An integrated fuzzy group decision-making model for construction enterprise contractor selection based on EDAS method and Information entropy

Abstract

With the rapid development of China’s construction industry, the competition in the construction industry is becoming increasingly fierce. Enterprises need to continuously improve their competitiveness in the market. Some non-core businesses can be outsourced to professional contractors. At present, contractors have more and more influence on the operation and development of enterprises. Whether it is the construction period or the quality of the project, it will have a greater impact on the operation of the construction project. In the environment of increasingly fierce market competition and increasing project quality requirements, for the construction project contracting enterprises, in order to achieve the goal of low cost and high quality, it is necessary to select the most suitable contractor on the basis of comprehensive consideration of multiple factors. The construction enterprise contractor selection is a classical multiple attribute group decision making (MAGDM) problem. In recent years, the MAGDM problem has become an important research field in modern decision science. This paper extends the EDAS method to the 2-tuple linguistic Pythagorean fuzzy sets (2TLPFSs). On the basis of the original EDAS method, 2-tuple linguistic Pythagorean fuzzy EDAS (2TLPF-EDAS) is built for MAGDM. Finally, a case study for construction enterprise contractor selection and some comparative analysis with the other methods show that the new method proposed in this paper is effective, reasonable and accurate.

Keywords

Multiple attribute group decision making (MAGDM)2-tuple linguistic Pythagorean fuzzy sets (2TLPFSs)EDAS method construction enterprise contractor selection

1 Introduction

Decision making is a process of action from a multitude of alternatives. Decision makers (DMs) must make decisions based on everything they know about certain situations, some of which they may not know [1 –5]. As the decision is often made in the choice of a variety of schemes, so the MAGDM has gradually entered the research scope of scholars [6 –9]. Then, Atanassov [10] proposed the intuitionistic fuzzy sets in 1986. Yager [11] proposed Pythagorean fuzzy sets (PHFSs) in 2013 and the methods related to the MAGDM have been fully applied to PHFSs environment, For example, EDAS method [12 –14], PRETHEE method [15], GRA method [16], MULTIMOORA method [17]. Furthermore, Deng, Wang, Wei and Lu [18] defined the 2-tuple linguistic Pythagorean fuzzy sets (2TLPFSs) based on Hamy Mean operators [19 –24]. Deng and Gao [25] proposed the TODIM method for MADM under 2TLPFSs. He, Zhang, Wei, Wang, Wu and Wei [26] proposed the CODAS method for MAGDM under 2TLPFSs. Deng, Wang, Wei and Wei [27] defined some Muirhead Mean (MM) operator [28 –35] under 2TLPFSs. Deng, Wei, Gao and Wang [36] connected the weighted BM (WBM) operator, the generalized WBM (GWBM) operator, and the dual GWBM operator under 2TLPFSs.

The general contracting mode of construction has been widely used in the international construction market, and it is also a key measure to transform the construction industry from extensive to intensive. At present, most domestic construction units mainly adopt the “subcontracting” and “outsourcing” mode in the construction process [37 –40]. However, with the rapid development of the economy, the construction contracting has become more and more perfect, and the general construction contracting has also achieved rapid development in the domestic market. Moreover, the general construction contracting mode is highly compatible with the idea of China’s economic transformation [41 –44]. Therefore, the rapid development of general construction contracting in the domestic construction market has become an inevitable trend. However, most of the existing domestic research focuses on the risks to be undertaken by the general contractor, while few researchers analyze how the construction unit should choose the general contractor to bring more comprehensive benefits to the construction unit [45 –48]. In recent years, the turnover of domestic construction enterprises has shown a trend of substantial growth. The average growth rate of turnover of enterprises with special and first-level qualifications has exceeded 30%, while the proportion of turnover completed by contractors has exceeded 70%. However, the general contractor’s dishonesty is a relatively common phenomenon, not only its economic interests will suffer serious losses, but also its social reputation will be adversely affected. It can be considered that the selection of the general contractor will have a decisive impact on the success of the project, which is particularly obvious for the general contract project [49 –52]. Therefore, the selection of the general contractor in line with its actual needs has a significant impact on the development of the enterprise. For the long-term and stable development of the national economy, the construction industry plays a very important role, is a very critical material production sector, and is also the basic industry for other industries to survive. Since China entered the World Trade Organization, the opening degree of the construction market has been deepening. Many large-scale foreign construction enterprises have entered the domestic market, and the number of large-scale projects invested by foreign investors in the domestic market has been increasing. In this case, the domestic construction general contractor needs to compete according to international management and international construction enterprises. Due to the influence of various factors, from the perspective of core competitiveness, there is a significant gap between domestic construction general contracting enterprises and developed countries such as Japan, Europe and the United States. If domestic construction general contracting enterprises want to achieve long-term and stable development and occupy a certain market share in the international market, they need to invest more resources to improve their core competitiveness [53 –55]. More importantly, the construction unit can reasonably select the general contractor, so as to achieve the goal of improving the project quality and comprehensive benefits of the project, so as to form a large-scale domestic enterprise with design, construction and other capabilities, and ultimately promote the development of the domestic construction general contractor team [56 –58]. Therefore, how to analyze the scientific and reasonable selection of the general contractor has become a problem that must be solved by the domestic engineering construction units in the current international market in a short time. With the method of reasonably selecting the general contractor, the ability of domestic general construction contractor can be continuously improved [59 –61].

