QUALIFLEX method for evaluating human factors in construction project management with Pythagorean 2-tuple linguistic information

Abstract

The overall quality evaluation of operation personnel helps contain site safety accidents, in this study, we proposed a combination of the Pythagorean 2-tuple linguistic fuzzy set and qualitative flexible multiple criteria (QUALIFLEX) method to evaluate comprehensive quality of operation personnel in engineering projects, Pythagorean 2-tuple linguistic fuzzy numbers to express decision makers’ evaluation on each scheme with original QUALIFLEX approach to decision making process. In the end, an example of the performance evaluation of operation personnel in the engineering project is provided to test the applicability and practicability of the method, comparison analysis for further elaboration.

Keywords

Multiple attribute group decision making (MAGDM)Pythagorean 2-tuple linguistic numbers QUALIFLEX method construction projects

1 Introduction

Engineering projects occupy an important position in Chinese economy. With the continuous expansion of the scale of engineering projects, frequent management problems have seriously affected the quality of engineering projects. In engineering project management, people play a very important role and effectively guide the results of engineering projects. In construction project construction process, therefore, must establish a set of suitable for construction safety evaluation system, and build a comprehensive evaluation model, and improve the building construction safety situation, ensure the security of the site construction personnel, to prevent and reduce safety accidents, the construction of safety science and control has important practical significance. In recent years, there have been a series of explorations and researches on construction management at home and abroad. Gebrehiwet and Luo [1] studied the methods used to assess the risk of delay in different life cycles of construction projects, using a comprehensive ranking model based on TOPSIS (technique for order preference by similarity to ideal solution) and FCE (fuzzy comprehensive evaluation), and verified the accuracy of this technology’s prediction with a practical case. Under incomplete weight information, based on interval-valued intuitionistic fuzzy number and CODAS model, Roy et al. [2] proposed a compound multi-attribute decision-making method and conducted an empirical study on the selection of bricks in sustainable construction projects. Due to the risk of construction project uncertainty, in order to reduce the incidence of risk, Biswas and Zaman [3] put forward a kind of method of triangular fuzzy number system combined with expert advice and historical data to calculate the risk value, compared with other fuzzy risk assessment methods, the most significant difference is to use an algorithm to deal with a single risk events involved in uncertainty. Wu et al. [4] expanded the Hamy mean (HM) operator with 2-tuple linguistic neutrosophic numbers (2TLNNs) to propose a 2-tuple linguistic neutrosophic Hamy mean (2TLNHM) operator et al. Then, form a multiple attribute decision-making (MADM) methods with these operators to assess the risk for construction engineering projects. Taylan et al. [5] first used the analytic hierarchy process to determine the weight of indicators, then the fuzzy TOPSIS method is applied to evaluate the performance of contractors in construction projects, and finally selected the best contractors. Ning and Wang [6] combined the intuitionistic fuzzy set with TOPSIS method to evaluate the construction site layout of a building project, which is conducive to improving the safety of the work environment. Lu [7] extended the TOPSIS model to the 2-tuple linguistic set and illustrated the applicability and effectiveness of the model with an example of evaluating project information management with 2-tuple linguistic set information. The above scholars studied the construction project management from different perspectives and methods. Most of the methods used were TOPSIS combined with other fuzzy set theories to evaluate the risks of construction projects, but the research on human factors evaluation of construction projects was scarce.

The fuzzy set theory [8] was first introduced to describe the uncertainty and fuzziness of things. In order to reflect the objective world as faithfully as possible, many people offered some extended forms of the fuzzy set, such as Interval-valued hesitant fuzzy set (IVHFS), Type-2 Fuzzy Set (T2FS), Intuitionistic fuzzy set (IFS) [9], etc. The IFS theory was proposed by Atanassov [9] in 1986 as an important extension of the classical fuzzy set theory. The research on its theory and application has achieved extensive research results in the field of fuzzy set theory and produced far-reaching influence [10 –16]. In intuitionistic fuzzy set, membership degree was defined as the degree of affirmation about the same concept and also non-membership degree as the degree of negation [17 –20]. However, when using intuitionistic fuzzy to make decision, the following situation may occur: membership degree plus non-membership degree of the scheme satisfying attributes given by the decision makers is greater than 1. Based on this, in 2013, Yager and Abbasov [21] proposed the Pythagorean fuzzy set, which satisfies that membership degree plus non-membership degree greater than 1, but the sum of squares does not exceed 1. Therefore, the decision maker does not need to modify the values of membership and non-membership, can be more accurate and detailed description of the reality. After the Pythagorean fuzzy set was proposed, a large number of researchers combined the Pythagorean fuzzy set [21] with various methods and applied these proposed methods to multiple-attribute decision-making (MADM). Ren et al. [22] provided a case of choosing the governor of Asian Infrastructure Investment Bank by using PF-TODIM method observe the feasibility of the model. Garg [23] studies the correlation properties of the hesitant Pythagorean fuzzy set combined with Maclaurin symmetric mean (MSM) operator. On the basis of The Pythagorean fuzzy set, Harish and Garg [24] studied some new logarithms operational laws based on real numbers and some properties. Zhang and Xu [25] first put forward the mathematical expression of Pythagorean fuzzy set, and then they tied the Pythagorean fuzzy set (PFS) and technique for order preference by similarity to ideal solution (TOPSIS) method together. Chen [26] defined a new VIseKriterijumska Optimizacija I KOmpromisno Resenje (VIKOR)-based method for MADM analysis containing Pythagorean fuzzy information. Chen [26] defined a new VIKOR-based method for MADM analysis containing Pythagorean fuzzy information. Based on the Pythagorean fuzzy set theory, Harish Garg [27] proposed the Neutrality operations based on Pythagorean fuzzy aggregation operators for multiple attribute group decision making process. Garg [28] defined the novel neutrality operation-based Pythagorean fuzzy geometric aggregation operators for multiple attribute group decision analysis. Discovering there are priorities for attributes and decision makers in actual decisions, based on Pythagorean fuzzy set, Khan et al. [29] developed Pythagorean fuzzy prioritized weighted average operator and Pythagorean fuzzy prioritized weighted geometric operator, and studied their properties. Ejegwa [30] gave the axiomatic definitions of distance and similarity measures for the Pythagorean fuzzy set. Ejegwa [31] studied the approach of max–min–max composite relation for Pythagorean fuzzy sets. In order to overcome the limitations of previous operators, Rahman and Ali [32] introduce the Pythagorean fuzzy Einstein hybrid geometric aggregation operator concept. Bolturk [33] expanded CODAS model to Pythagorean fuzzy environment. Ilbahar et al. [34] proposed three methods respectively to Fine Kinney, Pythagorean fuzzy analytic hierarchy process, and PFPRA. Tang et al. [35] defined the Pythagorean fuzzy Muirhead mean operators in MADM for evaluating of emerging technology commercialization. Li et al. [36] extended the Hamy mean (HM) operator and dual Hamy mean (DHM) operator [37] with Pythagorean fuzzy numbers (PFNs) to propose Pythagorean fuzzy Hamy mean (PFHM) operator, weighted Pythagorean fuzzy Hamy mean (WPFHM) operator, Pythagorean fuzzy dual Hamy mean (PFDHM) operator, weighted Pythagorean fuzzy dual Hamy mean (WPFDHM) operator. Zhang [38] presented a Pythagorean fuzzy QUALIFLEX method with the closeness index to address the layered MCDM issue under Pythagorean fuzzy environment on the basis of PFNs and IVPFNs. Khan et al. [39] presented an extension of TOPSIS under the interval value Pythagorean fuzzy context and IVPFCIG operator and distance formula based on Choquet integral to aggregate all fuzzy decision matrixes. Tang et al. [40] built the Models for multiple attribute decision making with interval-valued pythagorean fuzzy muirhead mean operators for green suppliers selection. A novel the linear programming technique for multidimensional analysis of preference (LINMAP) method was expanded by Xue et al. [41] to the fuzzy environment of Pythagorean fuzzy sets and interval-value Pythagorean fuzzy sets. Garg [42] connected the Pythagorean fuzzy set with the linguistic fuzzy set, and defined a new linguistic Pythagorean fuzzy set for processing fuzzy information. Arora and Garg [43] proposed a group decision-making method based on prioritized linguistic intuitionistic fuzzy aggregation operators. Huang and Wei [44] briefly introduced the definition of Pythagorean 2-tuple linguistic numbers (P2TLNs), the distance between two P2TLNs and the classic TODIM. On this basis, a new extended TODIM is put forward to deal with the MADM problem. He et al. [45] defined the Pythagorean 2-tuple linguistic VIKOR method for evaluating human factors in construction project management. He et al. [46] defined the Pythagorean 2-tuple linguistic Taxonomy method for supplier selection in medical instrument industries. Liu et al. [47] gives a linguistic connection number corresponding to linguistic intuitionistic fuzzy number, defines its cosine distance measure, and extends the TOPSIS method on this basis.

