Selection and application of building material suppliers with intuitionistic fuzzy multiple attribute decision making method

Abstract

The construction industry is the basic industry of the country. With the rapid development of the economy, the construction industry has grown rapidly and the competition in the construction market has become more intense. The competition in the construction market is not only between individual enterprises, but also between the whole supply chain that provides products. Therefore, it is imperative to introduce the idea of supply chain management, strengthen the cooperation with suppliers and improve competitiveness. Supplier evaluation and selection is one of the first issues to be solved for the development of supply chain management. The selection and application of building material suppliers is a classic multiple attribute decision making (MADM). In this paper, the intuitionistic fuzzy sets (IFSs) and Hamacher operations is introduced and the induced intuitionistic fuzzy Hamacher power ordered weighted average (I-IFHPOWA) operator is built. Meanwhile, the properties of built operator are analyzed. Then, the I-IFHPOWA operator is applied to solve the MADM under IFSs. Finally, an example for building material supplier selection is utilized to proof this built model.

Keywords

Multiple attribute decision making (MADM)intuitionistic fuzzy sets (IFSs)I-OWA operator I-IFHPOWA operator building material suppliers

1 Introduction

Multiple attribute decision making (MADM) refers to the ranking and selection of a finite number of solutions with multiple attributes[1 –5]. Due to the complexity of the decision environment and the differences in knowledge structure, judgment level and personal preferences of decision makers, uncertainty and ambiguity often exist in the decision process[6 –8]. Zadeh [9] formulated the fuzzy sets (FSs). The core idea of FSs is to extend the eigenfunctions that take values of 0 or 1 to subordination functions that can take any value in the closed interval [0,1]. In 1986, Atanassov [10, 11] formulated intuitionistic fuzzy sets (IFSs), which can describe uncertainty and vagueness more accurately by considering the information of both unaffiliated and hesitant dimensions in addition to the affiliation degree it is widely used in pattern recognition[12 –19], MADM[20 –24] and other fields. Joshi and Kumar [25] formulated the dissimilarity Jensen-Shannon-based divergence information measure for IFSs. Tian, Zhang, Wang and Wang [26] selected the green supplier through TOPSIS and Best-Worst algorithms under IFSs. Ye [27] formulated the intuitionistic fuzzy hybrid arithmetic & geometric fused operators. Li, Chen, Yang and Li [28] formulated the VIKOR for IF-MADM. Arya and Yadav [29] formulated the intuitionistic fuzzy super-efficiency with information measure. Garg [30] formulated the cosine similarity measure under IFSs. Su, Chen, Xia and Wang [31] constructed the interactive model for IF-MAGDM. Zhao et al. [32] formulated the interactive intuitionistic fuzzy MADM. Yu, Xu and Liu [33] formulated the derivatives & differentials with multiplicative intuitionistic fuzzy sets. Wu and Zhang [34] formulated IF-MADM pattern with weighted entropy. Niroomand [35] formulated linear programming with IFSs arguments. Li and Wan [36] expressed the minimum weighted minkowski distance for IF-MADM. Verma and Sharma [37] formulated some measures for IF-MADM. De and Sana [38] formulated random demand methods based on some Bonferroni mean under IFSs. Tan [39] formulated the Choquet-TOPSIS for IF-MADM. Li [40] formulated the GOWA for IF-MADM. Joshi et al. [41] defined the dissimilarity measure with IFSs. Yu et al. [33] formulated some derivatives and differentials for IFSs. Yu [42] formulated the generalized prioritized geometric under IFSs. Gong et al. [43] formulated intuitionistic fuzzy preference decision relations with utility functions. Wu et al. [44] formulated the MADM framework for project portfolio under IFSs.

With the increasingly fierce competition in the construction market, construction companies need to take a series of measures to overcome their disadvantages, pay more attention to management, enhance market competitiveness and improve profit margins[45 –49]. Developing strategic partners, including suppliers, is a new path in front of construction companies. The one-time nature of construction projects makes the construction supply chain temporary in nature. Construction enterprises need to organize project management departments for each project to form a new supply relationship based on the project, and after the project is completed, the corresponding project management department will be abolished and the temporary supply relationship will be dissolved. Thus, the purchaser and the supplier are usually in a temporary and transient trading relationship. It is because of this lack of coordination and communication in the short-term trading relationship, which makes the supply and demand sides fall into an unreasonable circle of competition rather than cooperation[50 –54]. Both parties seek to maximize personal short-term interests and avoid excessive support in the cooperative relationship to deal with the risk of speculative behavior, the purchaser uses price competition among suppliers to keep the price of materials as low as possible, while suppliers use the market situation of tight supply material prices or certain practical difficulties of the purchaser (such as emergency procurement) to raise prices, even at the expense of reducing the quality of materials. At the same time, how to choose the right supplier is very important, whether to establish a general trading relationship with suppliers or strategic partnership, need to build on the basis of the selection of suitable suppliers. In the face of today’s construction market competition, scientific and reasonable selection of material suppliers is important to ensure the construction progress and quality of construction projects, as well as to reduce costs and improve the economic benefits of the project[55 –59]. The construction supply chain is demand-driven, a supply chain that provides construction products and after-sales services for users according to their needs. It is longer, broader and involves more participants than the general supply chain, and the characteristics of the construction supply chain determine the complexity and diversity of its structure. Material suppliers occupy a pivotal position in the construction supply chain. They are located at the front end of the whole supply chain and are responsible for providing basic tools and raw materials for the whole construction project, and the quality of suppliers is directly related to the normal operation of the whole project. Therefore, choosing high-quality suppliers and ensuring high-efficiency and high-quality material supply can ensure the successful completion of construction projects. The problems of building material suppliers are the MADM [60 –68]. In this paper, the IFSs is listed and the I-IFHPOWA operator is devised on IOWA operator[69]. The good properties of I-IFHOWA operator are studied. Then, the I-IFHPOWA operator is formulated for MADM under defined IFSs. Finally, the real example for building material suppliers is formulated to test the I-IFHPOWA operator. In order to do such study, the IFSs is formulated in Sec. 2. the I-IFHPOWA is formulated on the IOWA in Sec. 3. The I-IFHPOWA operator is constructed for MADM under IFSs in Sec. 4. the building material suppliers is formulated to show the superiority in Sec. 5. some comparative studies are formulated in Sec. 6. The study conclusion is formulated in Sec. 7.