The construction enterprise contractor selection is a classical MAGDM problem. This paper extends the EDAS method to the 2-tuple linguistic Pythagorean fuzzy sets (2TLPFSs). On the basis of the original EDAS method [12], 2-tuple linguistic Pythagorean fuzzy EDAS (2TLPF-EDAS) is built for MAGDM. Finally, a case study for construction enterprise contractor selection and some comparative analysis with the other methods show that the new method proposed in this paper is effective, reasonable and accurate. In order to realize the research purpose, the research sequence of this paper is shown. In the Sect. 2, we introduce some concepts and operators about 2TLPFSs. In the Sect. 3, on the basics of the 2TLPFSs, we propose the new method 2TLPF-EDAS for MAGDM. In the Sect. 4, by a numerical case about construction enterprise contractor selection is given. In Sect. 5, we get some conclusion about the new method.

2 Preliminary knowledge

Deng, Wang, Wei and Lu [18] defined the 2-tuple linguistic Pythagorean fuzzy sets (2TLPFSs).

2.1 2TLSs

Definition 1 [62, 63]. Let hλ ={ hλ₁, hλ₂, …, hλ_T } be a LTSs. The label hλ_i depicts the linguistic variable, and hλ is defined: $h λ = {\begin{matrix} h λ_{0} = extremely poor, h λ_{1} = very poor, h λ_{2} = poor, h λ_{3} = medium, \\ h λ_{4} = good, h λ_{5} = very good, h λ_{6} = extremely good . \end{matrix}}$ (1)

2.2 2TLPFSs

Definition 2[18]. Let hδ_j (j = 1, 2, …, n) be a 2TLSs. If hδ =〈 (hl_t, hξ) , (hl_f, hζ) 〉 is defined for hl_t, , hl_f ∈ hλ, hξ, hζ ∈ [- 0.5, 0.5) , (hl_t, hξ) , (hl_f, hζ) depict the truth-membership and falsity-membership through 2TLSs, then the 2TLPFSs are: $h δ_{j} = 〈 ({hl}_{t_{j}}, h ξ_{j}), ({hl}_{f_{j}}, h ζ_{j}) 〉$ (2) where 0 ⩽ Δ^-1 (hl_{t
_j}, hξ_j) ⩽ T, 0 ⩽ Δ^-1 (hl_{f
_j}, hζ_j) ⩽ T, and $0 ⩽ {(\frac{Δ^{- 1} ({hl}_{t_{j}}, h ξ_{j})}{T})}^{2} + {(\frac{Δ^{- 1} ({hl}_{f_{j}}, h ζ_{j})}{T})}^{2} ⩽ 1 .$

Definition 3[18]. Let hδ =〈 (hl_t, hξ) , (hl_f, hζ) 〉, the defined score function SF (hδ) and accuracy function AF (hδ) are: $SF (h δ) = Δ (\frac{T (1 + {(\frac{Δ^{- 1} ({hl}_{t_{j}}, h ξ_{j})}{T})}^{2} - {(\frac{Δ^{- 1} ({hl}_{f_{j}}, h ζ_{j})}{T})}^{2})}{2})$ (3) $AF (h δ) = Δ (T ({(\frac{Δ^{- 1} ({hl}_{t_{j}}, h ξ_{j})}{T})}^{2} + {(\frac{Δ^{- 1} ({hl}_{f_{j}}, h ζ_{j})}{T})}^{2}))$ (4)

Definition 4[18]. Let hδ₁ =〈 (hl_{t
₁}, hξ₁) , (hl_{f
₁}, hζ₁) 〉 and hδ₂ =〈 (hl_{t
₂}, hξ₂) , (hl_{f
₂}, hζ₂) 〉, then (1) If SF (hδ₁) < SF (hδ₂), hδ₁ < hδ₂; (2) If SF (hδ₁) = SF (hδ₂) , AF (hδ₁) < AF (hδ₂), hδ₁ < hδ₂; (3) If SF (hδ₁) = SF (hδ₂) , AF (hδ₁) = AF (hδ₂), hδ₁ = hδ₂.

Definition 5[18]. Let hδ₁ =〈 (hl_{t
₁}, hξ₁) , (hl_{f
₁}, hζ₁) 〉, hδ₂ =〈 (hl_{t
₂}, hξ₂) , (hl_{f
₂}, hζ₂) 〉, hδ =〈 (hl_t, hξ) , (hl_f, hζ) 〉, then $h δ_{1} \oplus h δ_{2} = {\begin{matrix} Δ (T \sqrt{1 - (1 - {(\frac{Δ^{- 1} ({hl}_{t_{1}}, h ξ_{1})}{T})}^{2}) (1 - {(\frac{Δ^{- 1} ({hl}_{t_{2}}, h ξ_{2})}{T})}^{2})}) \\ Δ (T (\frac{Δ^{- 1} ({hl}_{f_{1}}, h ζ_{1})}{T}) (\frac{Δ^{- 1} ({hl}_{f_{2}}, h ζ_{2})}{T})) \end{matrix}};$ (5)

$h δ_{1} \otimes h δ_{2} = {\begin{matrix} Δ (T (\frac{Δ^{- 1} ({hl}_{t_{1}}, h ξ_{1})}{T}) (\frac{Δ^{- 1} ({hl}_{t_{2}}, h ξ_{2})}{T})) \\ Δ (T \sqrt{1 - (1 - {(\frac{Δ^{- 1} ({hl}_{f_{1}}, h ζ_{1})}{T})}^{2}) (1 - {(\frac{Δ^{- 1} ({hl}_{f_{2}}, h ζ_{2})}{T})}^{2})}) \end{matrix}}$ (6) $α h δ = {Δ (T \sqrt{1 - {(1 - {(\frac{Δ^{- 1} ({hl}_{t}, h ξ)}{T})}^{2})}^{a}}) Δ (T {(\frac{Δ^{- 1} ({hl}_{f}, h ζ)}{T})}^{a})}, α > 0;$ (7) $h δ^{α} = {Δ (T {(\frac{Δ^{- 1} ({hl}_{t}, h ξ)}{T})}^{a}), Δ (T \sqrt{1 - {(1 - {(\frac{Δ^{- 1} ({hl}_{f}, h ζ)}{T})}^{2})}^{a}})}, α > 0 .$ (8)