The QUALIFLEX (qualitative flexible multiple criteria method) was originally proposed by Paelinck [48]. Due to its flexibility in cardinality and ordinal information, it is effective to solve the MCDM problems. The QUALIFLEX method has recently been expanded into various fuzzy decision environments, for example, Demirel et al. [49] used the QUALIFLEX method in the Interval Type 2 Fuzzy Set (IT2FSs) environment. While QUALIFLEX can handle all possible variations of alternatives, IT2FS can address uncertainty. Liang and Chong [50] developed a framework for green supplier selection decision making by using the hesitant fuzzy qualitative multi-attribute method (QUALIFLEX). Liang et al. [51] used the traditional QUALFLEX approach to integrate with the ORESTE model to achieve mine rankings. Finally, the hesitant fuzzy ORESTE-QUALIFLEX method proposed was used to evaluate the performance of green mines. Song et al. [52] proposed an innovative integration approach. The proposed method combines the QUALIFLEX method and the interval rough number to evaluate the shelter, and illustrates the capability and feasibility of the method through a case study in Wenchuan County (a major earthquake on May 12, 2008). Combining quality function development (QFD) theory and qualitative flexible multiple criteria method (QUALIFLEX) under interval-valued Pythagorean uncertain linguistic context, Liu et al. [47] proposes a novel robot selection model and determines the most suitable robot.

Description based on the above research background, the purpose of this article is based on Pythagorean fuzzy sets and Pythagorean 2-Tuple linguistic fuzzy sets, use Pythagorean 2-Tuple linguistic fuzzy numbers to express decision makers of assessment information, and then use QUALIFLEX method for processing integrated evaluation information. Finally, the optimal solution (evaluation object) is obtained.

The remainder of this article is mainly as follows: Section 2: some basic definitions of P2TLNs; Section 3: the extending QUALIFLEX method with P2TLNs; Section 4: a case study of evaluating human factors in the process of construction project management and contrastive analysis; Section 5: conclusions.

2 Preliminaries

Wei et al.[44] proposed the Pythagorean 2-tuple linguistic sets (P2TLSs) based on the PFSs [53] and 2-tuple linguistic information [54].

Definition 1 [44]. A P2TLS O in X is given $O = {(s_{σ (x)}, ψ), (u_{o} (x), ν_{o} (x)), x \in X}$ (1)

Where s_σ(x) ∈ S, ψ ∈ [- 0.5 0.5) , u_o (x) ∈ [0, 1] and v_o (x) ∈ [0, 1], u_o (x) and ν_o (x) satisfy the following condition 0 ⩽ (u_o (x)) ² + (v_o (x)) ² ⩽ 1, ∀x ∈ X. The numbers u_o (x) , ν_o (x) represent the degree of membership and degree of non-membership of the element x to linguistic variable (s_σ(x), ψ).

O = 〈 (s_σ, ψ) , (u_o, v_o) 〉 is called a Pythagorean 2-tuple linguistic number (P2TLN).

