Optimizing building material supplier selection through integrated interval-valued intuitionistic fuzzy multi-attribute decision making

Abstract

In engineering construction, the unavoidable issue is how to choose suitable suppliers. The quality of suppliers has a direct impact on the progress, quality, and cost of the project. The selection of suppliers for construction projects in water conservancy and hydropower engineering is directly related to the cost of construction enterprises, and large-scale construction projects have stricter requirements for cost control. The investment in engineering construction is very huge, so cost control ability is a very important assessment indicator for construction project construction, and the cost of materials is a significant part of the construction project cost. Therefore, the research on the selection and optimization of building material suppliers is a topic that cannot be ignored. The building material supplier selection is a multi-attribute decision making (MADM). In this paper, some calculating laws on IVIFSs, Hamacher sum, Hamacher product are introduced, and the induced interval-valued intuitionistic fuzzy Hamacher interactive ordered weighted averaging (IVIFHIOWA) operators (I-IVIFHIOWA) operator is proposed. Meanwhile, some ideal properties of I-IVIFHIOWA operator are studied. Then, the I-IVIFHIOWA operator is employed to cope with the MADM under IVIFSs. Finally, an example for building material supplier selection is employed to test the I-IVIFHIOWA operator.

Keywords

Multi-attribute decision making (MADM)interval-valued intuitionistic fuzzy sets (IVIFSs)I-IVIFHIOWA operator building material supplier selection

1. Introduction

The construction industry is China’s pillar industry. In the next 20 years, the scale of fixed assets investment in China’s capital construction, technological transformation, real estate and other industries will remain at a high level [1, 2]. China’s construction market will also face important development opportunities in history. Urbanization, the development of the west, the revitalization of old industries, the rise of the central region, and ecological recovery will become the long-term strategic orientation of China’s capital construction, these will definitely provide broad development space for China’s construction market [3, 4, 5]. With China’s accession to the WTO, major foreign construction contractors entering the domestic construction market will undoubtedly intensify the competition and elimination of the domestic construction industry in the new situation [6, 7, 8]. Domestic construction contractors lack the practice and experience of competing with foreign contractors in the same environment and face challenges. So, how construction enterprises seize opportunities to meet challenges, build their own core competitiveness, change outdated management models, and become the focus of attention for enterprises, is also a question that more scholars and experts are pondering [9, 10]. Since the 1990s, the successful application of SCM in manufacturing practice has attracted the attention of many scholars and institutions in the field of architecture. In order to improve the competitiveness of the construction industry, many scholars at home and abroad have attempted to introduce advanced and successful supply chain management ideas from the manufacturing field into the construction industry, thus forming a research hotspot in construction supply chain management [11, 12]. According to the characteristics of the construction industry’s business activities, the construction supply chain refers to the goal of meeting the requirements of owners for construction projects, starting from the generation of project demands by owners, through project definition, project financing, project design, project construction The functional network structure composed of all relevant organizational structures involved in the construction process from the completion, acceptance, delivery, use, maintenance, and other stages of the project, until the reconstruction and expansion, and finally the demolition [13, 14]. Considering the current operating mechanism of the construction industry and the operability of CSCM, this paper gives a narrow definition of construction supply chain from the perspective of construction enterprises: construction supply chain refers to a construction network that connects material suppliers, engineering machinery equipment suppliers, subcontractors, designers and owners through information flow, logistics, and capital flow, starting from the owner’s Effective demand and taking contractors as the core enterprises [15, 16]. Supplier evaluation and selection are important aspects in the coordinated development of the supply chain [17, 18, 19, 20]. At present, some construction enterprises in our country still adopt traditional extensive methods when selecting material suppliers, or choose some building materials from small merchants and vendors without considering the various indicators of the materials. No wonder the quality of current construction projects often encounters problems? We have all heard of a Chinese proverb: “A skillful woman cannot cook without rice.” Even with good construction technology, without good building materials, the quality of construction projects cannot be guaranteed. It is not uncommon for serious problems such as delayed completion and poor quality of projects caused by errors in material supply, and the profit space in the fiercely competitive construction market is becoming smaller and smaller [21, 22]. In order for construction enterprises to seek significant development, they must integrate the supply chain. Under supply chain management, construction companies have increasingly high requirements for suppliers upstream of the supply chain in terms of material quality, delivery time, and cost [17, 23]. The quality of material suppliers not only affects the quality and cost of construction products, but also affects the entire business process of construction enterprises, and ultimately affects the competitive position of the supply chain [24, 25, 26]. How to correctly evaluate and select suitable building material suppliers has become one of the priority issues that need to be addressed in supply chain management [27, 28, 29, 30].

