑁𝑃𝐶}}_{ij}^M),({\text{𝑁𝑃𝐶}}_{ij}^U))\hfill \end{array}\right\}}},$(14)

Then, the Shannon entropy $\text{𝑇𝐹𝑁𝑁𝑆𝐸}=({\text{𝑇𝐹𝑁𝑁𝑆𝐸}}_1,{\text{𝑇𝐹𝑁𝑁𝑆𝐸}}_2,\mathrm{\dots },{\text{𝑇𝐹𝑁𝑁𝑆𝐸}}_n)$ is derived by Eq. (15):

${\text{𝑇𝐹𝑁𝑁𝑆𝐸}}_j=-{\displaystyle \frac{1}{\ln m}}{\displaystyle \sum _{i=1}^m}{\text{𝑁𝑁𝑅}}_{ij}\ln {\text{𝑁𝑁𝑅}}_{ij}$(15)

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

Then, the weights $PW=(P{W}_1,P{W}_2,\mathrm{\dots },P{W}_n)$ is derived:

$P{W}_j={\displaystyle \frac{1-{\text{𝑇𝑃𝑁𝑁𝑆𝐸}}_j}{\sum _{j=1}^n(1-{\text{𝑇𝑃𝑁𝑁𝑆𝐸}}_j)}},j=1,2,\mathrm{\dots },n.$(16)

Step 4. Compute the TFNN positive ideal solution (TFNNPIS) and the TFNN negative ideal solution (TFNNNIS) [52]:

$\text{𝑇𝑃𝑁𝑁𝑃𝐼𝑆}=\left\{\begin{array}{c}({({\text{𝑁𝑃𝐴}}_j^L)}^{+},{({\text{𝑁𝑃𝐴}}_j^M)}^{+},{({\text{𝑁𝑃𝐴}}_j^U)}^{+}),\hfill \\ \\ ({({\text{𝑁𝑃𝐵}}_j^L)}^{+},{({\text{𝑁𝑃𝐵}}_j^M)}^{+},{({\text{𝑁𝑃𝐵}}_j^U)}^{+}),\hfill \\ \\ ({({\text{𝑁𝑃𝐶}}_j^L)}^{+},{({\text{𝑁𝑃𝐶}}_j^M)}^{+},{({\text{𝑁𝑃𝐶}}_j^U)}^{+})\hfill \end{array}\right\}$(17) $\text{𝑇𝐹𝑁𝑁𝑁𝐼𝑆}=\left\{\begin{array}{c}({({\text{𝑁𝑃𝐴}}_j^L)}^{-},{({\text{𝑁𝑃𝐴}}_j^M)}^{-},{({\text{𝑁𝑃𝐴}}_j^U)}^{-}),\hfill \\ \\ ({({\text{𝑁𝑃𝐵}}_j^L)}^{-},{({\text{𝑁𝑃𝐵}}_j^M)}^{-},{({\text{𝑁𝑃𝐵}}_j^U)}^{+}),\hfill \\ \\ ({({\text{𝑁𝑃𝐶}}_j^L)}^{-},{({\text{𝑁𝑃𝐶}}_j^M)}^{-},{({\text{𝑁𝑃𝐶}}_j^U)}^{-})\hfill \end{array}\right\}$(18) $SF\left\{\begin{array}{c}({({\text{𝑁𝑃𝐴}}_j^L)}^{+},{({\text{𝑁𝑃𝐴}}_j^M)}^{+},{({\text{𝑁𝑃𝐴}}_j^U)}^{+}),\hfill \\ \\ ({({\text{𝑁𝑃𝐵}}_j^L)}^{+},{({\text{𝑁𝑃𝐵}}_j^M)}^{+},{({\text{𝑁𝑃𝐵}}_j^U)}^{+}),\hfill \\ \\ ({({\text{𝑁𝑃𝐶}}_j^L)}^{+},{({\text{𝑁𝑃𝐶}}_j^M)}^{+},{({\text{𝑁𝑃𝐶}}_j^U)}^{+})\hfill \end{array}\right\}$ (19) $=\underset{i}{max}SF\left\{\begin{array}{c}(({\text{𝑁𝑃𝐴}}_{ij}^L),({\text{𝑁𝑃𝐴}}_{ij}^M),({\text{𝑁𝑃𝐴}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐵}}_{ij}^L),({\text{𝑁𝑃𝐵}}_{ij}^M),({\text{𝑁𝑃𝐵}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐶}}_{ij}^L),({\text{𝑁𝑃𝐶}}_{ij}^M),({\text{𝑁𝑃𝐶}}_{ij}^U))\hfill \end{array}\right\}$ $SF\left\{\begin{array}{c}({({\text{𝑁𝑃𝐴}}_j^L)}^{-},{({\text{𝑁𝑃𝐴}}_j^M)}^{-},{({\text{𝑁𝑃𝐴}}_j^U)}^{-}),\hfill \\ \\ ({({\text{𝑁𝑃𝐵}}_j^L)}^{-},{({\text{𝑁𝑃𝐵}}_j^M)}^{-},{({\text{𝑁𝑃𝐵}}_j^U)}^{+}),\hfill \\ \\ ({({\text{𝑁𝑃𝐶}}_j^L)}^{-},{({\text{𝑁𝑃𝐶}}_j^M)}^{-},{({\text{𝑁𝑃𝐶}}_j^U)}^{-})\hfill \end{array}\right\}$ (20) $=\underset{i}{min}SF\left\{\begin{array}{c}(({\text{𝑁𝑃𝐴}}_{ij}^L),({\text{𝑁𝑃𝐴}}_{ij}^M),({\text{𝑁𝑃𝐴}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐵}}_{ij}^L),({\text{𝑁𝑃𝐵}}_{ij}^M),({\text{𝑁𝑃𝐵}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐶}}_{ij}^L),({\text{𝑁𝑃𝐶}}_{ij}^M),({\text{𝑁𝑃𝐶}}_{ij}^U))\hfill \end{array}\right\}$