Definition 2 [44]. Suppose that o = 〈 (s_σ, ψ) , (u_o, v_o) 〉 is a P2TLN, the score function of P2TLN can be depicted as follows: $\begin{matrix} S (o) = Δ (Δ^{- 1} (s_{σ (o)}, ψ) \frac{1 + {(u_{o})}^{2} - {(ν_{o})}^{2}}{2}), \\ Δ^{- 1} (S (o)) \in [0, L] . \end{matrix}$ (2)

Definition 3 [44]. Suppose that o = 〈 (s_σ, ψ) , (u_o, v_o) 〉 is a P2TLN, the accuracy function of P2TLN can be depicted as follows: $\begin{matrix} H (o) = Δ (Δ^{- 1} (s_{φ (o)}, ρ) \frac{{(u_{o})}^{2} + {(ν_{o})}^{2}}{2}), \\ Δ^{- 1} (H (o)) \in [0, L] . \end{matrix}$ (3)

Definition 4 [44]. Suppose that o₁ = 〈 (s_{σ
₁}, ψ₁) , (u_{o
₁}, v_{o
₁}) 〉 and o₂ = 〈 (s_{σ
₂}, ψ₂) , (u_{o
₂}, v_{o
₂}) 〉 are two P2TLNs. Respectively, the scores of o₁ and o₂ are $S (o_{1}) = Δ (Δ^{- 1} (s_{σ (o_{1})}, ψ_{1}) \cdot \frac{1 + {(u_{o_{1}})}^{2} - {(ν_{o_{1}})}^{2}}{2})$ and $S (o_{2}) = Δ (Δ^{- 1} (s_{σ (o_{2})}, ψ_{2}) \cdot \frac{1 + {(u_{o_{2}})}^{2} - {(ν_{o_{2}})}^{2}}{2})$ , and let $H (o_{1}) = Δ (Δ^{- 1} (s_{σ (o_{1})}, ψ_{1}) \cdot \frac{{(u_{o_{1}})}^{2} + {(ν_{o_{1}})}^{2}}{2})$ and $H (o_{2}) = Δ (Δ^{- 1} (s_{σ (o_{2})}, ψ_{2}) \cdot \frac{{(u_{o_{2}})}^{2} + {(ν_{o_{2}})}^{2}}{2})$ be the accuracy degrees of o₁ and o₂, then some operational laws of P2TLNs can be defined as follows:

$\begin{matrix} (1) if S (o_{1}) < S (o_{2}), o_{1} < o_{2}; \\ (2) if S (o_{1}) > S (o_{2}), o_{1} > o_{2}; \\ (3) if S (o_{1}) = S (o_{2}), H (o_{1}) < H (o_{2}), then o_{1} < o_{2}; \\ (4) if S (o_{1}) = S (o_{2}), H (o_{1}) > H (o_{2}), then o_{1} > o_{2}; \\ (5) if S (o_{1}) = S (o_{2}), H (o_{1}) = H (o_{2}), then o_{1} = o_{2}; \end{matrix}$

Definition 5 [44]. Suppose that o₁ = 〈 (s_{σ
₁}, ψ₁) , (u_{o
₁}, v_{o
₁}) 〉 and o₂ = 〈 (s_{σ
₂}, ψ₂) , (u_{o
₂}, v_{o
₂}) 〉 are two P2TLNs, the normalized Hamming distance (H_d) between a₁ and a₂ can be depicted below:

$\begin{matrix} H_{d} (o_{1}, o_{2}) \\ = \frac{1}{2 L} [| \begin{matrix} (1 + {(u_{o_{1}})}^{2} - {(v_{o_{1}})}^{2}) \cdot Δ^{- 1} (s_{σ_{1}}, ψ_{1}) - \\ (1 + {(u_{o_{2}})}^{2} - {(v_{o_{2}})}^{2}) \cdot Δ^{- 1} (s_{σ_{2}}, ψ_{2}) \end{matrix} |] \end{matrix}$ (4)

where L represents the length of the language scale. It is a numerical value.

Definition 6 [44]. Suppose that o₁ = 〈 (s_{σ
₁}, ψ₁) , (u_{o
₁}, v_{o
₁}) 〉 and o₂ = 〈 (s_{σ
₂}, ψ₂) , (u_{o
₂}, v_{o
₂}) 〉 are two P2TLNs, then $\begin{matrix} o_{1} \oplus o_{2} = 〈 Δ (Δ^{- 1} (s_{σ_{1}}, ψ_{1}) + Δ^{- 1} (s_{σ_{2}}, ψ_{2})), \\ (\sqrt{{(u_{o_{1}})}^{2} + {(u_{o_{2}})}^{2} - {(u_{o_{1}})}^{2} {(u_{o_{2}})}^{2}}, ν_{o_{1}} ν_{o_{2}}) 〉; \\ o_{1} \otimes o_{2} = 〈 Δ (Δ^{- 1} (s_{σ_{1}}, ψ_{1}) \cdot Δ^{- 1} (s_{σ_{2}}, ψ_{2})), \\ (u_{o_{1}} u_{o_{2}}, \sqrt{{(ν_{o_{1}})}^{2} + {(ν_{o_{2}})}^{2} - {(ν_{o_{1}})}^{2} {(ν_{o_{2}})}^{2}}) 〉; \\ λ o_{1} = 〈 Δ (λ Δ^{- 1} (s_{σ_{1}}, ψ_{1})), \\ (\sqrt{1 - {(1 - {(u_{o_{1}})}^{2})}^{λ}}, {(ν_{o_{1}})}^{λ}) 〉; \\ {(o_{1})}^{λ} = 〈 Δ ({(Δ^{- 1} (s_{σ_{1}}, ψ_{1}))}^{λ}), \\ ({(u_{o_{1}})}^{λ} \sqrt{1 - {(1 - {(ν_{o_{1}})}^{2})}^{λ}}) 〉 . \end{matrix}$

Theorem 1 [44]. For any two P2TLNs o₁ =〈 (s_{σ
₁}, ψ₁) , (u_{o
₁}, v_{o
₁}) 〉 and o₂ = 〈 (s_{σ
₂}, ψ₂) , (u_{o
₂}, v_{o
₂}) 〉, According to the Definition 6, naturally, we can get the following properties of the operation laws:

o₁ ⊕ o₂ = o₂ ⊕ o₁

o₁ ⊗ o₂ = o₂ ⊗ o₁

k (o₁ ⊕ o₂) = ko₁ ⊕ ko₂, 0 ⩽ k ⩽ 1

k₁o₁ ⊕ k₂o₁ = (k₁ ⊕ k₂) o₁, 0 ⩽ k₁, k₂, k₁ + k₂ ⩽ 1

$o_{1}^{k_{1}} \otimes o_{1}^{k_{2}} = {(o_{1})}^{k_{1} + k_{2}}, 0 ⩽ k_{1}, k_{2}, k_{1} + k_{2} ⩽ 1$

$o_{1}^{k_{1}} \otimes o_{2}^{k_{1}} = {(o_{1} \otimes o_{2})}^{k_{1}}, k_{1} ⩾ 0$