Definition 6 [25]. If hδ₁ =〈 (hl_{t
₁}, hξ₁) , (hl_{f
₁}, hζ₁) 〉 and hδ₂ =〈 (hl_{t
₂}, hξ₂) , (hl_{f
₂}, hζ₂) 〉 are two 2TLPFNs, the distance between hδ₁ =〈 (hl_{t
₁}, hξ₁) , (hl_{f
₁}, hζ₁) 〉 and hδ₂ =〈 (hl_{t
₂}, hξ₂) , (hl_{f
₂}, hζ₂) 〉 can be computed by the Hamming distance as follows. $d (h δ, h δ_{2}) = \frac{1}{2 T} (| Δ^{- 1} ({hl}_{t_{1}}, h ξ_{1}) - Δ^{- 1} ({hl}_{f_{1}}, h ζ_{1}) |)$ (9)

2.3 The 2TLPFWA and 2TLPFWG operators

Definition 7[64]. Let hδ_j ={ (hs_{t
_j}, hα_j) , (hs_{f
_j}, hχ_j) }, the 2TLPFWA operator is: $\begin{matrix} 2 TLPFWA (h δ_{1}, h δ_{2}, \dots, h δ_{n}) \\ = w_{1} (h δ_{1}) \oplus w_{2} (h δ_{2}) \oplus \dots \oplus w_{n} (h δ_{n}) = \oplus_{j = 1}^{n} (w_{j} h δ_{j}) \\ = {Δ (T \sqrt{1 - \prod_{j = 1}^{n} {(1 - {(\frac{Δ^{- 1} ({hs}_{t_{j}}, h α_{j})}{T})}^{2})}^{w_{j}}}) Δ (T \prod_{j = 1}^{n} {(\frac{Δ^{- 1} ({hs}_{f_{j}}, h χ_{j})}{T})}^{w_{j}})} \end{matrix}$ (10)

$\begin{matrix} 2 TLPFWG (h δ_{1}, h δ_{2}, \dots, h δ_{n}) \\ = {(h δ_{1})}^{w_{1}} \otimes {(h δ_{2})}^{w_{2}} \otimes \dots \otimes {(h δ_{n})}^{w_{n}} = \otimes_{j = 1}^{n} {(h δ_{j})}^{w_{j}} \\ = {Δ (T \prod_{j = 1}^{n} {(\frac{Δ^{- 1} ({hs}_{t_{j}}, h α_{j})}{T})}^{w_{j}}), Δ (T \sqrt{1 - \prod_{j = 1}^{n} {(1 - {(\frac{Δ^{- 1} ({hs}_{f_{j}}, h χ_{j})}{T})}^{2})}^{w_{j}}})} \end{matrix}$ (11) where w_j is weighting of hδ_j, j = 1, 2, …, n which meets $0 ⩽ w_{j} ⩽ 1, \sum_{j = 1}^{n} w_{j} = 1 .$

3 EDAS method for MAGDM with 2TLPFSs

Suppose that {HA₁, HA₂, … , HA_m } represents alternatives, {HO₁, HO₂, … , HO_n } represents attributes, {HD₁, HD₂, … , HD_t } represents the experts. Let call (wμ₁, wμ₂, ⋯ , wμ_n) for the weight of attributes and {dw₁, dw₂, ⋯ , dw_t } for the weight of experts. Then, build an evaluation matrix $HR = {[h φ_{ij}^{t}]}_{m \times n} = {({hs}_{t_{ij}}^{t}, h α_{ij}^{t}), ({hs}_{f_{ij}}^{t}, h χ_{ij}^{t})}_{m \times n}$ . It need satisfy some conditions which are $0 ⩽ Δ^{- 1} ({hs}_{t_{ij}}^{t}, h α_{ij}^{t}) ⩽ T$ , $0 ⩽ Δ^{- 1} ({hs}_{f_{ij}}^{t}, h χ_{ij}^{t}) ⩽ T$ and $0 ⩽ {(Δ^{- 1} ({hs}_{t_{ij}}^{t}, h α_{ij}^{t}))}^{2} + {(Δ^{- 1} ({hs}_{f_{ij}}^{t}, h χ_{ij}^{t}))}^{2} ⩽ T^{2}$ . Besides, $({hs}_{t_{ij}}^{t}, h α_{ij}^{t})$ represents the degree of the truth membership and $({hs}_{f_{ij}}^{t}, h χ_{ij}^{t})$ represents the falsity-membership. Based on the 2TLPF-EDAS, the calculating steps for MAGDM are listed below.

Step 1. The entropy weight method is an objective weight method because it only relies on the discreteness of the data itself. Therefore, this paper adopts the entropy weight method when calculating the attribute weight.

Firstly, we normalize decision matrix. ${SFHR}_{ij} = \frac{SF {({hs}_{t_{ij}}^{t}, h α_{ij}^{t}), ({hs}_{f_{ij}}^{t}, h χ_{ij}^{t})}}{\sum_{i = 1}^{m} SF {({hs}_{t_{ij}}^{t}, h α_{ij}^{t}), ({hs}_{f_{ij}}^{t}, h χ_{ij}^{t})}}$ (12)

Then, compute the degree of entropy. ${SFE}_{j} = - \frac{1}{ln m} \sum_{i = 1}^{m} {SFHR}_{ij} ln {SFHR}_{ij}$ (13) where, if SFHR_ij = 0, define ln SFHR_ij = 0.