2 Preliminaries

2.1 IFSs

Atanassov [11] produced the IFSs.

Definition 1 [11]. Let L be a fixed set, An IFS Q on L is:

$Q = {〈 l, {lm}_{q} (l), {lx}_{q} (l) 〉 | l \in L}$ (1) where 0 ⩽ lm_q (l) ⩽ 1 and 0 ⩽ lx_q (l) ⩽ 1, and 0 ⩽ lm_q (l) + lx_q (l) ⩽ 1,∀q ∈ Q.

Definition 2 [70]. Let lq = (lm, lx) be the IFN, the score is:

$S (lq) = lm + lm (1 - lm - lx), S (lp) \in [- 1, 1] .$ (2)

Definition 3 [70]. Let lq = (lm, lx) be the IFN, the accuracy degree is:

$H (lq) = lm + lx, H (lq) \in [0, 1] .$ (3)

Definition 4 [70]. Let lq₁ = (lm₁, lx₁) and lq₂ = (lm₂, lx₂) be IFNs:

$\begin{matrix} S ({lq}_{1}) = & {lm}_{1} + {lm}_{1} (1 - {lm}_{1} - {lx}_{1}), S ({lq}_{2}) \\ = {lm}_{2} + {lm}_{2} (1 - {lm}_{2} - {lx}_{2}) \end{matrix}$ (4)

$H ({lq}_{1}) = {lm}_{1} + {lx}_{1}, H ({lq}_{2}) = {lm}_{2} + {lx}_{2}$ (5)

For two IFNs lq₁ = (lm₁, lx₁) and lq₂ = (lm₂, lx₂), then $\begin{matrix} (1) if S ({lq}_{1}) < S ({lq}_{2}), then {lq}_{1} < {lq}_{2}; \\ (2) if S ({lq}_{1}) = S ({lq}_{2}), H ({lq}_{1}) < H ({lq}_{2}), then \\ {lq}_{1} < {lq}_{2}; if H ({lq}_{1}) = H ({lq}_{2}), then {lq}_{1} = {lq}_{2} . \end{matrix}$

2.2 Hamacher operations of IFSs

Huang [71] studied the Hamacher basic operations for IFSs. The Hamacher product (lq₁ ⊗ lq₂) and Hamacher sum (lq₁ ⊕ lq₂) on two IFSs lq₁ = (lm₁, lx₁) and lq₂ = (lm₂, lx₂) is listed. $\begin{matrix} {lq}_{1} \otimes {lq}_{2} \\ = (\frac{{lm}_{1} . {lm}_{2}}{1 + (1 - {lm}_{1}) (1 - {lm}_{2})}, \frac{{lx}_{1} + {lx}_{2}}{1 + {lx}_{1} {lx}_{2}}); \\ {lq}_{1} \oplus {lq}_{2} \\ = (\frac{{lm}_{1} + {lm}_{2}}{1 + {lm}_{1} {lm}_{2}}, \frac{{lx}_{1} {lx}_{2}}{1 + (1 - {lx}_{1}) (1 - {lx}_{2})}); \end{matrix}$

$\begin{matrix} λ {lq}_{1} = & (\frac{{(1 + {lm}_{1})}^{λ} - {(1 - {lm}_{1})}^{λ}}{{(1 + {lm}_{1})}^{λ} + {(1 - {lm}_{1})}^{λ}}, \frac{2 {lx}_{1}^{λ}}{(2 - {lx}_{1})^{λ} + {lx}_{1}^{λ}}), \\ λ > 0; \\ {({lq}_{1})}^{λ} = & (\frac{2 {lm}_{1}^{λ}}{(2 - {lm}_{1})^{λ} + {lm}_{1}^{λ}}, \frac{{(1 + {lx}_{1})}^{λ} - {(1 - {lx}_{1})}^{λ}}{{(1 + {lx}_{1})}^{λ} + {(1 - {lx}_{1})}^{λ}}), \\ λ > 0 . \end{matrix}$

3 I-IFHPOWA operator

Then, the IFHWA operator [71] is formulated.