((o₁) ^k
₁) ^k
₂ = (o₁) ^{k
₁
k
₂}

Definition 7 [44]. Assume that o_j = 〈 (s_{σ
_j}, ψ_j) , (u_{o
_j}, v_{o
_j}) 〉 (j = 1, 2, . . . , n) is a collection of P2TLNs. P2TLWA operator can be depicted as follows: $\begin{matrix} {P 2 TLWA}_{ω} (o_{1}, o_{2}, . . ., o_{n}) = \oplus_{j = 1}^{n} (ω_{j} o_{j}) \\ = 〈 Δ (\sum_{j = 1}^{n} ω_{j} Δ^{- 1} (s_{σ_{j}}, ψ_{j})), \\ (\sqrt{1 - \prod_{j = 1}^{n} {(1 - {(u_{o_{j}})}^{2})}^{ω_{j}}}, \prod_{j = 1}^{n} {(ν_{o_{j}})}^{ω_{j}}) 〉 \end{matrix}$ (5)

where ω = (ω₁, ω₂, . . . , ω_n) ^T is the weight vector of o_j (j = 1, 2, . . . , n) and $ω_{j} > 0, \sum_{j = 1}^{n} ω_{j} = 1$ .

Definition 8 [44]. Assume that o_j =  〈 (s_{σ
_j}, ψ_j) ,  (u_{o
_j}, v_{o
_j}) 〉 (j = 1, 2, . . . , n) is a collection of P2TLNs. P2TLWG operator can be depicted as follows: $\begin{matrix} {P 2 TLWG}_{ω} (o_{1}, o_{2}, . . ., o_{n}) = \otimes_{j = 1}^{n} (ω_{j} o_{j}) \\ = 〈 Δ (\prod_{j = 1}^{n} Δ^{- 1} {(s_{σ_{j}}, ψ_{j})}^{ω_{j}}), \\ (, \prod_{j = 1}^{n} {(u_{o_{j}})}^{ω_{j}} \sqrt{1 - \prod_{j = 1}^{n} {(1 - {(ν_{o_{j}})}^{2})}^{ω_{j}}}) 〉 \end{matrix}$ (6) where ω =  (ω₁, ω₂, . . . , ω_n) ^T is the weight vector of o_j (j = 1, 2, . . . , n) and $ω_{j} > 0, \sum_{j = 1}^{n} ω_{j} = 1$ .

Example 1. There are four P2TLNs p₁ = 〈 (s₁, 0) , (0.7, 0.4) 〉, p₂ = 〈 (s₂, 0) , (0.8, 0.6) 〉, p₃ = 〈 (s₃, 0) , (0.7, 0.5) 〉 and p₄ = 〈 (s₄, 0) , (0.6, 0.6) 〉 . Supposing that their weight are ω = (ω₁, ω₂, ω₃, ω₄) ^T = (0.18, 0.25, 0.23, 0.34), then according to (5) and (6), we have: $\begin{matrix} {P 2 TLWA}_{ω} (p_{1}, p_{2}, p_{3}, p_{4}) = ω_{1} p_{1} \oplus ω_{2} p_{2} \oplus ω_{3} p_{3} \oplus ω_{4} p_{4} \\ = 〈 \begin{matrix} Δ (0.18 \times 1 + 0.25 \times 2 + 0.23 \times 3 + 0.34 \times 5), \\ (\begin{matrix} \sqrt{1 - {(1 - 0 . 7^{2})}^{0.18} \times {(1 - 0 . 8^{2})}^{0.25} \times {(1 - 0 . 7^{2})}^{0.23} \times {(1 - 0 . 6^{2})}^{0.34}}, \\ 0 . 4^{0.18} \times 0 . 6^{0.25} \times 0 . 5^{0.23} \times 0 . 6^{0.34} \end{matrix}) \end{matrix} 〉 \\ = 〈 (s_{3}, - 0 . 27), (0 . 7036, 0 . 5349) 〉 \end{matrix}$ $\begin{matrix} {P 2 TLWG}_{ω} (p_{1}, p_{2}, p_{3}, p_{4}) = {(p_{1})}^{ω_{1}} \otimes {(p_{2})}^{ω_{2}} \otimes {(p_{3})}^{ω_{3}} \otimes {(p_{4})}^{ω_{4}} \\ = 〈 \begin{matrix} Δ (1^{0.18} \times 2^{0.25} \times 3^{0.23} \times 4^{0.34}), \\ (\begin{matrix} 0 . 7^{0.18} \times 0 . 8^{0.25} \times 0 . 7^{0.23} \times 0 . 6^{0.34}, \\ \sqrt{1 - {(1 - 0 . 4^{2})}^{0.18} \times {(1 - 0 . 6^{2})}^{0.25} \times {(1 - 0 . 5^{2})}^{0.23} \times {(1 - 0 . 6^{2})}^{0.34}}, \end{matrix}) \end{matrix} 〉 \\ = 〈 (s_{2}, 0 . 453), (0 . 6868, 0 . 5504) 〉 \end{matrix}$

3 QUALIFLEX Method for MADM Problems with P2TLNs

Suppose that N_i ={ N₁, N₂, … N_m } and κ_j = { κ₁, κ₂, … κ_n } are respectively m alternatives and n criteria. Let γ_j be the criteria’s weighting vector which satisfies ϑ_j ∈ [0, 1] and $\sum_{j = 1}^{n} ϑ_{j} = 1$ . Let Θ ={ Θ₁, Θ₂, … Θ_k } be the group of DMs, γ ={ γ₁, γ₂, … γ_k } be the weight of DMs, with γ_t ∈ [0, 1] and $\sum_{t = 1}^{k} γ_{t} = 1$ . Construct a decision matrix $X^{(t)} = {(x_{ij}^{(t)})}_{m \times n}$ , where $X^{(t)} = {(x_{ij}^{(t)})}_{m \times n} = 〈 (s_{σ_{ij}}^{(t)}, ψ_{ij}^{(t)}), (u_{r_{ij}}^{(t)}, v_{r_{ij}}^{(t)}) 〉_{m \times n}$ means the performance of the alternative N_i{ i = 1, 2 ⋯ , m } with respect to criteria κ_j{ j = 1, 2 ⋯ , n } by expert Θ^(t){ t = 1, 2, … k } λ using a P2TLN, $0 ⩽ u_{r_{ij}}^{(t)} ⩽ 1$ , $0 ⩽ v_{r_{ij}}^{(t)} ⩽ 1$ and $0 ⩽ {(u_{r_{ij}}^{(t)})}^{2} + {(v_{r_{ij}}^{(t)})}^{2} ⩽ 1$ , i = 1, 2 ⋯ , m, j = 1, 2 ⋯ , n, t = 1, 2 ⋯ , k.