Decision making refers to the analytical process in which people choose from several alternative solutions in order to achieve a certain goal [31, 32, 33, 34, 35, 36, 37]. It widely exists in all aspects of society, whether it is national policies and guidelines, enterprise operation management, data analysis and algorithm selection in the Internet Age, or personal daily life, all need to make decisions [38, 39, 40, 41]. Multi-attribute decision making (MADM) has always been one of the research hotspots in the field of decision analysis [31, 32, 33, 34, 35, 36]. In view of the shortcomings of the traditional precise mathematical theory [42, 43, 44], Zadeh [45] constructed the theory of fuzzy sets (FSs) to improve the accuracy of decision-making. Atanassov [46] constructed the intuitionistic fuzzy sets (IFSs). Atanassov and Gargov [47] constructed the interval valued IFSs (IVIFSs). Until now, no one has adopted interactive Hamacher information aggregating method to solve the decision-making of building material supplier selection. Consequently, the interactive Hamacher information aggregating method [48] was employed in this study. Thus, the objective of this paper is to present some series of information aggregation operators in an IVIFSs environment. For it, a new operational law on different IVIFNs was introduced by taking the interaction between the membership and non-membership functions. Based on these new operational laws, the induced interval-valued intuitionistic fuzzy Hamacher interactive ordered weighted averaging (IVIFHIOWA) operators (I-IVIFHIOWA) operator is proposed. Meanwhile, some ideal properties of I-IVIFHIOWA operator are studied. Then, the I-IVIFHIOWA operator is employed to cope with the MADM under IVIFSs. Finally, an example for building material supplier selection is employed to test the I-IVIFHIOWA operator.

The rest of this work is organized as follows: Part 2 gives a simple introduction of the IVIFSs; Part 3 builds the I-IVIFHIOWA operator; Part 4 builds the MADM model based on the I-IVIFHIOWA operator under IVIFSs; and Part 5 illustrates an example for building material supplier selection to prove the I-IVIFHIOWA operator. Part 6 ends this work with some conclusions.

2. Preliminaries

The IVIFSs is introduced [49].

Definition 1 [49]. The interval-valued IFSs (IVIFSs) on the $X$ is:

$\displaystyle MN=\{{\langle{x,MI(x),NI(x)}\rangle|{x\in X}}\}$ (1)

where $MI(x)\subset[{0,1}]$ is membership and $NI(x)\subset[{0,1}]$ is non-membership, and $MI(x),NI(x)$ meet condition: $0\leqslant\sup MI(x)+\sup NI(x)\leqslant 1$ , $\forall x\in X$ . For convenience, we call $MN=({[{ML,MR}],[{NL,NR}]})$ is an IVIFN.

Definition 2 [50]. Let $MN_{1}=({[{ML_{1},MR_{1}}],[{NL_{1},NR_{1}}]})$ and $MN_{2}=({[{ML_{2},MR_{2}}],[{NL_{2},NR_{2}}]})$ be two IVIFNs, the operation laws are established:

$\displaystyle MN_{1}\oplus MN_{2}=\left({\begin{array}[]{l}[ML_{1}+ML_{2}-ML_{% 1}ML_{2},MR_{1}+MR_{2}-MR_{1}MR_{2}],\\ {[}NL_{1}NL_{2},NR_{1}NR_{2}{]}\\ \end{array}}\right)$ (2) $\displaystyle MN_{1}\otimes MN_{2}=\left({\begin{array}[]{l}[{ML_{1}ML_{2},MR_% {1}MR_{2}}],\\ {[}{NL_{1}+NL_{2}-NL_{1}NL_{2},NR_{1}+NR_{2}-NR_{1}NR_{2}}{]}\\ \end{array}}\right)$ (3) $\displaystyle\lambda MN_{1}=({[{1-({1-ML_{1}})^{\lambda},1-({1-MR_{1}})^{% \lambda}}],[{({NL_{1}})^{\lambda},({NR_{1}})^{\lambda}}]}),\lambda>0$ (4) $\displaystyle MN_{1}^{\lambda}\mbox{ = }({[{({ML_{1}})^{\lambda},({MR_{1}})^{% \lambda}}],[{1-({1-NL_{1}})^{\lambda},1-({1-NR_{1}})^{\lambda}}]}),\lambda>0$ (5)

From the Definition 2, the following properties are established.

$\displaystyle(1)\;MN_{1}\oplus MN_{2}=MN_{2}\oplus MN_{1},MN_{1}\otimes MN_{2}% =MN_{2}\otimes MN_{1},$ $\displaystyle\qquad({({MN_{1}})^{\lambda_{1}}})^{\lambda_{2}}=({MN_{1}})^{% \lambda_{1}\lambda_{2}};$ $\displaystyle(2)\;\lambda({MN_{1}\oplus MN_{2}})=\lambda MN_{1}\oplus\lambda MN% _{2},({MN_{1}\otimes MN_{2}})^{\lambda}=({MN_{1}})^{\lambda}\otimes({MN_{2}})^% {\lambda};$ $\displaystyle(3)\;\lambda_{1}MN_{1}\oplus\lambda_{2}MN_{1}=({\lambda_{1}+% \lambda_{2}})MN_{1},({MN_{1}})^{\lambda_{1}}\otimes({MN_{1}})^{\lambda_{2}}=({% MN_{1}})^{({\lambda_{1}+\lambda_{2}})}.$

Definition 3 [51]. Let $MN_{1}=({[{ML_{1},MR_{1}}],[{NL_{1},NR_{1}}]})$ and $MN_{2}=({[{ML_{2},MR_{2}}],[{NL_{2},NR_{2}}]})$ be IVIFNs, the score and accuracy values of $MN_{1}$ and $MN_{2}$ are established:

$\displaystyle SV({MN_{1}})=\frac{ML_{1}+ML_{1}({1-ML_{1}-NL_{1}})+MR_{1}+MR_{1% }({1-MR_{1}-NR_{1}})}{2}$ (6) $\displaystyle SV({MN_{2}})=\frac{ML_{2}+ML_{2}({1-ML_{2}-NL_{2}})+MR_{2}+MR_{2% }({1-MR_{2}-NR_{2}})}{2}$ (7) $\displaystyle AV({MN_{1}})=\frac{ML_{1}+NL_{1}+MR_{1}+NR_{1}}{2},$ (8) $\displaystyle AV({MN_{2}})=\frac{ML_{2}+NL_{2}+MR_{2}+NR_{2}}{2}$ (9)

For two IVIFNs $MN_{1}$ and $MN_{2}$ , according to Definition 3, then

$\displaystyle(1)\;\text{if }SV({MN_{1}})<SV({MN_{2}}),MN_{1}<MN_{2};$ $\displaystyle(2)\;\text{if }SV({MN_{1}})>SV({MN_{2}}),MN_{1}>MN_{2};$ $\displaystyle(3)\;\text{if }SV({MN_{1}})=SV({MN_{2}}),AV({MN_{1}})<AV({MN_{2}}% ),MN_{1}<MN_{2};$ $\displaystyle(4)\;\text{if }SV({MN_{1}})=SV({MN_{2}}),AV({MN_{1}})=AV({MN_{2}}% ),MN_{1}=MN_{2}.$

Definition 4 [52]. Let $MN_{1}=({[{ML_{1},MR_{1}}],[{NL_{1},NR_{1}}]})$ and $MN_{2}=({[{ML_{2},MR_{2}}],[{NL_{2},NR_{2}}]})$ be IVIFNs, the Euclidean distance is established:

$\displaystyle\textit{IVIFED}({MN_{1},MN_{2}})=\sqrt{\frac{1}{4}\left[{\begin{% array}[]{l}({ML_{1}-ML_{2}})^{2}+({MR_{1}-MR_{2}})^{2}\\ {}+({NL_{1}-NL_{2}})^{2}+({NR_{1}-NR_{2}})^{2}\\ \end{array}}\right]}$ (10)

Garg, Agarwal [48] introduced the improved Hamacher operations on IVIFS.

Definition 5 [48]. Let $MN_{1}=({[{ML_{1},MR_{1}}],[{NL_{1},NR_{1}}]})$ and $MN_{2}=({[{ML_{2},MR_{2}}],[{NL_{2},NR_{2}}]})$ be IVIFNs, then, the new operational rules for these IVIFNs are defined as follows:

$\displaystyle MN_{1}\oplus MN_{2}=\left({\begin{array}[]{l}\left[{\begin{array% }[]{l}\frac{\prod\limits_{i=1}^{2}{({1+({\gamma-1})ML_{i}})-\prod\limits_{i=1}% ^{2}{({1-ML_{i}})}}}{\prod\limits_{i=1}^{2}{({1+({\gamma-1})ML_{i}})-({\gamma-% 1})\prod\limits_{i=1}^{2}{({1-ML_{i}})}}},\\ \frac{\prod\limits_{i=1}^{2}{({1+({\gamma-1})MR_{i}})-\prod\limits_{i=1}^{2}{(% {1-MR_{i}})}}}{\prod\limits_{i=1}^{2}{({1+({\gamma-1})MR_{i}})-({\gamma-1})% \prod\limits_{i=1}^{2}{({1-MR_{i}})}}}\\ \end{array}}\right],\\ \\ \left[{\begin{array}[]{l}\frac{\gamma\prod\limits_{i=1}^{2}{({1-ML_{i}})}-% \gamma\prod\limits_{i=1}^{2}{({1-ML_{i}-NL_{i}})}}{\prod\limits_{i=1}^{2}{({1+% ({\gamma-1})ML_{i}})-({\gamma-1})\prod\limits_{i=1}^{2}{({1-ML_{i}})}}},\\ \frac{\gamma\prod\limits_{i=1}^{2}{({1-MR_{i}})}-\gamma\prod\limits_{i=1}^{2}{% ({1-MR_{i}-NR_{i}})}}{\prod\limits_{i=1}^{2}{({1+({\gamma-1})MR_{i}})-({\gamma% -1})\prod\limits_{i=1}^{2}{({1-MR_{i}})}}}\\ \end{array}}\right]\\ \end{array}}\right);$ $\displaystyle MN_{1}\otimes MN_{2}=\left({\begin{array}[]{l}\left[{\begin{% array}[]{l}\frac{\gamma\prod\limits_{i=1}^{2}{({1-ML_{i}})}-\gamma\prod\limits% _{i=1}^{2}{({1-ML_{i}-NL_{i}})}}{\prod\limits_{i=1}^{2}{({1+({\gamma-1})ML_{i}% })-({\gamma-1})\prod\limits_{i=1}^{2}{({1-ML_{i}})}}},\\ \frac{\gamma\prod\limits_{i=1}^{2}{({1-MR_{i}})}-\gamma\prod\limits_{i=1}^{2}{% ({1-MR_{i}-NR_{i}})}}{\prod\limits_{i=1}^{2}{({1+({\gamma-1})MR_{i}})-({\gamma% -1})\prod\limits_{i=1}^{2}{({1-MR_{i}})}}}\\ \end{array}}\right],\\ \\ \left[{\begin{array}[]{l}\frac{\prod\limits_{i=1}^{2}{({1+({\gamma-1})ML_{i}})% -\prod\limits_{i=1}^{2}{({1-ML_{i}})}}}{\prod\limits_{i=1}^{2}{({1+({\gamma-1}% )ML_{i}})-({\gamma-1})\prod\limits_{i=1}^{2}{({1-ML_{i}})}}},\\ \frac{\prod\limits_{i=1}^{2}{({1+({\gamma-1})MR_{i}})-\prod\limits_{i=1}^{2}{(% {1-MR_{i}})}}}{\prod\limits_{i=1}^{2}{({1+({\gamma-1})MR_{i}})-({\gamma-1})% \prod\limits_{i=1}^{2}{({1-MR_{i}})}}}\\ \end{array}}\right]\\ \end{array}}\right);$ $\displaystyle\lambda MN_{1}=\left({\begin{array}[]{l}\left[{\begin{array}[]{l}% \frac{({1+({\gamma-1})ML_{i}})^{\lambda}-({1-ML_{i}})^{\lambda}}{({1+({\gamma-% 1})ML_{i}})^{\lambda}+({\gamma-1})({1-ML_{i}})^{\lambda}},\\ \\ \frac{({1+({\gamma-1})MR_{i}})^{\lambda}-({1-MR_{i}})^{\lambda}}{({1+({\gamma-% 1})MR_{i}})^{\lambda}+({\gamma-1})({1-MR_{i}})^{\lambda}}\\ \end{array}}\right],\\ \\ \left[{\begin{array}[]{l}\frac{\gamma({1-ML_{i}})^{\lambda}-\gamma({1-ML_{i}-% NL_{i}})^{\lambda}}{({1+({\gamma-1})ML_{i}})^{\lambda}+({\gamma-1})({1-ML_{i}}% )^{\lambda}},\\ \\ \frac{\gamma({1-MR_{i}})^{\lambda}-\gamma({1-MR_{i}-NR_{i}})^{\lambda}}{({1+({% \gamma-1})MR_{i}})^{\lambda}+({\gamma-1})({1-MR_{i}})^{\lambda}}\\ \end{array}}\right]\\ \end{array}}\right);$ $\displaystyle({MN_{1}})^{\lambda}=\left({\begin{array}[]{l}\left[{\begin{array% }[]{l}\frac{\gamma({1-ML_{i}})^{\lambda}-\gamma({1-ML_{i}-NL_{i}})^{\lambda}}{% ({1+({\gamma-1})ML_{i}})^{\lambda}+({\gamma-1})({1-ML_{i}})^{\lambda}},\\ \\ \frac{\gamma({1-MR_{i}})^{\lambda}-\gamma({1-MR_{i}-NR_{i}})^{\lambda}}{({1+({% \gamma-1})MR_{i}})^{\lambda}+({\gamma-1})({1-MR_{i}})^{\lambda}}\\ \end{array}}\right],\\ \\ \left[{\begin{array}[]{l}\frac{({1+({\gamma-1})ML_{i}})^{\lambda}-({1-ML_{i}})% ^{\lambda}}{({1+({\gamma-1})ML_{i}})^{\lambda}+({\gamma-1})({1-ML_{i}})^{% \lambda}},\\ \\ \frac{({1+({\gamma-1})MR_{i}})^{\lambda}-({1-MR_{i}})^{\lambda}}{({1+({\gamma-% 1})MR_{i}})^{\lambda}+({\gamma-1})({1-MR_{i}})^{\lambda}}\\ \end{array}}\right]\\ \end{array}}\right);$

3. I-IVIFHIOWA operator

Then, interval-valued intuitionistic fuzzy Hamacher interactive weighted averaging (IVIFHIWA) and interval-valued intuitionistic fuzzy Hamacher interactive ordered weighted averaging (IVIFHIOWA) operators are introduced [48].

Definition 6 [48]. Let $MN_{j}=({[{ML_{j},MR_{j}}],[{NL_{j},NR_{j}}]})({j=1,2,\ldots,n})$ be the collection of IVIFN, and let IVIFHIWA: $Q^{n}\to Q$ , if