Finally, we can get the entropy weight. $w μ_{j} = \frac{(1 - {SFE}_{j})}{\sum_{j = 1}^{n} (1 - {SFE}_{j})}$ (14)

Step 2. Normalize the evaluation matrix. For benefit attribute: ${[nh φ_{ij}^{t}]}_{m \times n} = {[h φ_{ij}^{t}]}_{m \times n} = {({hs}_{t_{ij}}^{t}, h α_{ij}^{t}), ({hs}_{f_{ij}}^{t}, h χ_{ij}^{t})}_{m \times n}$ . For cost attribute: ${[nh φ_{ij}^{t}]}_{m \times n} = {({hs}_{f_{ij}}^{t}, h χ_{ij}^{t}), ({hs}_{t_{ij}}^{t}, h α_{ij}^{t})}_{m \times n}$ . As a result, the evaluation matrix shifts from ${[h φ_{ij}^{t}]}_{m \times n} = {({hs}_{t_{ij}}^{t}, h α_{ij}^{t}), ({hs}_{f_{ij}}^{t}, h χ_{ij}^{t})}_{m \times n}$ to ${[nh φ_{ij}^{t}]}_{m \times n} = {({nhs}_{t_{ij}}^{t}, nh α_{ij}^{t}), ({nhs}_{f_{ij}}^{t}, nh χ_{ij}^{t})}_{m \times n}$ .

Step 3. According to the experts’ weighting vector, we can compute the overall matrix $NHR = {[nh φ_{ij}]}_{m \times n}$ by 2TLPFWA operator. $\begin{matrix} nh φ_{ij} = {({nhs}_{t_{ij}}, nh α_{ij}), ({nhs}_{f_{ij}}, nh χ_{ij})} \\ = 2 TLPFWA (h δ_{ij}^{1}, h δ_{ij}^{2}, \dots, h δ_{ij}^{t}) \\ = {dw}_{1} (h δ_{ij}^{1}) \oplus {dw}_{2} (h δ_{ij}^{2}) \oplus \dots \oplus {dw}_{t} (h δ_{ij}^{t}) = \oplus_{d = 1}^{t} ({dw}_{d} h φ_{ij}^{d}) \\ = {\begin{matrix} Δ (T \sqrt{1 - \prod_{d = 1}^{t} {(1 - {(\frac{Δ^{- 1} ({nhs}_{t_{ij}}^{t}, nh α_{ij}^{t})}{T})}^{2})}^{{dw}_{d}}}) \\ Δ (T \prod_{d = 1}^{t} {(\frac{Δ^{- 1} ({nhs}_{f_{ij}}^{t}, nh χ_{ij}^{t})}{T})}^{{dw}_{d}}) \end{matrix}} \end{matrix}$ (15)

Step 4. On the basics of attributes, calculate the average solution of each attribute. $\begin{matrix} {AV}_{j} = {({ahs}_{t_{ij}}, ah α_{ij}), ({ahs}_{f_{ij}}, ah χ_{ij})} \\ = \frac{\oplus_{i = 1}^{m} {({nhs}_{t_{ij}}, nh α_{ij}), ({nhs}_{f_{ij}}, nh χ_{ij})}}{m}, j = 1, \dots, m \end{matrix}$ (16)

According to the Definition 5, $\begin{matrix} \oplus_{i = 1}^{m} {({nhs}_{t_{ij}}, nh α_{ij}), ({nhs}_{f_{ij}}, nh χ_{ij})} \\ = {Δ (T \sqrt{1 - \prod_{i = 1}^{m} (1 - {(\frac{Δ^{- 1} ({nhs}_{t_{ij}}, nh α_{ij})}{T})}^{2})}) Δ (T \prod_{i = 1}^{m} (\frac{Δ^{- 1} ({nhs}_{f_{ij}}, nh χ_{ij})}{T}))} \end{matrix}$ (17)

$\begin{matrix} \frac{1}{m} \oplus_{i = 1}^{m} {({nhs}_{t_{ij}}, nh α_{ij}), ({nhs}_{f_{ij}}, nh χ_{ij})} \\ = {Δ (T \sqrt{1 - \prod_{i = 1}^{m} {(1 - {(\frac{Δ^{- 1} ({nhs}_{t_{ij}}, nh α_{ij})}{T})}^{2})}^{\frac{1}{m}}}) Δ (T \prod_{i = 1}^{m} {(\frac{Δ^{- 1} ({nhs}_{f_{ij}}, nh χ_{ij})}{T})}^{\frac{1}{m}})} \end{matrix}$ (18)

Step 5. Calculate the PDA and NDA under 2TLPFNs.

For the positive attributes, $2 {TLPFNPDA}_{ij} = \frac{max (0, d (\begin{matrix} {({nhs}_{t_{ij}}, nh α_{ij}), ({nhs}_{f_{ij}}, nh χ_{ij})}, \\ {({ahs}_{t_{ij}}, ah α_{ij}), ({ahs}_{f_{ij}}, ah χ_{ij})} \end{matrix}))}{SF {({ahs}_{t_{ij}}, ah α_{ij}), ({ahs}_{f_{ij}}, ah χ_{ij})}} .$ (19)

$2 {TLPFNNDA}_{ij} = \frac{max (0, - d (\begin{matrix} {({nhs}_{t_{ij}}, nh α_{ij}), ({nhs}_{f_{ij}}, nh χ_{ij})}, \\ {({ahs}_{t_{ij}}, ah α_{ij}), ({ahs}_{f_{ij}}, ah χ_{ij})} \end{matrix}))}{SF {({ahs}_{t_{ij}}, ah α_{ij}), ({ahs}_{f_{ij}}, ah χ_{ij})}},$ (20)