In view of both the P2TLNs theories and procedures from QUALIFLEX method, we put forward a P2TL-QUALIFLEX method to deal with the problem of MADM effectively. The new model can be shown below:

Step 1. Set up a decision-making group composed of several experts, choose the best attributes to measure alternatives, and finally get a series of P2TL fuzzy decision matrix $X^{(t)} = {(x_{ij}^{(t)})}_{m \times n}$ from each decision maker.

$\begin{matrix} X^{(t)} \\ = {[x_{ij}^{(t)}]}_{m \times n} = {[\begin{matrix} x_{11}^{(t)} & \dots & x_{1 j}^{(t)} & \dots & x_{1 n}^{(t)} \\ ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ x_{21}^{(t)} & \dots & x_{ij}^{(t)} & \dots & x_{in}^{(t)} \\ ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ x_{m 1}^{(t)} & \dots & x_{mj}^{(t)} & \dots & x_{mn}^{(t)} \end{matrix}]}_{m \times n} \end{matrix}$ (7) where $x_{ij}^{(t)}$ denotes the fuzzy performance value of i - th alternative (i = 1, 2, …, m) with respect to j - th criterion (j = 1, 2, …, n) and t - th decision-maker(t = 1, 2, …, k).

Step 2. Utilize P2TLWA operator or P2TLWG operator defined in Definition 7 and Definition 8 to fuse assessment information, then the group P2TL fuzzy decision matrix X = (x_ij) _m×n can be obtained by the calculation. $X_{ij} = {[x_{ij}]}_{m \times n} = {[\begin{matrix} x_{11} & \dots & x_{1 j} & \dots & x_{1 n} \\ ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ x_{21} & \dots & x_{ij} & \dots & x_{in} \\ ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ x_{m 1} & \dots & x_{mj} & \dots & x_{mn} \end{matrix}]}_{m \times n}$ (8) $\begin{matrix} x_{ij} = \oplus_{t = 1}^{k} x_{ij}^{(k)} = P 2 TLWA (x_{ij}^{(1)}, x_{ij}^{(2)}, . . ., x_{ij}^{(k)}) \\ = 〈 Δ (\sum_{t = 1}^{k} γ_{t} Δ^{- 1} (s_{σ_{ij}}^{(t)}, ψ_{ij}^{(t)})), \\ (\sqrt{1 - \prod_{t = 1}^{k} {(1 - {(u_{ij}^{(t)})}^{2})}^{γ_{t}}}, \prod_{t = 1}^{k} {(ν_{ij}^{(t)})}^{γ_{t}}) 〉 \end{matrix}$ (9)

Or $\begin{matrix} x_{ij} = \otimes_{t = 1}^{k} x_{ij}^{(k)} = P 2 TLWG (x_{ij}^{(1)}, x_{ij}^{(2)}, . . ., x_{ij}^{(k)}) \\ = 〈 Δ (\prod_{t = 1}^{k} Δ^{- 1} {(s_{σ_{ij}}^{(t)}, ψ_{ij}^{(t)})}^{γ_{t}}), \\ (\prod_{t = 1}^{k} {(u_{ij}^{(t)})}^{γ_{t}}, \sqrt{1 - \prod_{t = 1}^{k} {(1 - {(ν_{ij}^{(t)})}^{2})}^{γ_{t}}}) 〉 \end{matrix}$ (10) where x_ij means the average fuzzy performance value of i - th alternative in relative to j - th criterion.

Remark: P2TLWA operator or P2TLWG operator is used to fuse evaluation information of each expert (decision-maker) into a comprehensive evaluation matrix. See example 1 for the specific operation procedure.

Step 3. Normalize the decision-making matrix. $c_{ij} = {\begin{matrix} 〈 (s_{σ_{ij}}, ψ_{ij}), (u_{ij}, v_{ij}) 〉 \\ 〈 Δ (L - Δ^{- 1} (s_{σ_{ij}}, ψ_{ij})), (v_{ij}, u_{ij}) 〉 \end{matrix} \begin{matrix} if κ_{j} is benefit attribute, \\ if κ_{j} is cost attribute . \end{matrix}$ (11)

Remark: In Eq. (11), L means the length of the language scale. It is a numerical value. For example, if S = (s₀, s₁, s₂, s₃, s₄, s₅, s₆), then L = 6 -0 = 6.

Step 4. Determine all of the possible permutations of the alternatives.

List all the possible m ! permutations of the m alternatives that must be tested. Let P_δ denote the δth permutation as: $P_{δ} = (\dots, N_{ς}, \dots, N_{ξ}, \dots), δ = 1, 2, \dots m!$ (12)

Remark: where N_ς, N_ξ ∈ N_i, define the alternative N_ς is higher than or equal to N_ξ in the order in which they are placed. For example, there are three existing alternative, namely m = 3, the number of possible permutations is m ! =3 ! =6. $\begin{matrix} premutation 1 = (N_{1}, N_{2}, N_{3}), \\ premutation 2 = (N_{1}, N_{3}, N_{2}), \\ premutation 3 = (N_{2}, N_{1}, N_{3}), \\ premutation 4 = (N_{2}, N_{3}, N_{1}), \\ premutation 5 = (N_{3}, N_{1}, N_{2}), \\ premutation 6 = (N_{3}, N_{2}, N_{1}) . \end{matrix}$

Step 5: Calculate the concordance index/discordance index $ρ_{j}^{δ} (N_{ς}, N_{ξ})$ for each pair of alternatives (N_ς, N_ξ) at the level of preorder according to the jth criterion and the ranking corresponding to the δth permutation: $ρ_{j}^{δ} (N_{ς}, N_{ξ}) = H_{d} (x_{ξ j}, \overset{⌣}{x}) - H_{d} (x_{ς j}, \overset{⌣}{x})$ (13)

Remark: where $H_{d} (x_{ξ j}, \overset{⌣}{x})$ means the Hamming distance between x_ξjand $\overset{⌣}{x}$ , $\overset{⌣}{x}$ represents the ideal P2TLN. Here, we define the ideal P2TLN $\overset{⌣}{x} = 〈 (s_{6}, 0), (1, 0) 〉$ Similarly, the other is the same.