Step 5. Compute the grey rational coefficients (GRC) from the TFNNPIS and TFNNNIS as:

$\text{𝑇𝐹𝑁𝑁𝑃𝐼𝑆}({\xi }_{ij})$ (21) $={\displaystyle \frac{\left(\begin{array}{c}\underset{1⩽i⩽m}{min}\underset{1⩽j⩽m}{min}ED(\left\{\begin{array}{c}({({\text{𝑁𝑃𝐴}}_j^L)}^{+},{({\text{𝑁𝑃𝐴}}_j^M)}^{+},{({\text{𝑁𝑃𝐴}}_j^U)}^{+}),\hfill \\ ({({\text{𝑁𝑃𝐵}}_j^L)}^{+},{({\text{𝑁𝑃𝐵}}_j^M)}^{+},{({\text{𝑁𝑃𝐵}}_j^U)}^{+}),\hfill \\ ({({\text{𝑁𝑃𝐶}}_j^L)}^{+},{({\text{𝑁𝑃𝐶}}_j^M)}^{+},{({\text{𝑁𝑃𝐶}}_j^U)}^{+})\hfill \end{array}\right\},\hfill \\ \left\{\begin{array}{c}(({\text{𝑁𝑃𝐴}}_{ij}^L),({\text{𝑁𝑃𝐴}}_{ij}^M),({\text{𝑁𝑃𝐴}}_{ij}^U)),\hfill \\ (({\text{𝑁𝑃𝐵}}_{ij}^L),({\text{𝑁𝑃𝐵}}_{ij}^M),({\text{𝑁𝑃𝐵}}_{ij}^U)),\hfill \\ (({\text{𝑁𝑃𝐶}}_{ij}^L),({\text{𝑁𝑃𝐶}}_{ij}^M),({\text{𝑁𝑃𝐶}}_{ij}^U))\hfill \end{array}\right\})+\rho max{}_{1⩽i⩽m}max{}_{1⩽j⩽m}\hfill \\ ED(\left\{\begin{array}{c}({({\text{𝑁𝑃𝐴}}_j^L)}^{+},{({\text{𝑁𝑃𝐴}}_j^M)}^{+},{({\text{𝑁𝑃𝐴}}_j^U)}^{+}),\hfill \\ \\ ({({\text{𝑁𝑃𝐵}}_j^L)}^{+},{({\text{𝑁𝑃𝐵}}_j^M)}^{+},{({\text{𝑁𝑃𝐵}}_j^U)}^{+}),\hfill \\ \\ ({({\text{𝑁𝑃𝐶}}_j^L)}^{+},{({\text{𝑁𝑃𝐶}}_j^M)}^{+},{({\text{𝑁𝑃𝐶}}_j^U)}^{+})\hfill \end{array}\right\},\left\{\begin{array}{c}(({\text{𝑁𝑃𝐴}}_{ij}^L),({\text{𝑁𝑃𝐴}}_{ij}^M),({\text{𝑁𝑃𝐴}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐵}}_{ij}^L),({\text{𝑁𝑃𝐵}}_{ij}^M),({\text{𝑁𝑃𝐵}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐶}}_{ij}^L),({\text{𝑁𝑃𝐶}}_{ij}^M),({\text{𝑁𝑃𝐶}}_{ij}^U))\hfill \end{array}\right\})\hfill \end{array}\right)}{\left(\begin{array}{c}ED(\left\{\begin{array}{c}({({\text{𝑁𝑃𝐴}}_j^L)}^{+},{({\text{𝑁𝑃𝐴}}_j^M)}^{+},{({\text{𝑁𝑃𝐴}}_j^U)}^{+}),\hfill \\ ({({\text{𝑁𝑃𝐵}}_j^L)}^{+},{({\text{𝑁𝑃𝐵}}_j^M)}^{+},{({\text{𝑁𝑃𝐵}}_j^U)}^{+}),\hfill \\ ({({\text{𝑁𝑃𝐶}}_j^L)}^{+},{({\text{𝑁𝑃𝐶}}_j^M)}^{+},{({\text{𝑁𝑃𝐶}}_j^U)}^{+})\hfill \end{array}\right\},\hfill \\ \left\{\begin{array}{c}(({\text{𝑁𝑃𝐴}}_{ij}^L),({\text{𝑁𝑃𝐴}}_{ij}^M),({\text{𝑁𝑃𝐴}}_{ij}^U)),\hfill \\ (({\text{𝑁𝑃𝐵}}_{ij}^L),({\text{𝑁𝑃𝐵}}_{ij}^M),({\text{𝑁𝑃𝐵}}_{ij}^U)),\hfill \\ (({\text{𝑁𝑃𝐶}}_{ij}^L),({\text{𝑁𝑃𝐶}}_{ij}^M),({\text{𝑁𝑃𝐶}}_{ij}^U))\hfill \end{array}\right\})+\rho max{}_{1⩽i⩽m}max{}_{1⩽j⩽m}\hfill \\ ED(\left\{\begin{array}{c}({({\text{𝑁𝑃𝐴}}_j^L)}^{+},{({\text{𝑁𝑃𝐴}}_j^M)}^{+},{({\text{𝑁𝑃𝐴}}_j^U)}^{+}),\hfill \\ \\ ({({\text{𝑁𝑃𝐵}}_j^L)}^{+},{({\text{𝑁𝑃𝐵}}_j^M)}^{+},{({\text{𝑁𝑃𝐵}}_j^U)}^{+}),\hfill \\ \\ ({({\text{𝑁𝑃𝐶}}_j^L)}^{+},{({\text{𝑁𝑃𝐶}}_j^M)}^{+},{({\text{𝑁𝑃𝐶}}_j^U)}^{+})\hfill \end{array}\right\},\left\{\begin{array}{c}(({\text{𝑁𝑃𝐴}}_{ij}^L),({\text{𝑁𝑃𝐴}}_{ij}^M),({\text{𝑁𝑃𝐴}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐵}}_{ij}^L),({\text{𝑁𝑃𝐵}}_{ij}^M),({\text{𝑁𝑃𝐵}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐶}}_{ij}^L),({\text{𝑁𝑃𝐶}}_{ij}^M),({\text{𝑁𝑃𝐶}}_{ij}^U))\hfill \end{array}\right\})\hfill \end{array}\right)}}$ $\text{𝑇𝐹𝑁𝑁𝑁𝐼𝑆}({\xi }_{ij})$ (22) $={\displaystyle \frac{\left(\begin{array}{c}\underset{1⩽i⩽m}{min}\underset{1⩽j⩽m}{min}ED(\left\{\begin{array}{c}({({\text{𝑁𝑃𝐴}}_j^L)}^{-},{({\text{𝑁𝑃𝐴}}_j^M)}^{-},{({\text{𝑁𝑃𝐴}}_j^U)}^{-}),\hfill \\ ({({\text{𝑁𝑃𝐵}}_j^L)}^{-},{({\text{𝑁𝑃𝐵}}_j^M)}^{-},{({\text{𝑁𝑃𝐵}}_j^U)}^{+}),\hfill \\ ({({\text{𝑁𝑃𝐶}}_j^L)}^{-},{({\text{𝑁𝑃𝐶}}_j^M)}^{-},{({\text{𝑁𝑃𝐶}}_j^U)}^{-})\hfill \end{array}\right\},\hfill \\ \left\{\begin{array}{c}(({\text{𝑁𝑃𝐴}}_{ij}^L),({\text{𝑁𝑃𝐴}}_{ij}^M),({\text{𝑁𝑃𝐴}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐵}}_{ij}^L),({\text{𝑁𝑃𝐵}}_{ij}^M),({\text{𝑁𝑃𝐵}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐶}}_{ij}^L),({\text{𝑁𝑃𝐶}}_{ij}^M),({\text{𝑁𝑃𝐶}}_{ij}^U))\hfill \end{array}\right\})+\rho max{}_{1⩽i⩽m}max{}_{1⩽j⩽m}\hfill \\ ED(\left\{\begin{array}{c}({({\text{𝑁𝑃𝐴}}_j^L)}^{-},{({\text{𝑁𝑃𝐴}}_j^M)}^{-},{({\text{𝑁𝑃𝐴}}_j^U)}^{-}),\hfill \\ \\ ({({\text{𝑁𝑃𝐵}}_j^L)}^{-},{({\text{𝑁𝑃𝐵}}_j^M)}^{-},{({\text{𝑁𝑃𝐵}}_j^U)}^{+}),\hfill \\ \\ ({({\text{𝑁𝑃𝐶}}_j^L)}^{-},{({\text{𝑁𝑃𝐶}}_j^M)}^{-},{({\text{𝑁𝑃𝐶}}_j^U)}^{-})\hfill \end{array}\right\},\left\{\begin{array}{c}(({\text{𝑁𝑃𝐴}}_{ij}^L),({\text{𝑁𝑃𝐴}}_{ij}^M),({\text{𝑁𝑃𝐴}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐵}}_{ij}^L),({\text{𝑁𝑃𝐵}}_{ij}^M),({\text{𝑁𝑃𝐵}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐶}}_{ij}^L),({\text{𝑁𝑃𝐶}}_{ij}^M),({\text{𝑁𝑃𝐶}}_{ij}^U))\hfill \end{array}\right\})\hfill \end{array}\right)}{\left(\begin{array}{c}ED(\left\{\begin{array}{c}({({\text{𝑁𝑃𝐴}}_j^L)}^{-},{({\text{𝑁𝑃𝐴}}_j^M)}^{-},{({\text{𝑁𝑃𝐴}}_j^U)}^{-}),\hfill \\ \\ ({({\text{𝑁𝑃𝐵}}_j^L)}^{-},{({\text{𝑁𝑃𝐵}}_j^M)}^{-},{({\text{𝑁𝑃𝐵}}_j^U)}^{+}),\hfill \\ \\ ({({\text{𝑁𝑃𝐶}}_j^L)}^{-},{({\text{𝑁𝑃𝐶}}_j^M)}^{-},{({\text{𝑁𝑃𝐶}}_j^U)}^{-})\hfill \end{array}\right\},\hfill \\ \left\{\begin{array}{c}(({\text{𝑁𝑃𝐴}}_{ij}^L),({\text{𝑁𝑃𝐴}}_{ij}^M),({\text{𝑁𝑃𝐴}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐵}}_{ij}^L),({\text{𝑁𝑃𝐵}}_{ij}^M),({\text{𝑁𝑃𝐵}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐶}}_{ij}^L),({\text{𝑁𝑃𝐶}}_{ij}^M),({\text{𝑁𝑃𝐶}}_{ij}^U))\hfill \end{array}\right\})+\rho max{}_{1⩽i⩽m}max{}_{1⩽j⩽m}\hfill \\ ED(\left\{\begin{array}{c}({({\text{𝑁𝑃𝐴}}_j^L)}^{-},{({\text{𝑁𝑃𝐴}}_j^M)}^{-},{({\text{𝑁𝑃𝐴}}_j^U)}^{-}),\hfill \\ \\ ({({\text{𝑁𝑃𝐵}}_j^L)}^{-},{({\text{𝑁𝑃𝐵}}_j^M)}^{-},{({\text{𝑁𝑃𝐵}}_j^U)}^{+}),\hfill \\ \\ ({({\text{𝑁𝑃𝐶}}_j^L)}^{-},{({\text{𝑁𝑃𝐶}}_j^M)}^{-},{({\text{𝑁𝑃𝐶}}_j^U)}^{-})\hfill \end{array}\right\},\left\{\begin{array}{c}(({\text{𝑁𝑃𝐴}}_{ij}^L),({\text{𝑁𝑃𝐴}}_{ij}^M),({\text{𝑁𝑃𝐴}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐵}}_{ij}^L),({\text{𝑁𝑃𝐵}}_{ij}^M),({\text{𝑁𝑃𝐵}}_{ij}^U)),\hfill \\ \\ (({\text{𝑁𝑃𝐶}}_{ij}^L),({\text{𝑁𝑃𝐶}}_{ij}^M),({\text{𝑁𝑃𝐶}}_{ij}^U))\hfill \end{array}\right\})\hfill \end{array}\right)}}$