$\displaystyle\text{IVIFHIWA}_{m\omega}({MN_{1},MN_{2},\ldots,MN_{n}})=\mathop{% \oplus}\limits_{j=1}^{n}m\omega_{j}MN_{j}=\left({\begin{array}[]{l}\left[{% \begin{array}[]{l}\frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})ML_{j}})^{m% \omega_{j}}-\prod\limits_{j=1}^{n}{({1-ML_{j}})^{m\omega_{j}}}}}{\prod\limits_% {j=1}^{n}{({1+({\gamma-1})ML_{j}})^{m\omega_{j}}-({\gamma-1})\prod\limits_{j=1% }^{n}{({1-ML_{j}})^{m\omega_{j}}}}},\\ \frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})MR_{j}})^{m\omega_{j}}-\prod% \limits_{j=1}^{n}{({1-MR_{j}})^{m\omega_{j}}}}}{\prod\limits_{j=1}^{n}{({1+({% \gamma-1})MR_{j}})^{m\omega_{j}}-({\gamma-1})\prod\limits_{j=1}^{n}{({1-MR_{j}% })^{m\omega_{j}}}}}\\ \end{array}}\right],\\ \\ \left[{\begin{array}[]{l}\frac{\gamma\prod\limits_{j=1}^{n}{({1-ML_{j}})^{m% \omega_{j}}}-\gamma\prod\limits_{j=1}^{n}{({1-ML_{j}-NL_{j}})^{m\omega_{j}}}}{% \prod\limits_{j=1}^{n}{({1+({\gamma-1})ML_{j}})^{m\omega_{j}}-({\gamma-1})% \prod\limits_{j=1}^{n}{({1-ML_{j}})^{m\omega_{j}}}}},\\ \frac{\gamma\prod\limits_{j=1}^{n}{({1-MR_{j}})^{m\omega_{j}}}-\gamma\prod% \limits_{j=1}^{n}{({1-MR_{j}-NR_{j}})^{m\omega_{j}}}}{\prod\limits_{j=1}^{n}{(% {1+({\gamma-1})MR_{j}})^{m\omega_{j}}-({\gamma-1})\prod\limits_{j=1}^{n}{({1-% MR_{j}})^{m\omega_{j}}}}}\\ \end{array}}\right]\\ \end{array}}\right)$ (11)

where $m\omega=({m\omega_{1},m\omega_{2},\ldots,m\omega_{n}})^{T}$ be the weight information and $m\omega_{j}>0$ , $\sum\limits_{j=1}^{n}{m\omega_{j}}=1$ .

Garg, Agarwal [48] developed the IVIFHIOWA operator.

Definition 7 [48]. Let $MN_{j}=({[{ML_{j},MR_{j}}],[{NL_{j},NR_{j}}]})({j=1,2,\ldots,n})$ be the collection of IVIFNs, and let IVIFHIOWA: $Q^{n}\to Q$ , if

$\displaystyle\text{IVIFHIOWA}_{mw}({MN_{1},MN_{2},\ldots,MN_{n}})=\mathop{% \oplus}\limits_{j=1}^{n}mw_{j}MN_{\sigma(j)}=\left({\begin{array}[]{l}\left[{% \begin{array}[]{l}\frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})ML_{\sigma(j)}}% )^{mw_{j}}-\prod\limits_{j=1}^{n}{({1-ML_{\sigma(j)}})^{mw_{j}}}}}{\prod% \limits_{j=1}^{n}{({1+({\gamma-1})ML_{\sigma(j)}})^{mw_{j}}-({\gamma-1})\prod% \limits_{j=1}^{n}{({1-ML_{\sigma(j)}})^{mw_{j}}}}},\\ \frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})MR_{\sigma(j)}})^{mw_{j}}-\prod% \limits_{j=1}^{n}{({1-MR_{\sigma(j)}})^{mw_{j}}}}}{\prod\limits_{j=1}^{n}{({1+% ({\gamma-1})MR_{\sigma(j)}})^{mw_{j}}-({\gamma-1})\prod\limits_{j=1}^{n}{({1-% MR_{\sigma(j)}})^{mw_{j}}}}}\\ \end{array}}\right],\\ \\ \left[{\begin{array}[]{l}\frac{\gamma\prod\limits_{j=1}^{n}{({1-ML_{\sigma(j)}% })^{mw_{j}}}-\gamma\prod\limits_{j=1}^{n}{({1-ML_{j}-NL_{\sigma(j)}})^{mw_{j}}% }}{\prod\limits_{j=1}^{n}{({1+({\gamma-1})ML_{\sigma(j)}})^{mw_{j}}-({\gamma-1% })\prod\limits_{j=1}^{n}{({1-ML_{\sigma(j)}})^{mw_{j}}}}},\\ \frac{\gamma\prod\limits_{j=1}^{n}{({1-MR_{\sigma(j)}})^{mw_{j}}}-\gamma\prod% \limits_{j=1}^{n}{({1-MR_{\sigma(j)}-NR_{\sigma(j)}})^{mw_{j}}}}{\prod\limits_% {j=1}^{n}{({1+({\gamma-1})MR_{\sigma(j)}})^{mw_{j}}-({\gamma-1})\prod\limits_{% j=1}^{n}{({1-MR_{\sigma(j)}})^{mw_{j}}}}}\\ \end{array}}\right]\\ \end{array}}\right)$ (12)

where $mw=({mw_{1},mw_{2},\ldots,mw_{j}})^{T}$ is a weight information, $mw_{j}>0$ , $\sum\limits_{j=1}^{n}{mw_{j}}=1$ , $(\sigma(1),\sigma(2),\linebreak\ldots,\sigma(n))$ is a permutation of $({1,2,\ldots,n})$ .

Yager and Filev [53] established the induced OWA (IOWA) operator based on the OWA operator [54].

Definition 8 [53]. An IOWA operator is established through the following formula:

$\displaystyle\textit{IOWA}({\langle{mu_{1},mk_{1}}\rangle,\langle{mu_{2},mk_{2% }}\rangle,\ldots,\langle{mu_{n},mk_{n}}\rangle})=\sum\limits_{j=1}^{n}{mw_{j}% mk_{\sigma(j)}}$ (13)

$mk_{\sigma(j)}$ is the $mk_{j}$ of OWA pair $\langle{mu_{j},mk_{j}}\rangle$ having the j-th largest $mu_{j}({mu_{j}\in[{0,1}]})$ , and $mu_{j}$ in $\langle{mu_{j},mk_{j}}\rangle$ is established to as order inducing information and $mk_{j}$ are argument.