Definition 5 [71]. Let lq_a = (lm_a, lx_a) (a = 1, 2, ⋯ , c), IFHWA: Qⁿ → Q, if

$\begin{matrix} {IFHWA}_{l ω} ({lq}_{1}, {lq}_{2}, \dots, {lq}_{c}) = \oplus_{a = 1}^{c} (l ω_{a} {lq}_{a}) \\ = (\begin{matrix} \frac{\prod_{a = 1}^{c} {(1 + (l γ - 1) {lm}_{a})}^{l ω_{a}} - \prod_{a = 1}^{c} {(1 - {lm}_{a})}^{l ω_{a}}}{\prod_{a = 1}^{c} {(1 + (l γ - 1) {lm}_{a})}^{l ω_{a}} + (l γ - 1) \prod_{a = 1}^{c} {(1 - {lm}_{a})}^{l ω_{a}}}, \\ \frac{l γ \prod_{a = 1}^{c} {lx}_{a}^{l ω_{a}}}{\prod_{a = 1}^{c} {(1 + (l γ - 1) (1 - {lx}_{a}))}^{l ω_{a}} + (l γ - 1) \prod_{a = 1}^{c} {lx}_{a}^{l ω_{a}}} \end{matrix}) \end{matrix}$ (6)

where lω = (lω₁, lω₂, ⋯ , lω_c) ^T be weight, lω_a > 0, $\sum_{a = 1}^{c} l ω_{a} = 1$ .

Sun [72] formulated the IFHPWA operator.

Definition 6 [72]. Let lq_a = (lm_a, lx_a) (a = 1, 2, ⋯ , r), IFHPWA: Qⁿ → Q, if

$\begin{matrix} {IFHPWA}_{l ω} ({lq}_{1}, {lq}_{2}, \dots, {lq}_{r}) = \oplus_{a = 1}^{r} (\frac{l ω_{a} (1 + T ({lq}_{a})) {lq}_{a}}{\sum_{a = 1}^{c} l ω_{a} (1 + T ({lq}_{a}))}) \\ = (\begin{matrix} \frac{\prod_{a = 1}^{r} {(1 + (l γ - 1) {lm}_{a})}^{l ω_{a} (1 + T ({lq}_{a})) / \sum_{a = 1}^{r} l ω_{a} (1 + T ({lq}_{a}))} - \prod_{a = 1}^{r} {(1 - {lm}_{a})}^{l ω_{a} (1 + T ({lq}_{a})) / \sum_{a = 1}^{r} l ω_{a} (1 + T ({lq}_{a}))}}{\prod_{a = 1}^{r} {(1 + (l γ - 1) {lm}_{a})}^{l ω_{a} (1 + T ({lq}_{a})) / \sum_{a = 1}^{r} l ω_{a} (1 + T ({lq}_{a}))} + (l γ - 1) \prod_{a = 1}^{r} {(1 - {lm}_{a})}^{l ω_{a} (1 + T ({lq}_{a})) / \sum_{a = 1}^{r} l ω_{a} (1 + T ({lq}_{a}))}}, \\ \frac{l γ \prod_{a = 1}^{r} {lx}_{a}^{l ω_{a} (1 + T ({lq}_{a})) / \sum_{a = 1}^{r} l ω_{a} (1 + T ({lq}_{a}))}}{\prod_{a = 1}^{r} {(1 + (l γ - 1) (1 - {lx}_{a}))}^{l ω_{a} (1 + T ({lq}_{a})) / \sum_{a = 1}^{r} l ω_{a} (1 + T ({lq}_{a}))} + (l γ - 1) \prod_{a = 1}^{r} {lx}_{a}^{l ω_{a} (1 + T ({lq}_{a})) / \sum_{a = 1}^{r} l ω_{a} (1 + T ({lq}_{a}))}} \end{matrix}) \end{matrix}$ (7)

where lω = (lω₁, lω₂, ⋯ , lω_r) ^T be weight, lω_a > 0, $\sum_{a = 1}^{r} l ω_{a} = 1$ , and

$T ({lq}_{a}) = \sum_{\underset{a \neq b}{a = 1}}^{c} l ω_{a} Sup ({lq}_{b}, {lq}_{a})$ (8) and Sup (lq_b, lq_a) is information support for lq_b from lq_a, with formulated conditions: (1) Sup (lq_a, lq_b) ∈ [0, 1]; (2) Sup (lq_b, lq_a) = Sup (lq_a, lq_b); (3) Sup (lq_a, lq_b) ⩾ Sup (lq_s, lq_t). If d (lq_a, lq_b) ⩽ d (lq_s, lq_t), where d is a distance measure.

Yager and Filev [69] formulated the given IOWA operator with OWA [73, 74].

Definition 7 [69]. An IOWA is built:

$\begin{matrix} IOWA (〈 {lu}_{1}, {lq}_{1} 〉, 〈 {lu}_{2}, {lq}_{2} 〉, \dots, 〈 {lu}_{c}, {lq}_{c} 〉) \\ = \sum_{a = 1}^{c} w_{a} {lq}_{σ (a)} \end{matrix}$ (9)

lq_σ(a) is the lq_i of OWA pair 〈lu_a, lq_a〉 having the i-th biggest lu_a (lu_a ∈ [0, 1]), lu_i in 〈lu_i, lq_i〉 is built to as order induced argument and lq_i are argument.

Then, the I-IFHPOWA operator is formulated on IOWA operator[69].