From Equation (15), we can conclude that:

If $ρ_{j}^{δ} (N_{ς}, N_{ξ}) > 0$ , i.e, $H_{d} (x_{ς j}, \overset{⌣}{x}) < H_{d} (x_{ξ j}, \overset{⌣}{x})$ , then N_ς ranks over N_ξ under the jth criterion, thus, there is concordance between the Hamming distance-based ranking orders and the preorders of N_ς and N_ξ;

If $ρ_{j}^{δ} (N_{ς}, N_{ξ}) = 0$ , i.e, $H_{d} (x_{ς j}, \overset{⌣}{x}) = H_{d} (x_{ξ j}, \overset{⌣}{x})$ , then N_ς and N_ξ have the same rank;

If $ρ_{j}^{δ} (N_{ς}, N_{ξ}) > 0$ , i.e, $H_{d} (x_{ξ j}, \overset{⌣}{x}) > H_{d} (x_{ς j}, \overset{⌣}{x})$ , then N_ξ ranks over N_ς under the jth criterion, thus, there is discordance between the Hamming distance-based ranking orders and the preorders of N_ς and N_ξ.

Step 6: Calculate the weighted concordance/discordance index. $ρ^{δ} (N_{ς}, N_{ξ}) = \sum_{j = 1}^{n} ϑ_{j} ρ_{j}^{δ} (N_{ς}, N_{ξ})$ (14)

Step 7: Calculate the comprehensive concordance/discordance index.

Based on Step 6, the comprehensive index/discordance index ρ^δ of the permutation p_δ = (⋯ , N_ς, ⋯ , N_ξ, ⋯) , δ = 1, 2, ⋯ m ! can be obtained as follows: $ρ^{δ} = \sum_{N_{ς}, N_{ξ} \in N} ρ^{δ} (N_{ς}, N_{ξ})$ (15)

Step 8: Determine the ranking of all alternatives.

The bigger the comprehensive concordance index/discordance index, the better the final ranking result of the alternatives will be.

4 Numerical example and comparative analysis

4.1 Numerical example

In this section, we shall provide a numerical example to evaluate human factors in the process of construction project management by using P2TL-QUALIFLEX model. Assume that four possible construction projects N_i (i = 1, 2, 3, 4) to be selected and four evaluation criteria κ_j (j = 1, 2, 3, 4) to evaluate these construction projects: ① κ₁ is the workers’ inertia; ② κ₂ is the workers’ safety awareness; ③ κ₃ is the technical workers’ quality; ④ κ₄ is the workers’ emergency capacity. The four possible construction projects N_i (i = 1, 2, 3, 4) are to be evaluated through using P2TLNs with the four criteria by three experts Θ^k (expert’s weight γ_t = (0.31, 0 . 45, 0 . 24), attributes weight ϑ_j = (0 . 24, 0 . 17, 0 . 31, 0 . 28) ^T).

The following steps are used to evaluate the human factors associated with the four construction projects using the proposed P2TL-QUALIFLEX method:

Step 1: Construct the evaluation matrix $X^{(3)} = {(x_{ij}^{3})}_{4 \times 4} (i = 1, 2, 3, 4; j = 1, 2, 3, 4)$ of each DM as in Table 1 –3. Based on these tables and Equations (9) or (10), the group P2TLN decision matrix is computed and presented in Table 4.

Table 1
The P2TLN decision matrix by Θ⁽¹⁾

κ ₁ κ ₂ κ ₃ κ ₄

N ₁ < (s₃,0),(0.6,0.5)> < (s₆,0),(0.3,0.3)> < (s₁,0),(0.6,0.6)> < (s₃,0),(0.2,0.3)>

N ₂ < (s₁,0),(0.3,0.8)> < (s₆,0),(0.4,0.6)> < (s₁,0),(0.6,0.3)> < (s₀,0),(0.1,0.6)>

N ₃ < (s₃,0),(0.4,0.6)> < (s₁,0),(0.4,0.3)> < (s₅,0),(0.6,0.3)> < (s₃,0),(0.7,0.4)>

N ₄ < (s₄,0),(0.6,0.7)> < (s₆,0),(0.2,0.4)> < (s₆,0),(0.8,0.1)> < (s₁,0),(0.4,0.6)>

	κ ₁	κ ₂	κ ₃	κ ₄
N ₁	< (s₃,0),(0.6,0.5)>	< (s₆,0),(0.3,0.3)>	< (s₁,0),(0.6,0.6)>	< (s₃,0),(0.2,0.3)>
N ₂	< (s₁,0),(0.3,0.8)>	< (s₆,0),(0.4,0.6)>	< (s₁,0),(0.6,0.3)>	< (s₀,0),(0.1,0.6)>
N ₃	< (s₃,0),(0.4,0.6)>	< (s₁,0),(0.4,0.3)>	< (s₅,0),(0.6,0.3)>	< (s₃,0),(0.7,0.4)>
N ₄	< (s₄,0),(0.6,0.7)>	< (s₆,0),(0.2,0.4)>	< (s₆,0),(0.8,0.1)>	< (s₁,0),(0.4,0.6)>

Table 2

The P2TLN decision matrix by Θ⁽²⁾

	κ ₁	κ ₂	κ ₃	κ ₄
N ₁	< (s₆,0),(0.8,0.7)>	< (s₁,0),(0.8,0.5)>	< (s₆,0),(0.3,0.3)>	< (s₅,0),(0.1,0.7)>
N ₂	< (s₀,0),(0.2,0.1)>	< (s₀,0),(0.2,0.2)>	< (s₃,0),(0.8,0.6)>	< (s₃,0),(0.5,0.3)>
N ₃	< (s₃,0),(0.4,0.2)>	< (s₀,0),(0.8,0.1)>	< (s₁,0),(0.6,0.6)>	< (s₅,0),(0.1,0.7)>
N ₄	< (s₄,0),(0.7,0.7)>	< (s₅,0),(0.1,0.2)>	< (s_4,0),(0.7,0.1)>	< (s₂,0),(0.8,0.5)>