Then, the induced IVIFHIOWA (I-IVIFHIOWA) operator is established based on the IOWA operator [53] and IVIFHIOWA operator [48].

Definition 9. Let $\langle{mu_{j},MN_{j}}\rangle({j=1,2,\ldots,n})$ be a set of 2-tuples, then the I-IVIFHIOWA operator is established:

where $mw=({mw_{1},mw_{2},\ldots,mw_{j}})^{T}$ is a weight information, $mw_{j}>0$ , $\sum\limits_{j=1}^{n}{mw_{j}}=1$ , $MN_{\sigma(j)}$ is the $MN_{j}$ of I-IVIFHIOWA pair $\langle{mu_{j},MN_{j}}\rangle$ having the j-th largest $mu_{j}({mu_{j}\in[{0,1}]})$ , and $mu_{j}$ in $\langle{mu_{j},MN_{j}}\rangle$ is established to as order inducing information and $MN_{j}$ are IVIFNs.

Theorem 1. Let $\langle{mu_{j},MN_{j}}\rangle({j=1,2,\ldots,n})$ be a set of 2-tuples, then its fused value by I-IVIFHIOWA operator is also an IVIFNs, and

Now some special cases of the I-IVIFHIOWA operator are investigated:

(1) If $mu_{j}=MN_{j}$ for all $j$ , then I-IVIFHIOWA operator reduces the IVIFHIOWA operator:

$\displaystyle\text{I-IVIFHIOWA}_{mw}({\langle{mu_{1},MN_{1}}\rangle,\langle{mu% _{2},MN_{2}}\rangle,\ldots,\langle{mu_{n},MN_{n}}\rangle})=\mathop{\oplus}% \limits_{j=1}^{n}mw_{j}MN_{\sigma(j)}=\left({\begin{array}[]{l}\left[{\begin{% array}[]{l}\frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})ML_{\sigma(j)}})^{mw_{% j}}-\prod\limits_{j=1}^{n}{({1-ML_{\sigma(j)}})^{mw_{j}}}}}{\prod\limits_{j=1}% ^{n}{({1+({\gamma-1})ML_{\sigma(j)}})^{mw_{j}}-({\gamma-1})\prod\limits_{j=1}^% {n}{({1-ML_{\sigma(j)}})^{mw_{j}}}}},\\ \frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})MR_{\sigma(j)}})^{mw_{j}}-\prod% \limits_{j=1}^{n}{({1-MR_{\sigma(j)}})^{mw_{j}}}}}{\prod\limits_{j=1}^{n}{({1+% ({\gamma-1})MR_{\sigma(j)}})^{mw_{j}}-({\gamma-1})\prod\limits_{j=1}^{n}{({1-% MR_{\sigma(j)}})^{mw_{j}}}}}\\ \end{array}}\right],\\ \\ \left[{\begin{array}[]{l}\frac{\gamma\prod\limits_{j=1}^{n}{({1-ML_{\sigma(j)}% })^{mw_{j}}}-\gamma\prod\limits_{j=1}^{n}{({1-ML_{j}-NL_{\sigma(j)}})^{mw_{j}}% }}{\prod\limits_{j=1}^{n}{({1+({\gamma-1})ML_{\sigma(j)}})^{mw_{j}}-({\gamma-1% })\prod\limits_{j=1}^{n}{({1-ML_{\sigma(j)}})^{mw_{j}}}}},\\ \frac{\gamma\prod\limits_{j=1}^{n}{({1-MR_{\sigma(j)}})^{mw_{j}}}-\gamma\prod% \limits_{j=1}^{n}{({1-MR_{\sigma(j)}-NR_{\sigma(j)}})^{mw_{j}}}}{\prod\limits_% {j=1}^{n}{({1+({\gamma-1})MR_{\sigma(j)}})^{mw_{j}}-({\gamma-1})\prod\limits_{% j=1}^{n}{({1-MR_{\sigma(j)}})^{mw_{j}}}}}\\ \end{array}}\right]\\ \end{array}}\right)\text{IVIFHIOWA}_{mw}({MN_{1},MN_{2},\ldots,MN_{n}})=% \mathop{\oplus}\limits_{j=1}^{n}mw_{j}MN_{\sigma(j)}=\left({\begin{array}[]{l}% \left[{\begin{array}[]{l}\frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})ML_{% \sigma(j)}})^{mw_{j}}-\prod\limits_{j=1}^{n}{({1-ML_{\sigma(j)}})^{mw_{j}}}}}{% \prod\limits_{j=1}^{n}{({1+({\gamma-1})ML_{\sigma(j)}})^{mw_{j}}-({\gamma-1})% \prod\limits_{j=1}^{n}{({1-ML_{\sigma(j)}})^{mw_{j}}}}},\\ \frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})MR_{\sigma(j)}})^{mw_{j}}-\prod% \limits_{j=1}^{n}{({1-MR_{\sigma(j)}})^{mw_{j}}}}}{\prod\limits_{j=1}^{n}{({1+% ({\gamma-1})MR_{\sigma(j)}})^{mw_{j}}-({\gamma-1})\prod\limits_{j=1}^{n}{({1-% MR_{\sigma(j)}})^{mw_{j}}}}}\\ \end{array}}\right],\\ \\ \left[{\begin{array}[]{l}\frac{\gamma\prod\limits_{j=1}^{n}{({1-ML_{\sigma(j)}% })^{mw_{j}}}-\gamma\prod\limits_{j=1}^{n}{({1-ML_{j}-NL_{\sigma(j)}})^{mw_{j}}% }}{\prod\limits_{j=1}^{n}{({1+({\gamma-1})ML_{\sigma(j)}})^{mw_{j}}-({\gamma-1% })\prod\limits_{j=1}^{n}{({1-ML_{\sigma(j)}})^{mw_{j}}}}},\\ \frac{\gamma\prod\limits_{j=1}^{n}{({1-MR_{\sigma(j)}})^{mw_{j}}}-\gamma\prod% \limits_{j=1}^{n}{({1-MR_{\sigma(j)}-NR_{\sigma(j)}})^{mw_{j}}}}{\prod\limits_% {j=1}^{n}{({1+({\gamma-1})MR_{\sigma(j)}})^{mw_{j}}-({\gamma-1})\prod\limits_{% j=1}^{n}{({1-MR_{\sigma(j)}})^{mw_{j}}}}}\\ \end{array}}\right]\\ \end{array}}\right)$ (16)