Definition 8. Let 〈lu_a, lq_a 〉 (a = 1, 2, ⋯ , c) be a group of 2-tuples, the I-IFHPOWA is:

$\begin{matrix} {I - IFHPOWA}_{lw} (〈 {lu}_{1}, {lq}_{1} 〉, 〈 {lu}_{2}, {lq}_{2} 〉, \dots, 〈 {lu}_{c}, {lq}_{c} 〉) \\ = \oplus_{a = 1}^{c} (\frac{{lw}_{a} (1 + T ({lq}_{σ (a)})) {lq}_{σ (a)}}{\sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))}) \end{matrix}$ (10)

where lw = (lw₁, lw₂, ⋯ , lw_c) ^T is a weight, lw_a > 0, $\sum_{a = 1}^{c} {lw}_{a} = 1$ , lq_σ(a) is lq_a of IFHPOWA pair 〈lu_i, lq_i〉 having the j-th biggest lu_i (lu_i ∈ [0, 1]), lu_i is order inducing values and lq_i are IFNs.

Theorem 1. The fused information for I-IFHPOWA is still the IFNs, and

$\begin{matrix} \begin{matrix} {I - IFHPOWA}_{lw} (〈 {lu}_{1}, {lq}_{1} 〉, 〈 {lu}_{2}, {lq}_{2} 〉, \dots, 〈 {lu}_{c}, {lq}_{c} 〉) \\ = \oplus_{a = 1}^{c} (\frac{{lw}_{a} (1 + T ({lq}_{σ (a)})) {lq}_{σ (a)}}{\sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))}) \\ = (\begin{matrix} \frac{\prod_{a = 1}^{c} {(1 + (l γ - 1) {lm}_{σ (a)})}^{{lw}_{a} (1 + T ({lq}_{σ (a)})) / \sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))} - \prod_{a = 1}^{c} {(1 - {lm}_{σ (a)})}^{{lw}_{a} (1 + T ({lq}_{σ (a)})) / \sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))}}{\prod_{a = 1}^{c} {(1 + (l γ - 1) {lm}_{σ (a)})}^{{lw}_{a} (1 + T ({lq}_{σ (a)})) / \sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))} + (l γ - 1) \prod_{a = 1}^{c} {(1 - {lm}_{σ (a)})}^{{lw}_{a} (1 + T ({lq}_{σ (a)})) / \sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))}}, \\ \frac{l γ \prod_{a = 1}^{c} {lx}_{σ (a)}^{{lw}_{a} (1 + T ({lq}_{σ (a)})) / \sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))}}{\prod_{a = 1}^{c} {(1 + (l γ - 1) (1 - {lx}_{σ (a)}))}^{{lw}_{a} (1 + T ({lq}_{σ (a)})) / \sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))} + (l γ - 1) \prod_{a = 1}^{c} {lx}_{σ (a)}^{{lw}_{a} (1 + T ({lq}_{σ (a)})) / \sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))}} \end{matrix}) \end{matrix} \end{matrix}$ (11)

where lw = (lw₁, lw₂, ⋯ , lw_c) ^T is a weight, lw_a > 0, $\sum_{a = 1}^{c} {lw}_{a} = 1$ , lq_σ(a) is lq_a of IFHPOWA pair 〈lu_i, lq_i〉 which has the j-th biggest lu_i (lu_i ∈ [0, 1]), lu_i in 〈lu_i, lq_i〉 is order inducing values and lq_i are IFNs.

Now, the special cases of the I-IFHPOWA are formulated:

(1) If lu_j = lq_j for all j, then I-IFHPOWA becomes the IFHPOWA:

$\begin{matrix} \begin{matrix} = {IFHPOWA}_{lw} (〈 {lu}_{1}, {lq}_{1} 〉, 〈 {lu}_{2}, {lq}_{2} 〉, \dots, 〈 {lu}_{c}, {lq}_{c} 〉) \\ = \oplus_{a = 1}^{c} (\frac{{lw}_{a} (1 + T ({lq}_{σ (a)})) {lq}_{σ (a)}}{\sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))}) \\ = (\begin{matrix} \frac{\prod_{a = 1}^{c} {(1 + (l γ - 1) {lm}_{σ (a)})}^{{lw}_{a} (1 + T ({lq}_{σ (a)})) / \sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))} - \prod_{a = 1}^{c} {(1 - {lm}_{σ (a)})}^{{lw}_{a} (1 + T ({lq}_{σ (a)})) / \sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))}}{\prod_{a = 1}^{c} {(1 + (l γ - 1) {lm}_{σ (a)})}^{{lw}_{a} (1 + T ({lq}_{σ (a)})) / \sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))} + (l γ - 1) \prod_{a = 1}^{c} {(1 - {lm}_{σ (a)})}^{{lw}_{a} (1 + T ({lq}_{σ (a)})) / \sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))}}, \\ \frac{l γ \prod_{a = 1}^{c} {lx}_{σ (a)}^{{lw}_{a} (1 + T ({lq}_{σ (a)})) / \sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))}}{\prod_{a = 1}^{c} {(1 + (l γ - 1) (1 - {lx}_{σ (a)}))}^{{lw}_{a} (1 + T ({lq}_{σ (a)})) / \sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))} + (l γ - 1) \prod_{a = 1}^{c} {lx}_{σ (a)}^{{lw}_{a} (1 + T ({lq}_{σ (a)})) / \sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))}} \end{matrix}) \end{matrix} \end{matrix}$ (12)

here (σ (1) , σ (2) , ⋯ , σ (c)) is an any permutation of (1, 2, ⋯ , c), lq_σ(a-1) ⩾ lq_σ(a) for all a = 2, ⋯ , c.