Table 3

The P2TLN decision matrix by Θ⁽³⁾

	κ ₁	κ ₂	κ ₃	κ ₄
N ₁	< (s₁,0),(0.4,0.3)>	< (s₀,0),(0.6,0.2)>	< (s₅,0),(0.8,0.6)>	< (s₀,0),(0.2,0.1)>
N ₂	< (s₀,0),(0.2,0.6)>	< (s₂,0),(0.5,0.5)>	< (s₃,0),(0.6,0.6)>	< (s₂,0),(0.7,0.5)>
N ₃	< (s₁,0),(0.3,0.8)>	< (s₅,0),(0.5,0.6)>	< (s₀,0),(0.1,0.5)>	< (s₀,0),(0.4,0.7)>
N ₄	< (s₆,0),(0.1,0.5)>	< (s₂,0),(0.7,0.7)>	< (s₄,0),(0.5,0.3)>	< (s₁,0),(0.6,0.6)>

Table 4

The group Pythagorean 2-tuple linguistic decision matrixX_ij

	κ ₁	κ ₂
N ₁	< (s₃,–0.03),(0.5629,0.638)>	< (s₂,0.31),(0.5963,0.4975)>
N ₂	< (s₂,–0.28),(0.4618,0.6476)>	< (s₁,0.1),(0.3632,0.4104)>
N ₃	< (s₃,–0.48),(0.3791,0.4594)>	< (s₃,–0.14),(0.6646,0.3139)>
N ₄	< (s₃,0.34),(0.5379,0.7212)>	< (s₄,–0.34),(0.2394,0.4449)>
	κ ₃	κ ₄
N ₁	< (s₄,0.21),(0.5603,0.534)>	< (s₄,0.21),(0.1631,0.4901)>
N ₂	< (s₃,0.31),(0.7113,0.7029)>	< (s₂,–0.17),(0.5048,0.4926)>
N ₃	< (s₂,0),(0.4091,0.6729)>	< (s₄,0.14),(0.4745,0.7818)>
N ₄	< (s₅,–0.38),(0.6079,0.2658)>	< (s₁,0.45),(0.6801,0.6476)>

Step 3. Eqs.(11) is used to normalize the decision-making matrix.

Step 4: List the ranking of alternatives involving 24 permutations: $\begin{matrix} P_{1} = (N_{1}, N_{2}, N_{3}, N_{4}), \\ P_{2} = (N_{1}, N_{2}, N_{4}, N_{3}), \\ \dots \dots \\ P_{23} = (N_{4}, N_{3}, N_{2}, N_{1}), \\ P_{24} = (N_{4}, N_{3}, N_{1}, N_{2}) . \end{matrix}$

Step 5: Calculate the concordance index/discordance index $ρ_{j}^{δ} (N_{ς}, N_{ξ})$ for each pair of alternatives (N_ς, N_ξ) in the permutation P_δ with respect to thejth criterion by employing Equation (13). The ideal P2TLN is $\overset{⌣}{x} = 〈 (s_{6}, 0), (1, 0) 〉$ Due to space limitations, only some calculation results are listed here.

Step 6: Calculate the weighted concordance index/discordance index by utilizing Eq. (14). Take P₁ for example: $\begin{matrix} ρ^{1} (N_{1}, N_{2}) = \sum_{j = 1}^{4} ϑ_{j} ρ_{j}^{1} (N_{1}, N_{2}) = 0 . 0412 \\ ρ^{1} (N_{1}, N_{3}) = \sum_{j = 1}^{4} ϑ_{j} ρ_{j}^{1} (N_{1}, N_{3}) = 0 . 0670 \\ ρ^{1} (N_{1}, N_{4}) = \sum_{j = 1}^{4} ϑ_{j} ρ_{j}^{1} (N_{1}, N_{4}) = - 0 . 0117 \\ ρ^{1} (N_{2}, N_{3}) = \sum_{j = 1}^{4} ϑ_{j} ρ_{j}^{1} (N_{2}, N_{3}) = 0 . 0258 \\ ρ^{1} (N_{2}, N_{4}) = \sum_{j = 1}^{4} ϑ_{j} ρ_{j}^{1} (N_{2}, N_{4}) = - 0 . 0528 \\ ρ^{1} (N_{3}, N_{4}) = \sum_{j = 1}^{4} ϑ_{j} ρ_{j}^{1} (N_{3}, N_{4}) = - 0 . 0787 \end{matrix}$

Step 7: Calculate the comprehensive concordance/discordance index by using Eq. (15) as follows:

$\begin{matrix} ρ^{1} = - 0 . 01, ρ^{2} = 0 . 15, ρ^{3} = - 0 . 06, ρ^{4} = 0 . 07, \\ ρ^{5} = 0 . 25, ρ^{6} = 0 . 20, ρ^{7} = - 0 . 09, ρ^{8} = - 0 . 23, \\ ρ^{9} = - 0 . 20, ρ^{10} = - 0 . 04, ρ^{11} = 0 . 07, ρ^{12} = 0 . 09 \\ ρ^{13} = - 0 . 19, ρ^{14} = - 0 . 28, ρ^{15} = - 0 . 25, \\ ρ^{16} = - 0 . 09 ρ^{17} = - 0 . 07, ρ^{18} = - 0 . 11, \\ ρ^{19} = 0 . 28, ρ^{20} = 0 . 23, ρ^{21} = 0 . 19, ρ^{22} = 0 . 06, \\ ρ^{23} = 0 . 01, ρ^{24} = 0 . 09 \end{matrix}$

Step 8: The largest optimal comprehensive concordance/ discordance index value is ρ¹⁹ = 0 . 28, and the best order is P₁₉ = (N₄, N₁, N₂, N₃). So, the highest evaluation of the human factors in these four construction projects is N₄.

4.2 Comparative analysis

A comparative analysis is also performed in this section to demonstrate the stability of the ranking result. we will compare our proposed P2TL-QUALIFLEX model with the P2TLWA and P2TLWG operators defined by Wei [44], P2TL-TODIM [44] method and P2TL-CODAS [55] method.

The comparison results of different methods are as follows.

It is clear from Table 8 that the results are slightly different in ranking of alternatives but the best alternative is always N₄ by comparing the values of our proposed P2TL-QUALIFLEX method with P2TLWA / P2TLWG operators, P2TL-TODIM method and P2TL-CODAS method. Notably, in practical MADM problems, P2TL-QUALIFLEX method is very suitable for situations where there are more attributes than the number of alternatives, such as the evaluation of financial status and the choice of green supply chain. However, this approach becomes quite complicated when a large number of alternatives need to be evaluated.