In generally, distinguishing coefficient $\rho$ is between 0 and 1, usually 0.5.

Step 6. Obtain the grey relation degree (GRD) from given TFNNPIS and TFNNNIS:

$\text{𝑇𝐹𝑁𝑁𝑃𝐼𝑆𝐺𝑅𝐷}({\xi }_i)={\displaystyle \sum _{j=1}^n}P{W}_j\times \text{𝑇𝐹𝑁𝑁𝑃𝐼𝑆}({\xi }_{ij})$(23) $\text{𝑇𝐹𝑁𝑁𝑁𝐼𝑆𝐺𝑅𝐷}({\xi }_i)={\displaystyle \sum _{j=1}^n}P{W}_j\times \text{𝑇𝐹𝑁𝑁𝑁𝐼𝑆}({\xi }_{ij})$(24)

Step 7. Obtain the TFNN relative relational degree (TFNNRRD) from TFNNPIS:

$\text{𝑇𝐹𝑁𝑁𝑅𝑅𝐷}({\xi }_i)={\displaystyle \frac{\text{𝑇𝐹𝑁𝑁𝑃𝐼𝑆𝐺𝑅𝐷}({\xi }_i)}{\text{𝑇𝐹𝑁𝑁𝑃𝐼𝑆𝐺𝑅𝐷}({\xi }_i)+\text{𝑇𝐹𝑁𝑁𝑁𝐼𝑆𝐺𝑅𝐷}({\xi }_i)}}$(25)

Step 8. Obtain the optimal alternative with highest $\text{𝑇𝑃𝑁𝑁𝑅𝑅𝐷}({\xi }_i)$ .

4. The numerical case and comparative analysis

4.1 Numerical case

The EPC project mainly has the following key characteristics, which also determine that the EPC project can be scientifically and reasonably implemented: firstly, as the owner, we should entrust all related work such as engineering design, procurement, and construction services to the general contractor of the engineering project as much as possible. In this case, the owner only needs to pay attention to the overall, target control, and management. Secondly, the owner only signs the contract with the general contractor of the engineering project. After all the signing is completed, as the general contractor of the project, some design, procurement, and construction related service work should be subcontracted to subcontractors to complete separately, and they do not need to worry about either side. As for subcontractors, before the implementation of the project, they should sign relevant contracts with the general contractor, especially subcontracts, To avoid signing relevant contracts with the owner, and as a subcontractor, all of its work should be the responsibility of the general contractor. Once again, special management organizations organized by owners can also entrust some specialized engineering project management companies to implement overall, objective control and management of engineering projects as owners. In addition, as owners, they should make every effort to transfer the management risks inherent in the EPC model to the general contractor, while the general contractors of all engineering projects need to analyze specific problems and bear certain unexpected risks and responsibilities based on their own situation. At this time, the general contractor of the engineering project will have the opportunity to gain profits. The involvement of owners in certain specific organizations is not deep enough, therefore, under the EPC mode, the general contractor of engineering projects can exert their subjective initiative in certain aspects and also use their management experience to strive for more benefits for owners or contractors, if conditions permit. Finally, there are several derivative models for the contracting boundary of EPC project general contracting, for example, as a design contracting, it should start from multiple scheme designs. It should be noted that as a procurement department, the core of its work needs to be entrusted to the entire instrument company. The engineering procurement construction project quality evaluation can be regarded as MADM. In current section a numerical case for engineering procurement construction project quality evaluation through TFNNGRA. Assume that five engineering procurement construction projects $P{X}_i(i=1,2,3,4,5)$ to be evaluated through four attributes:

⟀ FG₁ is quality management in design. The general contracting of power EPC projects is a very systematic project, which includes a series of links such as market analysis, market sales, project evaluation, project bidding, engineering design, equipment procurement, and construction stage. Most of the current EPC general contractors in our country are transformed from survey and design enterprises in various regions. In response to this special situation, it is important to be clear in understanding that there are significant differences between EPC general contracting projects and previous pure design work. The design work in the EPC general contracting project is not the entire project, it is only a relatively important stage in the entire project chain. This stage requires preparation for various tasks during project initiation, and also provides specific requirements and data parameters for the subsequent equipment procurement and construction stages. The key points of quality management in the design phase are as follows: management of basic documents, management of calculation sheets, management of design changes, supplier drawing evaluation, design review, project completion review, and other contents.

⟁ FG₂ is the quality management in procurement: The procurement of power equipment and materials is a very important task, which is not only related to project cost and construction progress, but also directly related to project quality. Quality management in procurement includes supplier qualification assessment, implementation of inspection and testing plans, and other tasks. Most power equipment is non-standard products, so it is very important to have excellent equipment suppliers in all aspects. Many large foreign engineering companies have established a database of equipment subcontractors, and in principle, only existing manufacturers in the database are qualified for supply. Through the database, staff can quickly obtain information about all equipment suppliers they have previously worked with.

⟂ FG₃ is quality management during construction. The most important link in the quality management of EPC general contracting projects is the quality management during the construction phase, which directly reflects the success of the work in the design and procurement stages. Effective quality management can not only ensure the quality of the project, but also control engineering and achieve the expected goals, which is also conducive to reducing costs. The management of construction quality plan includes numerous contents such as inspection and testing plan (ITP), training phase, relevant supervision work, and project completion handover. It is a special part of the ITP content of the sample project. In order to ensure the quality control method in the ITP, a sample project shall be made before the commencement of important projects. After verification, all similar projects shall be executed according to the quality standard of the sample project. This method can effectively control the quality of the project.

⟃ FG₄ is environmental factor quality management. The geographical environment of engineering projects and the natural environment of construction have a very important impact on project quality, such as the engineering geological environment may affect the design scheme, temperature and humidity may affect the selection of construction materials, and so on. During the project implementation process, it is necessary to attach great importance to the impact of environmental factors and take practical and effective preventive measures based on the actual situation of the engineering project. Control can be carried out from the following aspects: firstly, in the initial stage of the project, a comprehensive and detailed investigation should be conducted on the geographical and natural environment of the project, an evaluation report should be issued, reasonable design proposals should be proposed, and construction suggestions should be made to ensure the quality of the project. The second is to develop special measures when carrying out concrete engineering and underwater engineering construction under harsh conditions such as winter rain and hot weather. Thirdly, during the construction process, measures such as improving construction techniques should be taken to minimize environmental pollution to the greatest extent possible.

The five possible engineering procurement construction projects $P{X}_i(i=1,2,3,4,5)$ are to be evaluated by TFNNs under the four attributes. The TFNNGRA method is used for engineering procurement construction project quality evaluation.

Step 1. Build the TFNNmatrix $PR={[P{R}_{ij}]}_{m\times n}$ (see Table 1).