For the negative attributes, $2 {TLPFNPDA}_{ij} = \frac{max (0, - d (\begin{matrix} {({nhs}_{t_{ij}}, nh α_{ij}), ({nhs}_{f_{ij}}, nh χ_{ij})}, \\ {({ahs}_{t_{ij}}, ah α_{ij}), ({ahs}_{f_{ij}}, ah χ_{ij})} \end{matrix}))}{SF {({ahs}_{t_{ij}}, ah α_{ij}), ({ahs}_{f_{ij}}, ah χ_{ij})}},$ (21)

$2 {TLPFNNDA}_{ij} = \frac{max (0, d (\begin{matrix} {({nhs}_{t_{ij}}, nh α_{ij}), ({nhs}_{f_{ij}}, nh χ_{ij})}, \\ {({ahs}_{t_{ij}}, ah α_{ij}), ({ahs}_{f_{ij}}, ah χ_{ij})} \end{matrix}))}{SF {({ahs}_{t_{ij}}, ah α_{ij}), ({ahs}_{f_{ij}}, ah χ_{ij})}},$ (22)

Step 6. Based on the weight of the attributes, compute the weighted 2TLPFNPDA and 2TLPFNNDA. $2 {TLPFNSP}_{i} = \sum_{j = 1}^{n} 2 {TLPFNPDA}_{ij} \cdot w μ_{j}, i = 1, 2, \dots, m$ (23)

$2 {TLPFNSN}_{i} = \sum_{j = 1}^{n} 2 {TLPFNNDA}_{ij} \cdot w μ_{j}, i = 1, 2, \dots, m$ (24)

Step 7. Compute the value of the weighted normalized 2TLPFNPDA and 2TLPFNNDA. $2 {TLPFNNSP}_{i} = \frac{2 {TLPFNSP}_{i}}{max_{i} (2 {TLPFNSP}_{i})}, i = 1, 2, \dots, m$ (25) $2 {TLPFNNSN}_{i} = 1 - \frac{2 {TLPFNSN}_{i}}{max_{i} (2 {TLPFNSN}_{i})}, i = 1, 2, \dots, m$ (26)

Step 8. On the basics of $2 {TLPFNNSP}_{i} = \frac{2 {TLPFNSP}_{i}}{max_{i} (2 {TLPFNSP}_{i})}$ and $2 {TLPFNNSN}_{i} = 1 - \frac{2 {TLPFNSN}_{i}}{max_{i} (2 {TLPFNSN}_{i})}$ , we can compute the scores of 2TLPFNAS_i. $2 {TLPFNAS}_{i} = \frac{1}{2} (2 {TLPFNNSP}_{i} + 2 {TLPFNNSN}_{i}), i = 1, 2, \dots, m$ (27)

Step 9. Based on the values of 2TLPFNAS_i, rank the final scores of alternatives, and choose the best corresponding alternative.

4 Numerical example and comparative analysis

4.1 Numerical example

The construction industry plays an important role in promoting the development of the national economy, is an important material production sector of the national economy, and is also the basic leading industry on which all industries rely for development. In today’s changing environment, “outsourcing” has become a strategic adjustment tool that has become popular around the world in recent years, that is, to delegate some important but non-core business functions to senior contractors outside the enterprise, and to focus the knowledge and resources inside the enterprise on those core businesses with competitive advantages. Since 2001, the turnover of Chinese construction enterprises has increased rapidly, especially the turnover of enterprises with special and first-level qualifications has increased by more than 30% on average, and more than 70% of the turnover is achieved through subcontractors. However, the contractor’s dishonesty has occurred from time to time, causing great losses to its social reputation and economic interests. It can be said that the selection of contractors determines the success or failure of a project, so the reasonable selection of contractors is crucial to the development of enterprises. When faced with multiple potential contractors, how to quickly and effectively select the appropriate project contractors among these contractors has become an important problem to be solved at this stage. On the basis of the comprehensive evaluation index system of contractors in construction projects, the contractors are investigated and relevant information about contractors is collected, including all aspects such as past project quality and industry reputation. On this basis, the experts of the evaluation team use the existing evaluation index system and evaluation method to comprehensively evaluate the contractors. The selection process of contractors can be further subdivided into two stages: rough selection and fine selection. In the rough selection stage, contractors are screened according to the requirements of the project, and their quantity is controlled within the required range. In the fine selection stage, the rough selected contractors will be evaluated by common evaluation methods and the most suitable contractors will be selected. The construction enterprise contractor selection is a classical MAGDM problem. In this section, we provide a numerical example for MHE evaluation of college students by using 2TLPF-EDAS and PT method. Assume that five possible construction enterprise contractors HA_i (i = 1, 2, 3, 4, 5) to be selected and four attributes HO_j (j = 1, 2, 3, 4) to assess the construction enterprise contractors: contractor’s engineering management cost, contractor’s social influence, contractor’s market development and marketing ability, contractor’s financing and financial ability. And, the first attribute is the cost index. In addition, three experts conduct whether the company selects a new construction enterprise contractor selection by 2TLPFNs. The weight of experts is dw = (0.27, 0.42, 0.31) ^T. Specific decision matrixes are depicted in Tables 1 –3.