Table 5
The normalized group Pythagorean 2-tuple linguistic decision matrix C_ij

κ ₁ κ ₂

N ₁ < (s₄,0.03),(0.638,0.5629)> < (s₂,0.31),(0.5963,0.4975)>

N ₂ < (s₅,0.28),(0.6476,0.4618)> < (s₁,0.1),(0.3632,0.4104)>

N ₃ < (s₄,0.48),(0.4594,0.3791)> < (s₃,–0.14),(0.6646,0.3139)>

N ₄ < (s₄,–0.34),(0.7212,0.5379)> < (s₄,–0.34),(0.2394,0.4449)>

κ ₃ κ ₄

N ₁ < (s₄,0.21),(0.5603,0.534)> < (s₄,0.21),(0.1631,0.4901)>

N ₂ < (s₃,0.31),(0.7113,0.7029)> < (s₂,–0.17),(0.5048,0.4926)>

N ₃ < (s₂,0),(0.4091,0.6729)> < (s₄,0.14),(0.4745,0.7818)>

N ₄ < (s₅,–0.38),(0.6079,0.2658)> < (s₁,0.45),(0.6801,0.6476)>

	κ ₁	κ ₂
N ₁	< (s₄,0.03),(0.638,0.5629)>	< (s₂,0.31),(0.5963,0.4975)>
N ₂	< (s₅,0.28),(0.6476,0.4618)>	< (s₁,0.1),(0.3632,0.4104)>
N ₃	< (s₄,0.48),(0.4594,0.3791)>	< (s₃,–0.14),(0.6646,0.3139)>
N ₄	< (s₄,–0.34),(0.7212,0.5379)>	< (s₄,–0.34),(0.2394,0.4449)>
	κ ₃	κ ₄
N ₁	< (s₄,0.21),(0.5603,0.534)>	< (s₄,0.21),(0.1631,0.4901)>
N ₂	< (s₃,0.31),(0.7113,0.7029)>	< (s₂,–0.17),(0.5048,0.4926)>
N ₃	< (s₂,0),(0.4091,0.6729)>	< (s₄,0.14),(0.4745,0.7818)>
N ₄	< (s₅,–0.38),(0.6079,0.2658)>	< (s₁,0.45),(0.6801,0.6476)>

Table 6

The concordance index/discordance index $ρ_{j}^{1} (N_{ς}, N_{ξ})$ under P₁

P ₁	κ ₁	κ ₂	κ ₃	κ ₄
$ρ_{j}^{1} (N_{1}, N_{2})$	–0.1646	0.1250	0.0818	0.1215
$ρ_{j}^{1} (N_{1}, N_{3})$	–0.0324	–0.1068	0.2418	0.0641
$ρ_{j}^{1} (N_{1}, N_{4})$	–0.0093	–0.0488	–0.1391	0.1498
$ρ_{j}^{1} (N_{2}, N_{3})$	0.1322	–0.2318	0.1600	–0.0574
$ρ_{j}^{1} (N_{2}, N_{4})$	0.1553	–0.1738	–0.2210	0.0283
$ρ_{j}^{1} (N_{3}, N_{4})$	0.0231	0.0580	–0.3809	0.0857

Table 7

The concordance index/discordance index $ρ_{j}^{24} (N_{ς}, N_{ξ})$ underP₂₄

P ₂₄	κ ₁	κ ₂	κ ₃	κ ₄
$ρ_{j}^{24} (N_{4}, N_{1})$	0.0093	0.0488	0.1391	–0.1498
$ρ_{j}^{24} (N_{4}, N_{2})$	–0.1553	0.1738	0.2210	–0.0283
$ρ_{j}^{24} (N_{4}, N_{3})$	–0.0231	–0.0580	0.3809	–0.0857
$ρ_{j}^{24} (N_{1}, N_{2})$	–0.1646	0.1250	0.0818	0.1215
$ρ_{j}^{24} (N_{3}, N_{2})$	–0.1322	0.2318	–0.1600	0.0574
$ρ_{j}^{24} (N_{3}, N_{1})$	0.0324	0.1068	–0.2418	–0.0641

Table 8

Rank of Alternatives

Methods	Attributes weight vector	Order
P2TLWA	(0 . 24, 0 . 17, 0 . 31, 0 . 28) ^T	N₄ > N₁ > N₃ > N₂
P2TLWG	(0 . 24, 0 . 17, 0 . 31, 0 . 28) ^T	N₄ > N₁ = N₂ = N₃
P2TL-TODIM	(0 . 24, 0 . 17, 0 . 31, 0 . 28) ^T	N₄ > N₁ > N₃ > N₂
P2TL-CODAS	(0 . 24, 0 . 17, 0 . 31, 0 . 28) ^T	N₄ > N₁ > N₂ > N₃
P2TL-QUALIFLEX	(0 . 24, 0 . 17, 0 . 31, 0 . 28) ^T	N₄ > N₁ > N₂ > N₃

5 Conclusion

Human factors are not only the leading factors affecting the quality of construction projects, but also the most basic and core factors in the quality assurance system, so the evaluation of human factors in construction projects is particularly critical. The contribution of this paper lies in the design of an evaluation method which uses the method of P2TL-QUALIFLEX to qualitatively evaluate human factors in construction projects under Pythagorean 2-tuple Linguistic sets. The method has significant advantages, human factor evaluation in construction projects is a MADM problem, and the decision-able information is vague or uncertain in nature. Therefore, we used language variables to express the preferences of experts. The Pythagorean 2-tuple linguistic sets (P2TLSs) can well reflect uncertain or fuzzy information and solve this kind of problems, and the original QUALIFLEX method can retain more information, become more flexible and applicable, and is especially suitable for solving multi-attribute group decision problems where the number of standards is significantly higher than the number of alternatives. Naturally, we combined the Pythagorean 2-tuple linguistic sets with QUALIFLEX, and the recommended method was systematically applied to the human factor evaluation of four construction projects to find an optimal construction project. The comparative study shows that the proposed MADM algorithm is very effective and useful for decision making. In the future, we will continue to study the MADM problem, and apply the developed method to construction project selection, Supply chain management and many other fields to derive more scientific and reasonable decision making results [56 –64].

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