(1) If lu_a = No . a for all a, if a is ordered information of 〈lu_a, lq_a〉, then I-IFHPOWA reduces to the IFHPWA:

$\begin{matrix} {I - IFHPOWA}_{lw} (〈 {lu}_{1}, {lq}_{1} 〉, 〈 {lu}_{2}, {lq}_{2} 〉, \dots, 〈 {lu}_{c}, {lq}_{c} 〉) = \oplus_{a = 1}^{c} (\frac{{lw}_{a} (1 + T ({lq}_{σ (a)})) {lq}_{σ (a)}}{\sum_{a = 1}^{c} {lw}_{a} (1 + T ({lq}_{σ (a)}))}) \\ {IFHPWA}_{l ω} ({lq}_{1}, {lq}_{2}, \dots, {lq}_{c}) = \oplus_{a = 1}^{c} (\frac{l ω_{a} (1 + T ({lq}_{a})) {lq}_{a}}{\sum_{a = 1}^{c} l ω_{a} (1 + T ({lq}_{a}))}) \\ = (\begin{matrix} \frac{\prod_{a = 1}^{c} {(1 + (l γ - 1) {lm}_{a})}^{l ω_{a} (1 + T ({lq}_{a})) / \sum_{a = 1}^{c} l ω_{a} (1 + T ({lq}_{a}))} - \prod_{a = 1}^{c} {(1 - i_{a})}^{l ω_{a} (1 + T ({lq}_{a})) / \sum_{a = 1}^{c} l ω_{a} (1 + T ({lq}_{a}))}}{\prod_{a = 1}^{c} {(1 + (l γ - 1) i_{a})}^{l ω_{a} (1 + T ({lq}_{a})) / \sum_{a = 1}^{c} l ω_{a} (1 + T ({lq}_{a}))} + (l γ - 1) \prod_{a = 1}^{c} {(1 - i_{a})}^{l ω_{a} (1 + T ({lq}_{a})) / \sum_{a = 1}^{c} l ω_{a} (1 + T ({lq}_{a}))}}, \\ \frac{l γ \prod_{a = 1}^{c} {lx}_{a}^{l ω_{a} (1 + T ({lq}_{a})) / \sum_{a = 1}^{c} l ω_{a} (1 + T ({lq}_{a}))}}{\prod_{a = 1}^{c} {(1 + (l γ - 1) (1 - {lx}_{a}))}^{l ω_{a} (1 + T ({lq}_{a})) / \sum_{a = 1}^{c} l ω_{a} (1 + T ({lq}_{a}))} + (l γ - 1) \prod_{a = 1}^{c} {lx}_{a}^{l ω_{a} (1 + T ({lq}_{a})) / \sum_{a = 1}^{c} l ω_{a} (1 + T ({lq}_{a}))}} \end{matrix}) \end{matrix}$ (13)

where lw = (lw₁, lw₂, ⋯ , lw_c) ^T is a weight, lw_a > 0, $\sum_{a = 1}^{c} {lw}_{a} = 1$ .

It is easily formulated that the I-IFHPOWA has given basic properties.

Theorem 2 (Idempotency). If all lq_a (a = 1, 2, ⋯ , c) are equal, i.e. lq_a = lq for all a, then

$\begin{matrix} {I - IFHPOWA}_{lw} (〈 {lu}_{1}, {lq}_{1} 〉, 〈 {lu}_{2}, {lq}_{2} 〉, \\ \dots, 〈 {lu}_{c}, {lq}_{c} 〉) = lq \end{matrix}$ (14)

Theorem 3 (Boundedness). Let 〈lu_a, lq_a 〉 (a = 1, 2, ⋯ , c), and let ${lq}^{-} = min_{j} {lq}_{j}, {lq}^{+} = max_{a} {lq}_{a}$

Then

$\begin{matrix} {lq}^{-} ⩽ & {I - IFHPOWA}_{lw} (〈 {lu}_{1}, {lq}_{1} 〉, 〈 {lu}_{2}, {lq}_{2} 〉, \\ \dots, 〈 {lu}_{c}, {lq}_{c} 〉) ⩽ {lq}^{+} \end{matrix}$ (15)

Theorem 4 (Monotonicity). Let 〈lu_a, lq_a 〉 (a = 1, 2, ⋯ , c), $〈 {lq}_{a}^{'}, {lq}_{a}^{'} 〉 (a = 1, 2, \dots, c)$ , if ${lq}_{a} ⩽ {lq}_{a}^{'}$ , for all a, then

$\begin{matrix} {I - IFHPOWA}_{lw} (〈 {lu}_{1}, {lq}_{1} 〉, 〈 {lu}_{2}, {lq}_{2} 〉, \dots, 〈 {lu}_{c}, {lq}_{c} 〉) \\ ⩽ {I - IFHPOWA}_{lw} (〈 {lu}_{1}^{'}, {lq}_{1}^{'} 〉, 〈 {lu}_{2}^{'}, {lq}_{2}^{'} 〉, \dots, \\ 〈 {lu}_{c}^{'}, {lq}_{c}^{'} 〉) \end{matrix}$ (16)

Theorem 5 (Commutativity). Let 〈lu_a, lq_a 〉 (a = 1, 2, ⋯ , c), $〈 {lq}_{a}^{'}, {lq}_{a}^{'} 〉 (a = 1, 2, \dots, c)$ , then