Table 1
The evaluation matrix HD¹

HO ₁ HO ₂ HO ₃ HO ₄

HA ₁ { (hλ₄, 0) , (hλ₃, 0)} { (hλ₂, 0) , (hλ₄, 0)} { (hλ₂, 0) , (hλ₄, 0)} { (hλ₁, 0) , (hλ₂, 0)}

HA ₂ { (hλ₃, 0) , (hλ₁, 0)} { (hλ₃, 0) , (hλ₂, 0)} { (hλ₄, 0) , (hλ₃, 0)} { (hλ₂, 0) , (hλ₄, 0)}

HA ₃ { (hλ₂, 0) , (hλ₁, 0)} { (hλ₃, 0) , (hλ₄, 0)} { (hλ₁, 0) , (hλ₃, 0)} { (hλ₄, 0) , (hλ₃, 0)}

HA ₄ { (hλ₁, 0) , (hλ₃, 0)} { (hλ₄, 0) , (hλ₂, 0)} { (hλ₃, 0) , (hλ₁, 0)} { (hλ₂, 0) , (hλ₄, 0)}

HA ₅ { (hλ₂, 0) , (hλ₄, 0)} { (hλ₃, 0) , (hλ₁, 0)} { (hλ₄, 0) , (hλ₂, 0)} { (hλ₁, 0) , (hλ₃, 0)}

	HO ₁	HO ₂	HO ₃	HO ₄
HA ₁	{ (hλ₄, 0) , (hλ₃, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}	{ (hλ₁, 0) , (hλ₂, 0)}
HA ₂	{ (hλ₃, 0) , (hλ₁, 0)}	{ (hλ₃, 0) , (hλ₂, 0)}	{ (hλ₄, 0) , (hλ₃, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}
HA ₃	{ (hλ₂, 0) , (hλ₁, 0)}	{ (hλ₃, 0) , (hλ₄, 0)}	{ (hλ₁, 0) , (hλ₃, 0)}	{ (hλ₄, 0) , (hλ₃, 0)}
HA ₄	{ (hλ₁, 0) , (hλ₃, 0)}	{ (hλ₄, 0) , (hλ₂, 0)}	{ (hλ₃, 0) , (hλ₁, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}
HA ₅	{ (hλ₂, 0) , (hλ₄, 0)}	{ (hλ₃, 0) , (hλ₁, 0)}	{ (hλ₄, 0) , (hλ₂, 0)}	{ (hλ₁, 0) , (hλ₃, 0)}

Table 2

The evaluation matrix HD²

	HO ₁	HO ₂	HO ₃	HO ₄
HA ₁	{ (hλ₃, 0) , (hλ₁, 0)}	{ (hλ₄, 0) , (hλ₃, 0)}	{ (hλ₁, 0) , (hλ₂, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}
HA ₂	{ (hλ₄, 0) , (hλ₂, 0)}	{ (hλ₁, 0) , (hλ₂, 0)}	{ (hλ₃, 0) , (hλ₁, 0)}	{ (hλ₃, 0) , (hλ₂, 0)}
HA ₃	{ (hλ₂, 0) , (hλ₃, 0)}	{ (hλ₃, 0) , (hλ₄, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}	{ (hλ₄, 0) , (hλ₁, 0)}
HA ₄	{ (hλ₂, 0) , (hλ₄, 0)}	{ (hλ₂, 0) , (hλ₃, 0)}	{ (hλ₄, 0) , (hλ₃, 0)}	{ (hλ₁, 0) , (hλ₃, 0)}
HA ₅	{ (hλ₂, 0) , (hλ₁, 0)}	{ (hλ₄, 0) , (hλ₂, 0)}	{ (hλ₃, 0) , (hλ₁, 0)}	{ (hλ₃, 0) , (hλ₄, 0)}

Table 3

The evaluation matrix HD³

	HO ₁	HO ₂	HO ₃	HO ₄
HA ₁	{ (hλ₂, 0) , (hλ₁, 0)}	{ (hλ₃, 0) , (hλ₄, 0)}	{ (hλ₄, 0) , (hλ₂, 0)}	{ (hλ₃, 0) , (hλ₂, 0)}
HA ₂	{ (hλ₃, 0) , (hλ₄, 0)}	{ (hλ₁, 0) , (hλ₃, 0)}	{ (hλ₂, 0) , (hλ₃, 0)}	{ (hλ₁, 0) , (hλ₂, 0)}
HA ₃	{ (hλ₂, 0) , (hλ₁, 0)}	{ (hλ₄, 0) , (hλ₁, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}	{ (hλ₃, 0) , (hλ₄, 0)}
HA ₄	{ (hλ₃, 0) , (hλ₄, 0)}	{ (hλ₂, 0) , (hλ₁, 0)}	{ (hλ₁, 0) , (hλ₂, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}
HA ₅	{ (hλ₂, 0) , (hλ₁, 0)}	{ (hλ₄, 0) , (hλ₃, 0)}	{ (hλ₂, 0) , (hλ₃, 0)}	{ (hλ₄, 0) , (hλ₁, 0)}

The 2TLPF-EDAS method is used to solve the construction enterprise contractor selection.

Step 1. Calculate the attributes’ weight by entropy weight method in Table 4.

Table 4

The attributes weight

	HO ₁	HO ₂	HO ₃	HO ₄
Weight	0.3114	0.2491	0.2148	0.2247

Step 2. The normalized decision matrix is derived in Tables 5 –7.