Table 1

The TFNN information

	PG1	PG2
PX1	$\left\{\begin{array}{c}(0.51,0.65,0.75),\hfill \\ (0.33,0.54,0.61),\hfill \\ (0.42,0.63,0.81)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.45,0.62,0.95),\hfill \\ (0.51,0.63,0.87),\hfill \\ (0.43,0.58,0.61)\hfill \end{array}\right\}$
PX2	$\left\{\begin{array}{c}(0.53,0.65,0.80),\hfill \\ (0.65,0.74,0.82),\hfill \\ (0.41,0.58,0.76)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.34,0.52,0.76),\hfill \\ (0.39,0.44,0.59),\hfill \\ (0.25,0.41,0.68)\hfill \end{array}\right\}$
PX3	$\left\{\begin{array}{c}(0.45,0.61,0.76),\hfill \\ (0.59,0.71,0.92),\hfill \\ (0.43,0.65,0.89)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.53,0.66,0.81),\hfill \\ (0.52,0.67,0.83),\hfill \\ (0.26,0.49,0.72)\hfill \end{array}\right\}$
PX4	$\left\{\begin{array}{c}(0.62,0.83,0.92),\hfill \\ (0.42,0.57,0.69),\hfill \\ (0.35,0.46,0.58)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.51,0.68,0.71),\hfill \\ (0.38,0.49,0.58),\hfill \\ (0.29,0.35,0.43)\hfill \end{array}\right\}$
PX5	$\left\{\begin{array}{c}(0.61,0.73,0.95),\hfill \\ (0.45,0.51,0.64),\hfill \\ (0.52,0.61,0.75)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.47,0.52,0.89),\hfill \\ (0.23,0.51,0.68),\hfill \\ (0.26,0.48,0.73)\hfill \end{array}\right\}$
	PG3	PG4
PX1	$\left\{\begin{array}{c}(0.60,0.89,0.93),\hfill \\ (0.28,0.39,0.55),\hfill \\ (0.25,0.33,0.49)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.35,0.59,0.71),\hfill \\ (0.44,0.58,0.69),\hfill \\ (0.29,0.47,0.56)\hfill \end{array}\right\}$
PX2	$\left\{\begin{array}{c}(0.58,0.62,0.83),\hfill \\ (0.45,0.71,0.92),\hfill \\ (0.51,0.70,0.85)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.45,0.54,0.61),\hfill \\ (0.38,0.51,0.76),\hfill \\ (0.42,0.58,0.69)\hfill \end{array}\right\}$
PX3	$\left\{\begin{array}{c}(0.42,0.59,0.66),\hfill \\ (0.41,0.65,0.91),\hfill \\ (0.73,0.81,0.93)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.29,0.41,0.63),\hfill \\ (0.35,0.58,0.89),\hfill \\ (0.61,0.78,0.93)\hfill \end{array}\right\}$
PX4	$\left\{\begin{array}{c}(0.49,0.55,0.87),\hfill \\ (0.43,0.59,0.65),\hfill \\ (0.36,0.57,0.69)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.35,0.45,0.82),\hfill \\ (0.58,0.68,0.78),\hfill \\ (0.57,0.63,0.81)\hfill \end{array}\right\}$
PX5	$\left\{\begin{array}{c}(0.63,0.74,0.86),\hfill \\ (0.26,0.58,0.69),\hfill \\ (0.35,0.48,0.73)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.55,0.61,0.79),\hfill \\ (0.51,0.66,0.83),\hfill \\ (0.48,0.59,0.72)\hfill \end{array}\right\}$

Step 2. Normalize the matrix $PR={[P{R}_{ij}]}_{m\times n}$ to $\text{𝑁𝑃𝑅}={[{\text{𝑁𝑃𝑅}}_{ij}]}_{m\times n}$ . All the attributes are benefit, the $\text{𝑁𝑃𝑅}={[{\text{𝑁𝑃𝑅}}_{ij}]}_{m\times n}$ is same to $PR={[P{R}_{ij}]}_{m\times n}$ (see Table 2).

Table 2

The normalized TFNN information

	PG1	PG2
PX1	$\left\{\begin{array}{c}(0.49,0.64,0.75),\hfill \\ (0.58,0.79,0.93),\hfill \\ (0.48,0.64,0.88)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.58,0.64,0.82),\hfill \\ (0.56,0.92,0.81),\hfill \\ (0.28,0.48,0.79)\hfill \end{array}\right\}$
PX2	$\left\{\begin{array}{c}(0.64,0.81,0.72),\hfill \\ (0.49,0.55,0.67),\hfill \\ (0.38,0.44,0.59)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.55,0.67,0.72),\hfill \\ (0.37,0.48,0.59),\hfill \\ (0.28,0.34,0.46)\hfill \end{array}\right\}$
PX3	$\left\{\begin{array}{c}(0.62,0.75,0.94),\hfill \\ (0.47,0.52,0.62),\hfill \\ (0.58,0.66,0.71)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.47,0.51,0.84),\hfill \\ (0.24,0.55,0.66),\hfill \\ (0.29,0.42,0.72)\hfill \end{array}\right\}$
PX4	$\left\{\begin{array}{c}(0.53,0.67,0.79),\hfill \\ (0.32,0.55,0.62),\hfill \\ (0.45,0.67,0.84)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.42,0.66,0.98),\hfill \\ (0.52,0.66,0.89),\hfill \\ (0.42,0.56,0.68)\hfill \end{array}\right\}$
PX5	$\left\{\begin{array}{c}(0.52,0.68,0.82),\hfill \\ (0.63,0.71,0.81),\hfill \\ (0.43,0.56,0.72)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.32,0.51,0.75),\hfill \\ (0.38,0.45,0.58),\hfill \\ (0.29,0.47,0.66)\hfill \end{array}\right\}$
	PG3	PG4
PX1	$\left\{\begin{array}{c}(0.51,0.82,0.36),\hfill \\ (0.46,0.64,0.97),\hfill \\ (0.72,0.91,0.83)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.27,0.42,0.65),\hfill \\ (0.32,0.56,0.87),\hfill \\ (0.62,0.91,0.33)\hfill \end{array}\right\}$
PX2	$\left\{\begin{array}{c}(0.42,0.56,0.86),\hfill \\ (0.45,0.81,0.63),\hfill \\ (0.38,0.53,0.67)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.33,0.96,0.42),\hfill \\ (0.53,0.64,0.75),\hfill \\ (0.51,0.65,0.82)\hfill \end{array}\right\}$
PX3	$\left\{\begin{array}{c}(0.61,0.73,0.85),\hfill \\ (0.28,0.54,0.68),\hfill \\ (0.37,0.44,0.75)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.72,0.81,0.32),\hfill \\ (0.56,0.65,0.82),\hfill \\ (0.42,0.54,0.64)\hfill \end{array}\right\}$
PX4	$\left\{\begin{array}{c}(0.61,0.82,0.94),\hfill \\ (0.22,0.37,0.59),\hfill \\ (0.26,0.35,0.48)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.32,0.51,0.75),\hfill \\ (0.65,0.56,0.67),\hfill \\ (0.25,0.48,0.59)\hfill \end{array}\right\}$
PX5	$\left\{\begin{array}{c}(0.56,0.60,0.81),\hfill \\ (0.44,0.75,0.97),\hfill \\ (0.52,0.73,0.81)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.41,0.53,0.62),\hfill \\ (0.35,0.52,0.74),\hfill \\ (0.46,0.51,0.68)\hfill \end{array}\right\}$