where $({\sigma(1),\sigma(2),\ldots,\sigma(n)})$ is a permutation of $({1,2,\ldots,n})$ , such that $MN_{\sigma({j-1})}\geqslant MN_{\sigma(j)}$ for all $j=2,\ldots,n$ .

(2) If $mu_{j}=No.j$ for all $j$ , if $j$ is ordered position of $\langle{mu_{j},MN_{j}}\rangle$ , then I-IVIFHIOWA operator becomes the IVIFHIWA operator:

$\displaystyle\text{I-IVIFHIOWA}_{mw}({\langle{mu_{1},MN_{1}}\rangle,\langle{mu% _{2},MN_{2}}\rangle,\ldots,\langle{mu_{n},MN_{n}}\rangle})=\mathop{\oplus}% \limits_{j=1}^{n}mw_{j}MN_{\sigma(j)}=\left({\begin{array}[]{l}\left[{\begin{% array}[]{l}\frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})ML_{\sigma(j)}})^{mw_{% j}}-\prod\limits_{j=1}^{n}{({1-ML_{\sigma(j)}})^{mw_{j}}}}}{\prod\limits_{j=1}% ^{n}{({1+({\gamma-1})ML_{\sigma(j)}})^{mw_{j}}-({\gamma-1})\prod\limits_{j=1}^% {n}{({1-ML_{\sigma(j)}})^{mw_{j}}}}},\\ \frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})MR_{\sigma(j)}})^{mw_{j}}-\prod% \limits_{j=1}^{n}{({1-MR_{\sigma(j)}})^{mw_{j}}}}}{\prod\limits_{j=1}^{n}{({1+% ({\gamma-1})MR_{\sigma(j)}})^{mw_{j}}-({\gamma-1})\prod\limits_{j=1}^{n}{({1-% MR_{\sigma(j)}})^{mw_{j}}}}}\\ \end{array}}\right],\\ \\ \left[{\begin{array}[]{l}\frac{\gamma\prod\limits_{j=1}^{n}{({1-ML_{\sigma(j)}% })^{mw_{j}}}-\gamma\prod\limits_{j=1}^{n}{({1-ML_{j}-NL_{\sigma(j)}})^{mw_{j}}% }}{\prod\limits_{j=1}^{n}{({1+({\gamma-1})ML_{\sigma(j)}})^{mw_{j}}-({\gamma-1% })\prod\limits_{j=1}^{n}{({1-ML_{\sigma(j)}})^{mw_{j}}}}},\\ \frac{\gamma\prod\limits_{j=1}^{n}{({1-MR_{\sigma(j)}})^{mw_{j}}}-\gamma\prod% \limits_{j=1}^{n}{({1-MR_{\sigma(j)}-NR_{\sigma(j)}})^{mw_{j}}}}{\prod\limits_% {j=1}^{n}{({1+({\gamma-1})MR_{\sigma(j)}})^{mw_{j}}-({\gamma-1})\prod\limits_{% j=1}^{n}{({1-MR_{\sigma(j)}})^{mw_{j}}}}}\\ \end{array}}\right]\\ \end{array}}\right)\text{IVIFHIWA}_{m\omega}({MN_{1},MN_{2},\ldots,MN_{n}})=% \mathop{\oplus}\limits_{j=1}^{n}m\omega_{j}MN_{j}=\left({\begin{array}[]{l}% \left[{\begin{array}[]{l}\frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})ML_{j}})% ^{m\omega_{j}}-\prod\limits_{j=1}^{n}{({1-ML_{j}})^{m\omega_{j}}}}}{\prod% \limits_{j=1}^{n}{({1+({\gamma-1})ML_{j}})^{m\omega_{j}}-({\gamma-1})\prod% \limits_{j=1}^{n}{({1-ML_{j}})^{m\omega_{j}}}}},\\ \frac{\prod\limits_{j=1}^{n}{({1+({\gamma-1})MR_{j}})^{m\omega_{j}}-\prod% \limits_{j=1}^{n}{({1-MR_{j}})^{m\omega_{j}}}}}{\prod\limits_{j=1}^{n}{({1+({% \gamma-1})MR_{j}})^{m\omega_{j}}-({\gamma-1})\prod\limits_{j=1}^{n}{({1-MR_{j}% })^{m\omega_{j}}}}}\\ \end{array}}\right],\\ \\ \left[{\begin{array}[]{l}\frac{\gamma\prod\limits_{j=1}^{n}{({1-ML_{j}})^{m% \omega_{j}}}-\gamma\prod\limits_{j=1}^{n}{({1-ML_{j}-NL_{j}})^{m\omega_{j}}}}{% \prod\limits_{j=1}^{n}{({1+({\gamma-1})ML_{j}})^{m\omega_{j}}-({\gamma-1})% \prod\limits_{j=1}^{n}{({1-ML_{j}})^{m\omega_{j}}}}},\\ \frac{\gamma\prod\limits_{j=1}^{n}{({1-MR_{j}})^{m\omega_{j}}}-\gamma\prod% \limits_{j=1}^{n}{({1-MR_{j}-NR_{j}})^{m\omega_{j}}}}{\prod\limits_{j=1}^{n}{(% {1+({\gamma-1})MR_{j}})^{m\omega_{j}}-({\gamma-1})\prod\limits_{j=1}^{n}{({1-% MR_{j}})^{m\omega_{j}}}}}\\ \end{array}}\right]\\ \end{array}}\right)$ (17)