$\begin{matrix} {I - IFHPOWA}_{lw} (〈 {lu}_{1}, {lq}_{1} 〉, 〈 {lu}_{2}, {lq}_{2} 〉, \dots, 〈 {lu}_{c}, {lq}_{c} 〉) \\ = {I - IFHPOWA}_{lw} (〈 {lu}_{1}^{'}, {lq}_{1}^{'} 〉, 〈 {lu}_{2}^{'}, {lq}_{2}^{'} 〉, \dots, \\ 〈 {lu}_{c}^{'}, {lq}_{c}^{'} 〉) \end{matrix}$ (17)

where 〈lu_a, lq_a〉 is permutation of $〈 {lq}_{a}^{'}, {lq}_{a}^{'} 〉 (a = 1, 2, \dots, c)$ .

4 Method for MADM problem with IFNs

In such formulated section, the MADM are formulated with I-IFHOWA operator. Let LA ={ LA₁, LA₂, ⋯ , LA_m } be a group of possible solutions, and LG ={ LG₁, LG₂, ⋯ , LG_n } be the group of criterions. Let lw = (lw₁, lw₂, ⋯ , lw_c) be weight information, where lw_a ⩾ 0, $\sum_{a = 1}^{c} {lw}_{a} = 1$ . Suppose that Q = (lq_uv) _m×n = (lm_uv, lx_uv) _m×n is IF-matrix, lm_uv ∈ [0, 1], lx_uv ∈ [0, 1], lm_uv + lx_uv ⩽ 1, u = 1, 2, ⋯ , m, v = 1, 2, ⋯ , n.

Then, the I-IFHPOWA operator is formulated for MADM with IFNs.

Step 1. Utilize the Q = (lq_uv) _m×n = (lm_uv, lx_uv) _m×n and I-IFHPOWA

${lq}_{u} = ({lm}_{u}, {lx}_{u}) {= I - IFHPOWA}_{lw} (〈 {lu}_{u 1}, {lq}_{u 1} 〉, 〈 {lu}_{u 2}, {lq}_{u 2} 〉, \dots, 〈 {lu}_{uv}, {lq}_{uv} 〉)$

$\begin{matrix} = \oplus_{v = 1}^{n} (\frac{{lw}_{a} (1 + T ({lq}_{σ (uv)})) {lq}_{σ (uv)}}{\sum_{v = 1}^{n} {lw}_{a} (1 + T ({lq}_{σ (uv)}))}) \\ = (\begin{matrix} \frac{\prod_{v = 1}^{n} {(1 + (l γ - 1) {lm}_{σ (uv)})}^{{lw}_{a} (1 + T ({lq}_{σ (uv)})) / \sum_{v = 1}^{n} {lw}_{a} (1 + T ({lq}_{σ (uv)}))} - \prod_{v = 1}^{n} {(1 - {lm}_{σ (uv)})}^{{lw}_{a} (1 + T ({lq}_{σ (uv)})) / \sum_{v = 1}^{n} {lw}_{a} (1 + T ({lq}_{σ (uv)}))}}{\prod_{a = 1}^{c} {(1 + (l γ - 1) {lm}_{σ (uv)})}^{{lw}_{a} (1 + T ({lq}_{σ (uv)})) / \sum_{v = 1}^{n} {lw}_{a} (1 + T ({lq}_{σ (uv)}))} + (l γ - 1) \prod_{a = 1}^{c} {(1 - {lm}_{σ (uv)})}^{{lw}_{a} (1 + T ({lq}_{σ (uv)})) / \sum_{v = 1}^{n} {lw}_{a} (1 + T ({lq}_{σ (uv)}))}}, \\ \frac{l γ \prod_{v = 1}^{n} {lx}_{σ (uv)}^{{lw}_{a} (1 + T ({lq}_{σ (uv)})) / \sum_{v = 1}^{n} {lw}_{a} (1 + T ({lq}_{σ (uv)}))}}{\prod_{v = 1}^{n} {(1 + (l γ - 1) (1 - {lx}_{σ (uv)}))}^{{lw}_{a} (1 + T ({lq}_{σ (uv)})) / \sum_{v = 1}^{n} {lw}_{a} (1 + T ({lq}_{σ (uv)}))} + (l γ - 1) \prod_{v = 1}^{n} {lx}_{σ (uv)}^{{lw}_{a} (1 + T ({lq}_{σ (uv)})) / \sum_{v = 1}^{n} {lw}_{a} (1 + T ({lq}_{σ (uv)}))}} \end{matrix}) \end{matrix}$ (18)

to formulated the overall values lq_u (u = 1, 2, ⋯ , m), lw = (lw₁, lw₂, ⋯ , lw_c) is the associated vector of the I-IFHPOWA operator, lw_v > 0 and $\sum_{v = 1}^{n} {lw}_{v} = 1$ .

Step 2. Derive the S (lq_u) and H (lq_u).

Step 3. Rank the LA_u (u = 1, 2, ⋯ , m) and derive the best choice with S (lq_u) and H (lq_u).

Step 4. End.