Step 3. Compute the weight values through information entropy:

$P{W}_1=0.1606,P{W}_2=0.3118,P{W}_3=0.2897,P{W}_4=0.2379.$

Step 4. Obtain the TFNNPIS and TFNNNIS (see Table 3).

Table 3

The TFNNPIS and TFNNNIS

	PG1	PG2
TFNNPIS	$\left\{\begin{array}{c}(0.64,0.81,0.72),\hfill \\ (0.49,0.55,0.67),\hfill \\ (0.38,0.44,0.59)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.58,0.64,0.82),\hfill \\ (0.56,0.92,0.81),\hfill \\ (0.28,0.48,0.79)\hfill \end{array}\right\}$
TFNNNIS	$\left\{\begin{array}{c}(0.49,0.64,0.75),\hfill \\ (0.58,0.79,0.93),\hfill \\ (0.48,0.64,0.88)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.32,0.51,0.75),\hfill \\ (0.38,0.45,0.58),\hfill \\ (0.29,0.47,0.66)\hfill \end{array}\right\}$
	PG3	PG4
TFNNPIS	$\left\{\begin{array}{c}(0.61,0.73,0.85),\hfill \\ (0.28,0.54,0.68),\hfill \\ (0.37,0.44,0.75)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.72,0.81,0.32),\hfill \\ (0.56,0.65,0.82),\hfill \\ (0.42,0.54,0.64)\hfill \end{array}\right\}$
TFNNNIS	$\left\{\begin{array}{c}(0.42,0.56,0.86),\hfill \\ (0.45,0.81,0.63),\hfill \\ (0.38,0.53,0.67)\hfill \end{array}\right\}$	$\left\{\begin{array}{c}(0.27,0.42,0.65),\hfill \\ (0.32,0.56,0.87),\hfill \\ (0.62,0.91,0.33)\hfill \end{array}\right\}$

Step 5. Obtain the $\text{𝑇𝐹𝑁𝑁𝑃𝐼𝑆}({\xi }_{ij})$ and $\text{𝑇𝐹𝑁𝑁𝑁𝐼𝑆}({\xi }_{ij})$ (see Tables 4 and 5).

Table 4

The $\text{𝑇𝐹𝑁𝑁𝑃𝐼𝑆}({\xi }_{ij})$

Alternatives	PG1	PG2	PG3	PG4
PX1	0.6171	1.0000	0.4753	0.5612
PX2	1.0000	0.7702	0.5545	0.5483
PX3	0.5873	0.5608	0.5805	1.0000
PX4	0.7951	0.4332	1.0000	0.5867
PX5	0.6164	0.4919	0.6917	0.5842

Table 5

The $\text{𝑇𝐹𝑁𝑁𝑁𝐼𝑆}({\xi }_{ij})$

Alternatives	PG1	PG2	PG3	PG4
PX1	1.0000	0.4014	0.4767	1.0000
PX2	0.8917	0.5190	1.0000	0.9402
PX3	0.5042	0.4204	0.3839	0.4511
PX4	0.5887	0.5887	0.8078	0.4757
PX5	0.4901	1.0000	0.5515	0.5393

Step 6. Obtain the $\text{𝑇𝐹𝑁𝑁𝑃𝐼𝑆𝐺𝑅𝐷}({\xi }_i)$ and $\text{𝑇𝐹𝑁𝑁𝑁𝐼𝑆𝐺𝑅𝐷}({\xi }_i)$ (see Table 6).

Table 6

The $\text{𝑇𝐹𝑁𝑁𝑃𝐼𝑆𝐺𝑅𝐷}({\xi }_i)$ and $\text{𝑇𝐹𝑁𝑁𝑁𝐼𝑆𝐺𝑅𝐷}({\xi }_i)$

	$\text{𝑇𝐹𝑁𝑁𝑃𝐼𝑆𝐺𝑅𝐷}({\xi }_i)$	$\text{𝑇𝐹𝑁𝑁𝑁𝐼𝑆𝐺𝑅𝐷}({\xi }_i)$
PX1	0.6821	0.6617
PX2	0.6918	0.8184
PX3	0.6752	0.4306
PX4	0.6920	0.6253
PX5	0.5917	0.6786

Step 7. Obtain the $\text{𝑇𝐹𝑁𝑁𝑅𝑅𝐷}({\xi }_i)$ (see Table 7).