Table 5

The evaluation matrix HD¹

	HO ₁	HO ₂	HO ₃	HO ₄
HA ₁	{ (hλ₃, 0) , (hλ₄, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}	{ (hλ₁, 0) , (hλ₂, 0)}
HA ₂	{ (hλ₁, 0) , (hλ₃, 0)}	{ (hλ₃, 0) , (hλ₂, 0)}	{ (hλ₄, 0) , (hλ₃, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}
HA ₃	{ (hλ₁, 0) , (hλ₂, 0)}	{ (hλ₃, 0) , (hλ₄, 0)}	{ (hλ₁, 0) , (hλ₃, 0)}	{ (hλ₄, 0) , (hλ₃, 0)}
HA ₄	{ (hλ₃, 0) , (hλ₁, 0)}	{ (hλ₄, 0) , (hλ₂, 0)}	{ (hλ₃, 0) , (hλ₁, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}
HA ₅	{ (hλ₄, 0) , (hλ₂, 0)}	{ (hλ₃, 0) , (hλ₁, 0)}	{ (hλ₄, 0) , (hλ₂, 0)}	{ (hλ₁, 0) , (hλ₃, 0)}

Table 6

The evaluation matrix HD²

	HO ₁	HO ₂	HO ₃	HO ₄
HA ₁	{ (hλ₁, 0) , (hλ₃, 0)}	{ (hλ₄, 0) , (hλ₃, 0)}	{ (hλ₁, 0) , (hλ₂, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}
HA ₂	{ (hλ₂, 0) , (hλ₄, 0)}	{ (hλ₁, 0) , (hλ₂, 0)}	{ (hλ₃, 0) , (hλ₁, 0)}	{ (hλ₃, 0) , (hλ₂, 0)}
HA ₃	{ (hλ₃, 0) , (hλ₂, 0)}	{ (hλ₃, 0) , (hλ₄, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}	{ (hλ₄, 0) , (hλ₁, 0)}
HA ₄	{ (hλ₄, 0) , (hλ₂, 0)}	{ (hλ₂, 0) , (hλ₃, 0)}	{ (hλ₄, 0) , (hλ₃, 0)}	{ (hλ₁, 0) , (hλ₃, 0)}
HA ₅	{ (hλ₁, 0) , (hλ₂, 0)}	{ (hλ₄, 0) , (hλ₂, 0)}	{ (hλ₃, 0) , (hλ₁, 0)}	{ (hλ₃, 0) , (hλ₄, 0)}

Table 7

The evaluation matrix HD³

	HO ₁	HO ₂	HO ₃	HO ₄
HA ₁	{ (hλ₁, 0) , (hλ₂, 0)}	{ (hλ₃, 0) , (hλ₄, 0)}	{ (hλ₄, 0) , (hλ₂, 0)}	{ (hλ₃, 0) , (hλ₂, 0)}
HA ₂	{ (hλ₄, 0) , (hλ₃, 0)}	{ (hλ₁, 0) , (hλ₃, 0)}	{ (hλ₂, 0) , (hλ₃, 0)}	{ (hλ₁, 0) , (hλ₂, 0)}
HA ₃	{ (hλ₁, 0) , (hλ₂, 0)}	{ (hλ₄, 0) , (hλ₁, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}	{ (hλ₃, 0) , (hλ₄, 0)}
HA ₄	{ (hλ₄, 0) , (hλ₃, 0)}	{ (hλ₂, 0) , (hλ₁, 0)}	{ (hλ₁, 0) , (hλ₂, 0)}	{ (hλ₂, 0) , (hλ₄, 0)}
HA ₅	{ (hλ₁, 0) , (hλ₂, 0)}	{ (hλ₄, 0) , (hλ₃, 0)}	{ (hλ₂, 0) , (hλ₃, 0)}	{ (hλ₄, 0) , (hλ₁, 0)}

Step 3. The overall evaluation matrix is derived by 2TLPFWA operator (See Table 8).

Table 8

The overall evaluation matrix

	HO ₁	HO ₂	HO ₃	HO ₄
HA ₁	${\begin{matrix} (h λ_{3}, 0.2122), \\ (h λ_{2}, 0.3126) \end{matrix}}$	${\begin{matrix} (h λ_{3}, 0.2026), \\ (h λ_{1}, 0.3128) \end{matrix}}$	${\begin{matrix} (h λ_{2}, 0.4723), \\ (h λ_{2}, 0.3005) \end{matrix}}$	${\begin{matrix} (h λ_{3}, 0.15468), \\ (h λ_{2}, 0.4016) \end{matrix}}$
HA ₂	${\begin{matrix} (h λ_{2}, 0.3217), \\ (h λ_{2}, 0.2012) \end{matrix}}$	${\begin{matrix} (h λ_{1}, 0.3613), \\ (h λ_{2}, 0.4156) \end{matrix}}$	${\begin{matrix} (h λ_{3}, 0.4226), \\ (h λ_{2}, 0.4204) \end{matrix}}$	${\begin{matrix} (h λ_{2}, 0.1248), \\ (h λ_{1}, 0.4017) \end{matrix}}$
HA ₃	${\begin{matrix} (h λ_{2}, 0.4002), \\ (h λ_{1}, 0.2141) \end{matrix}}$	${\begin{matrix} (h λ_{3}, 0.3253), \\ (h λ_{1}, 0.4209) \end{matrix}}$	${\begin{matrix} (h λ_{2}, 0.1216), \\ (h λ_{1}, 0.2139) \end{matrix}}$	${\begin{matrix} (h λ_{3}, 0.3369), \\ (h λ_{2}, 0.5235) \end{matrix}}$
HA ₄	${\begin{matrix} (h λ_{1}, 0.2189), \\ (h λ_{2}, 0.0326) \end{matrix}}$	${\begin{matrix} (h λ_{2}, 0.3582), \\ (h λ_{2}, 0.3036) \end{matrix}}$	${\begin{matrix} (h λ_{1}, 0.4071), \\ (h λ_{1}, 0.4387) \end{matrix}}$	${\begin{matrix} (h λ_{3}, 0.3481), \\ (h λ_{2}, 0.4286) \end{matrix}}$
HA ₅	${\begin{matrix} (h λ_{2}, 0.1049), \\ (h λ_{2}, 0.3482) \end{matrix}}$	${\begin{matrix} (h λ_{2}, 0.1203), \\ (h λ_{1}, 0.4827) \end{matrix}}$	${\begin{matrix} (h λ_{3}, 0.3546), \\ (h λ_{3}, 0.0231) \end{matrix}}$	${\begin{matrix} (h λ_{1}, 0.4073), \\ (h λ_{1}, 0.4409) \end{matrix}}$

Step 4. Compute the average solution (See Table 9).