5 Numerical example

The construction industry is the basic industry of the country, China’s construction industry due to low technology content, low production efficiency and other issues leading to low competitiveness. The competitiveness of the construction industry is low due to low technology and low production efficiency. Introducing the idea of supply chain management into the construction industry is one of the important ways to improve the competitiveness of construction enterprises. One of the major issues to be solved is the supplier evaluation for the implementation of supply chain management. In view of the extreme importance of material cost in the project cost, this paper focuses on the evaluation of material suppliers. The evaluation of material suppliers is characterized by many indicators, large amount of data, high information correlation, low transparency, etc., with highly unstructured It is difficult to solve the problem by a single method because of its complex decision-making process. One of the effective means to solve this problem is to apply modern One of the effective means to solve this problem is to apply modern evaluation theories and new algorithms to the practice of construction material supplier evaluation to optimize the efficiency of evaluation. The second is to use advanced information processing technology to establish an intelligent decision support system for construction material supplier evaluation. The selection of suppliers is crucial for construction companies. A quality supplier is an effective guarantee to improve the quality of the enterprise products, enhance the credibility of the market and improve the competitiveness of the enterprise market. First of all, by delivering high quality raw materials, it can effectively improve the various strengths of construction products and enhance their durability. At the same time, it can also ensure the timeliness, continuity and controllability of raw material supply, effectively avoiding various conditions arising from material supply problems, such as missed work and delays caused by untimely supply. Product performance is becoming increasingly complex, and suppliers with strong strength have strong new product development capabilities to meet the individual needs of users for products, thus providing high quality and high performance products. Finally, the long-term cooperation, mutual trust and information sharing between the two parties will improve the market competitiveness of both companies and realize the long-term benefits of the enterprises. The selection and application of building material suppliers is the MADM. This section presents useful numerical example to select the building material suppliers. There is a panel with five possible building material suppliers LA_u (u = 1, 2, 3, 4, 5) to evaluate. The decision department selects four attributes to depict the five possible building material suppliers: ding172LG₁ is the product quality; ding173LG₂ is the delivery capacity; ding174LG₃ is the service level; ding175LG₄ is the enterprise performance. The five possible building material suppliers LA_u (u = 1, 2, 3, 4, 5) are to be evaluated through the IFNs (whose weight information w = (0.2, 0.3, 0.3, 0.2) ^T), as depicted in Table 1.

Table 1
Intuitionistic fuzzy matrix

Alternatives LG₁ LG₂ LG₃ LG₄

LA₁ (0.522, 0.373) (0.632, 0.254) (0.421, 0.576) (0.821, 0.176)

LA₂ (0.252, 0.427) (0.565, 0.332) (0.654, 0.321) (0.478, 0.186)

LA₃ (0.832, 0.146) (0.687, 0.165) (0.476, 0.398) (0.722, 0.276)

LA₄ (0.765, 0.187) (0.343, 0.476) (0.598, 0.387) (0.642, 0.399)

LA₅ (0.721, 0.276) (0.598, 0.254) (0.712, 0.267) (0.647, 0.179)

Alternatives	LG₁	LG₂	LG₃	LG₄
LA₁	(0.522, 0.373)	(0.632, 0.254)	(0.421, 0.576)	(0.821, 0.176)
LA₂	(0.252, 0.427)	(0.565, 0.332)	(0.654, 0.321)	(0.478, 0.186)
LA₃	(0.832, 0.146)	(0.687, 0.165)	(0.476, 0.398)	(0.722, 0.276)
LA₄	(0.765, 0.187)	(0.343, 0.476)	(0.598, 0.387)	(0.642, 0.399)
LA₅	(0.721, 0.276)	(0.598, 0.254)	(0.712, 0.267)	(0.647, 0.179)

Then, the I-IFHPOWA operator is applied to building material supplier selection.

Step 1. If the I-IFHPOWA operator is applied to MADM with IFNs, the weight of I-IFHPOWA operator is supposed that: w = (0.30, 0 . 20, 0 . 40, 0 . 10).

Step 2. The invited famous experts use induced arguments to depict given complex opinions of board members. The complex opinions are always formulated in Table 2.

Table 2

Inducing arguments

	LG₁	LG₂	LG₃	LG₄
LA₁	17	13	19	12
LA₂	23	24	20	13
LA₃	18	13	21	15
LA₄	12	9	15	13
LA₅	19	22	11	20

Step 3. The decision values lq_u formulated with I-IFHPOWA of the college training teachers is: $\begin{matrix} {lq}_{1} = (0 . 3476, 0 . 5459), {lq}_{2} = (0 . 5267, 0 . 3379), \\ {lq}_{2} = (0 . 4785, 0 . 4879) \\ {lq}_{4} = (0 . 4429, 0 . 2988), {lq}_{4} = (0 . 3819, 0 . 4907) \end{matrix}$

Step 4. Calculate the given scores S (lq_u) $\begin{matrix} S ({lq}_{1}) = - 0.0948, S ({lq}_{2}) = 0.4752, \\ S ({lq}_{3}) = 0.2708 \\ S ({lq}_{4}) = 0.3979, S ({lq}_{5}) = - 0.0754 \end{matrix}$

Step 5. Rank all the LA_u (u = 1, 2, 3, 4, 5) through S (lq_u): LA₂ ≻ LA₄ ≻ LA₃ ≻ LA₅ ≻ LA₁, and the most desirable foreign college is LA₂.