Table 7

The $\text{𝑇𝑃𝑁𝑁𝑅𝑅𝐷}({\xi }_i)$

	$\text{𝑇𝑃𝑁𝑁𝑅𝑅𝐷}({\xi }_i)$	Order
PX1	0.5076	3
PX2	0.4581	5
PX3	0.6106	1
PX4	0.5253	2
PX5	0.4658	4

Step 8. According to the $\text{𝑇𝑃𝑁𝑁𝑅𝑅𝐷}({\xi }_i)$ , the order is: $P{X}_3>P{X}_4>P{X}_1>P{X}_5>P{X}_2$ and $P{X}_3$ is the best engineering procurement construction project

4.2 Compare TFNNGRA with information operators

Then, the TFNN-GRA is compared with TFNNWA operator [35], TFNNWG operator [35], single-valued triangular Neutrosophic Dombi prioritized normalized Bonferroni mean (SVTNDPNBM) operator [59], triangular fuzzy neutrosophic number cross-entropy (TFNN-CE) method [60], TFNN-VIKOR method [38], TFNN-MABAC method [39] and TFN-EDAS method [61]. The order of these methods is given in Table 8.

Table 8

The order for different methods

	Order
TFNNWA operator [35]	$P{X}_3>P{X}_4>P{X}_1>P{X}_5>P{X}_2$
TFNNWG operator [35]	$P{X}_3>P{X}_5>P{X}_1>P{X}_4>P{X}_2$
SVTNDPNBM operator [59]	$P{X}_3>P{X}_5>P{X}_1>P{X}_4>P{X}_2$
TFNN-CE method [60]	$P{X}_3>P{X}_4>P{X}_1>P{X}_5>P{X}_2$
TFNN-VIKOR method [38]	$P{X}_3>P{X}_4>P{X}_1>P{X}_5>P{X}_2$
TFNN-MABAC method [39]	$P{X}_3>P{X}_4>P{X}_1>P{X}_5>P{X}_2$
TFN-EDAS method [61]	$P{X}_3>P{X}_4>P{X}_1>P{X}_5>P{X}_2$
TFNN-GRA method	$P{X}_3>P{X}_4>P{X}_1>P{X}_5>P{X}_2$

In accordance with WS coefficients [62, 63], the WS coefficient calculation between TFNNWA operator [35], TFNNWG operator [35], SVTNDPNBM operator [59], TFNN-CE method [60], TFNN-VIKOR method [38], TFNN-MABAC method [39], TFNEDAS method [61] and the proposed TFNN-GRA method is 1.0000, 0.9714, 0.9714 1.0000, 1.0000, 1.0000, 1.0000, respectively. From the above detailed analysis, it could be seen that these models have the same optimal engineering procurement construction project and worst engineering procurement construction project. This verifies the TFNN-GRA method is reasonable and effective. The main difference of between the built TFNN-GRA method and the existing TFNN-GRA method [52] are outlined: (1) the built TFNN-GRA method employed the objective weight information, while the existing TFNN-GRA method [52] employed the subjective weight information; (2) How to obtain the weight information: the built TFNN-GRA method employed the information entropy and score function to obtain the objective weight information, while the existing TFNN-GRA method [52] employed the subjective weight information without method for determining weight information; (3) the built TFNN-GRA method employed the Euclidean distance to obtain the grey rational coefficients and grey rational degree, while the existing TFNN-GRA method [52] employed the Hamming distance to obtain the grey rational coefficients and grey rational degree.

5. Conclusion

In the process of steady development, China’s construction industry has gradually become a pillar industry of the country. In order to better regulate the construction industry and enable it to enter the international market, the government is actively promoting the application and promotion of the general contracting model in the construction field. The GB 50358-2017 “Management Standards for General Contracting of Construction Projects”, which came into effect on January 1, 2018, specifies the standardized management of the EPC general contracting model and advocates, promotes, and encourages its implementation. Compared to the traditional contracting model, especially for projects with high implementation difficulty, short construction period, and large scale, the EPC model can better coordinate and achieve the three major goals of project quality, progress, and investment. Among them, project quality objectives are the top priority of the project and the lifeline that runs through the entire EPC project. The author believes that only from the perspective of contractors or construction industry workers can the quality of projects under the EPC mode be guaranteed, in order to exceed the needs and expectations of users and thereby achieve the social value of contractors. For engineering construction under the EPC mode, we should not simply separate the design, procurement, and construction of EPC projects, but should understand the integration and integration of the entire industry chain of engineering construction. As the domestic construction industry gradually integrates with the international market, the quality of engineering directly affects the success or failure of enterprise development. With the continuous improvement of quality control measures, engineering quality will inevitably be better controlled. The engineering procurement construction project quality evaluation is looked as the MADM. In this paper, the triangular fuzzy neutrosophic number GRA (TFNN-GRA) method is put up under triangular fuzzy neutrosophic sets (TFNSs) with completely unknown weight information. The information entropy is employed to obtain the weight values under TFNSs. Then, commenting GRA method with TFNSs, the TFNN-GRA is designed and the decision steps for MADM are constructed. Finally, a numerical example for engineering procurement construction project quality evaluation was given and some comparative analysis is employed to verify the advantages of TFNN-GRA method. In future research, we should strengthen the methods design of the above aspects, so that the evaluation system can be improved and perfected through different decision methods [64, 65, 66, 67, 68, 69]. At the same time, more and more methods for determining weights could be also studied in our future study works [70, 71, 72, 73, 74, 75].