Table 9

The average solution

	The average solution
HO ₁	{ (hλ₂, 0.7542) , (hλ₃, 0.2606)}
HO ₂	{ (hλ₂, 0.7203) , (hλ₁, 0.8324)}
HO ₃	{ (hλ₂, 0.6745) , (hλ₂, 0.3462)}
HO ₄	{ (hλ₃, 0.1276) , (hλ₂, 0.3623)}

Step 5. The 2TLPFPDA and 2TLPFNDA are derived in Tables 10, 11.

Table 10

The 2TLPFPDA

	HO ₁	HO ₂	HO ₃	HO ₄
HA ₁	0.0000	0.0000	0.0000	0.0000
HA ₂	0.0000	0.0000	0.0000	0.0000
HA ₃	0.0494	0.0367	0.0000	0.0000
HA ₄	0.0000	0.0000	0.0000	0.0000
HA ₅	0.0000	0.0453	0.0000	0.0000

Table 11

The 2TLPFNDA

	HO ₁	HO ₂	HO ₃	HO ₄
HA ₁	0.0783	0.0572	0.0852	0.0795
HA ₂	0.0846	0.0615	0.0712	0.0996
HA ₃	0.0000	0.0000	0.0226	0.0329
HA ₄	0.0356	0.0978	0.0735	0.0812
HA ₅	0.0528	0.0000	0.2256	0.0672

Step 6. Calculate the 2TLPFNSP and 2TLPFNSN (See Table 12).

Table 12

The values of 2TLPFNSP and 2TLPFNSN

	2TLPFNSP	2TLPFNSN
HA ₁	0.0000	0.0748
HA ₂	0.0000	0.0793
HA ₃	0.0245	0.0122
HA ₄	0.0000	0.0695
HA ₅	0.0113	0.0800

Step 7. Calculate 2TLPFNNSP and 2TLPFNNSP in Table 13.

Table 13

The values of 2TLPFNNSP and 2TLPFNNSP

	2TLPFNNSP	2TLPFNNSP
HA ₁	0.0000	0.0651
HA ₂	0.0000	0.0083
HA ₃	1.0000	0.8469
HA ₄	0.0000	0.1315
HA ₅	0.4601	0.0000

Step 8. Calculate the 2TLPFNAS (See Table 14).

Table 14

The 2TLPFNAS

	2TLPFNAS	Order
HA ₁	0.0325	4
HA ₂	0.0041	5
HA ₃	0.9235	1
HA ₄	0.0657	3
HA ₅	0.2301	2

Step 9. Relying on the 2TLPFNAS, the order of all chosen alternatives is: HA₃ > HA₅ > HA₄ > HA₁ > HA₂ and HA₃ is the best construction enterprise contractor.

4.2 comparative analysis

In this part, the 2TLPF-EDAS method is compared with 2TLPFWA operator [18], 2TLPFWG operator [18], 2TLPFWBM operator [36] and 2TLPFWGBM operator [36]. The decision results are listed in Table 15.

Table 15
Evaluation results of these different methods

Methods Ranking order The optimal alternative

2TLPFWA operator [18] HA₃ > HA₅ > HA₄ > HA₁ > HA₂ HA ₃

2TLPFWG operator [18] HA₃ > HA₅ > HA₄ > HA₁ > HA₂ HA ₃

2TLPFWBM operator [36] HA₃ > HA₅ > HA₄ > HA₁ > HA₂ HA ₃

2TLPFWGBM operator [36] HA₃ > HA₅ > HA₄ > HA₁ > HA₂ HA ₃

2TLPF-EDAS method HA₃ > HA₅ > HA₄ > HA₁ > HA₂ HA ₃

Methods	Ranking order	The optimal alternative
2TLPFWA operator [18]	HA₃ > HA₅ > HA₄ > HA₁ > HA₂	HA ₃
2TLPFWG operator [18]	HA₃ > HA₅ > HA₄ > HA₁ > HA₂	HA ₃
2TLPFWBM operator [36]	HA₃ > HA₅ > HA₄ > HA₁ > HA₂	HA ₃
2TLPFWGBM operator [36]	HA₃ > HA₅ > HA₄ > HA₁ > HA₂	HA ₃
2TLPF-EDAS method	HA₃ > HA₅ > HA₄ > HA₁ > HA₂	HA ₃

From the above detailed analysis in Table 15, it could be seen that these models have the same decision order. This verifies the 2TLPF-EDAS method is reasonable and effective.

5 Conclusion

According to the requirements of the national “going out” strategy, China’s construction contractors have made remarkable achievements after years of active overseas expansion. In recent years, both the number of enterprises participating in the international market bidding and the amount of contracts signed have increased rapidly, the business scale has continued to expand, the technical content has continued to improve, the international competitiveness has significantly improved, and a number of excellent multinational enterprises have emerged. In the changing international competitive market environment, opportunities and challenges coexist. International engineering contractors must be more cautious in expanding their overseas businesses, see the situation clearly, and strive to make themselves more stable in the international market. However, it is easy to “go out” and difficult to “stand firm”. How to maintain and cultivate its unique competitive advantage is particularly important. The construction enterprise contractor selection is a classical MAGDM problem. In this paper extends the EDAS method to the 2TLPFSs. On the basis of the original EDAS method, the 2TLPF-EDAS is built for MAGDM. Finally, a case study for construction enterprise contractor selection is given and some comparative analysis with the other methods show that the new method proposed in this paper is effective, reasonable and accurate.

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