6 Compare analysis

Firstly, compared with IFWA [75] & IFWG [76]. For IFWA, the calculation is: S (lq₁) = 0 . 0798, S (lq₂) = 0 . 1436, S (lq₃) = 0 . 3345, S (lq₄) = 0 . 0495, S (lq₅) = 0 . 0479 . Thus, the ranking order is LA₂ ≻ LA₄ ≻ LA₃ ≻ LA₅ ≻ LA₁. For IFWG, the calculation is S (lq₁) = -0.0187, S (lq₂) = 0 . 1349, S (lq₃) = 0 . 3365, S (lq₄) = 0 . 0379, S (lq₅) = 0 . 0087 . So the ranking order is LA₂ ≻ LA₄ ≻ LA₃ ≻ LA₅ ≻ LA₁.

What’s more, compared with IF-VIKOR method [77]. The value is: IFQ₁ = 1 . 0000, IFQ₂ = 0 . 0000, IFQ₃ = 0 . 7987, IFQ₄ = 0 . 5346, IFQ₅ = 0 . 6909. The order is: LA₂ ≻ LA₄ ≻ LA₅ ≻ LA₃ ≻ LA₁.

Then, compared with IF-CODAS method [78]. The assessment score (AS) is: AS₁ = -0 . 8033, AS₂ = 1 . 5569, AS₃ = -0 . 3663, AS₄ = 0 . 1856 AS₅ = -0 . 4675. Therefore, the order is LA₂ ≻ LA₄ ≻ LA₅ ≻ LA₃ ≻ LA₁.

Then, the results of the different methods are in Table 3.

Table 3
Compare analysis for these methods

Methods Order The best choice The worst choice

IFWA [75] LA₂ ≻ LA₄ ≻ LA₃ ≻ LA₅ ≻ LA₁ LA ₂ LA ₁

IFWG [76] LA₂ ≻ LA₄ ≻ LA₃ ≻ LA₅ ≻ LA₁ LA ₂ LA ₁

IF-VIKOR [77] LA₂ ≻ LA₄ ≻ LA₅ ≻ LA₃ ≻ LA₁ LA ₂ LA ₁

IF-CODAS [78] LA₂ ≻ LA₄ ≻ LA₅ ≻ LA₃ ≻ LA₁ LA ₂ LA ₁

The developed model LA₂ ≻ LA₄ ≻ LA₃ ≻ LA₅ ≻ LA₁ LA ₂ LA ₁

Methods	Order	The best choice	The worst choice
IFWA [75]	LA₂ ≻ LA₄ ≻ LA₃ ≻ LA₅ ≻ LA₁	LA ₂	LA ₁
IFWG [76]	LA₂ ≻ LA₄ ≻ LA₃ ≻ LA₅ ≻ LA₁	LA ₂	LA ₁
IF-VIKOR [77]	LA₂ ≻ LA₄ ≻ LA₅ ≻ LA₃ ≻ LA₁	LA ₂	LA ₁
IF-CODAS [78]	LA₂ ≻ LA₄ ≻ LA₅ ≻ LA₃ ≻ LA₁	LA ₂	LA ₁
The developed model	LA₂ ≻ LA₄ ≻ LA₃ ≻ LA₅ ≻ LA₁	LA ₂	LA ₁

From built Table 3, the best is LA₂, while the worst is LA₂. Different given methods may formulate the MADM from different angles. These four formulated decision methods have their decision advantages. Therefore, the I-IFHPOWA operator is useful and effective than existing decision operators.

7 Conclusion

The construction industry is becoming increasingly competitive and construction companies need to enhance their competitiveness through cooperation, while the current level of cooperation between general contracting companies and suppliers in China needs to be improved. In order to adapt to the development of the construction industry, construction enterprises should adopt appropriate procurement strategies to establish certain cooperative relationships with suppliers, and different relationships should be established with different suppliers. The building material suppliers selection is viewed as the MADM issue. In this paper, the IFSs is formulated and I-IFHPOWA operator is formulated on given IOWA[69]. The properties of I-IFHOWA is formulated. Then, the I-IFHPOWA is formulated for MADM with IFSs. Finally, an example for building material suppliers selection is formulated to test the I-IFHPOWA operator.

Due to limited resources and capacity, the research in this paper has some shortcomings. This paper only investigates the construction material supplier evaluation index system, supplier selection method, and in the course of the research. In the process of research, it may not be comprehensive. In the future, we should focus on the following issues for further research: (1) we will continue working in the development and application of the built operators to other uncertain and fuzzy domains[79 –85]. (2) The established supplier selection evaluation index system may not be comprehensive and detailed enough, and needs to be further improved. The evaluation index system is only for construction projects, but in different sub-sectors, such as petroleum and petrochemical industry, water conservancy and hydropower industry, the evaluation index system is not comprehensive enough. petroleum and petrochemical industry, water conservancy and hydropower industry and so on, the evaluation index system is not fully applicable and has a certain limitation. It has some limitations. In the future, the above evaluation index system can be further improved by combining the characteristics of the industry and products to make it more rigorous. And in the future, the above evaluation index system can be further improved by combining the characteristics of industries and products, so as to make it more rigorous, scientific and detailed. (3) In terms of the model construction method, this paper adopted the I-IFHPOWA-based method and improves it, which is only one of many methods, and further research is needed in the future to come up with better and This is only one of the many methods, and further research is needed to find a better and more effective selection method. (4) With the deepening of the green construction concept, more consideration should be given to the green construction concept when selecting evaluation indexes in the future. green construction concept, and integrate the green construction concept into the evaluation and selection of construction material suppliers.

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