where $m\omega=({m\omega_{1},m\omega_{2},\ldots,m\omega_{n}})^{T}$ be the weight information and $m\omega_{j}>0$ , $\sum\limits_{j=1}^{n}{m\omega_{j}}=1$ .

It is easily proved that the I-IVIFHIOWA operator has the following properties.

Theorem 2. (Idempotency) If all $MN_{j}=({[{ML_{j},MR_{j}}],[{NL_{j},NR_{j}}]})({j=1,2,\ldots,n})$ are equal, i.e. $MN_{j}=MN$ for all $a$ , then

$\displaystyle\text{I-IVIFHIOWA}_{mw}({\langle{mu_{1},MN_{1}}\rangle,\langle{mu% _{2},MN_{2}}\rangle,\ldots,\langle{mu_{n},MN_{n}}\rangle})=MN$ (18)

Theorem 3. (Boundedness) Let $MN_{j}=({[{ML_{j},MR_{j}}],[{NL_{j},NR_{j}}]})$ be a set of IVIFNs, and let

$\displaystyle MN^{-}=\mathop{\min}\limits_{j}MN_{j},MN^{+}=\mathop{\max}% \limits_{j}MN_{j}$

Then

$\displaystyle MN^{-}\leqslant\text{I-IVIFHIOWA}_{mw}({\langle{mu_{1},MN_{1}}% \rangle,\langle{mu_{2},MN_{2}}\rangle,\ldots,\langle{mu_{n},MN_{n}}\rangle})% \leqslant MN^{+}$ (19)

Theorem 4. (Monotonicity) Let $\langle{mu_{j},MN_{j}}\rangle({j=1,2,\ldots,n})$ and $\langle{mu^{\prime}_{j},MN^{\prime}_{j}}\rangle({j=1,2,\ldots,n})$ be two set of IVIFNs, if $MN_{j}\leqslant MN^{\prime}_{j}$ , for all $j$ , then

$\displaystyle\text{I-IVIFHIOWA}_{mw}({\langle{mu_{1},MN_{1}}\rangle,\langle{mu% _{2},MN_{2}}\rangle,\ldots,\langle{mu_{n},MN_{n}}\rangle})$ (20) $\displaystyle\quad\leqslant\text{I-IVIFHIOWA}_{mw}({\langle{mu^{\prime}_{1},MN% ^{\prime}_{1}}\rangle,\langle{mu^{\prime}_{2},MN^{\prime}_{2}}\rangle,\ldots,% \langle{mu^{\prime}_{n},MN^{\prime}_{n}}\rangle})$

Theorem 5. (Commutativity) Let $\langle{mu_{j},MN_{j}}\rangle({j=1,2,\ldots,n})$ and $\langle{mu^{\prime}_{j},MN^{\prime}_{j}}\rangle({j=1,2,\ldots,n})$ be two set of IVIFNs, then

$\displaystyle\text{I-IVIFHIOWA}_{mw}({\langle{mu_{1},MN_{1}}\rangle,\langle{mu% _{2},MN_{2}}\rangle,\ldots,\langle{mu_{n},MN_{n}}\rangle})$ (21) $\displaystyle\quad=\text{I-IVIFHIOWA}_{mw}({\langle{mu^{\prime}_{1},MN^{\prime% }_{1}}\rangle,\langle{mu^{\prime}_{2},MN^{\prime}_{2}}\rangle,\ldots,\langle{% mu^{\prime}_{n},MN^{\prime}_{n}}\rangle})$

where $\langle{mu^{\prime}_{j},MN^{\prime}_{j}}\rangle({j=1,2,\ldots,n})$ is any permutation of $\langle{mu_{j},MN_{j}}\rangle({j=1,2,\ldots,n})$ .

4. Method for MADM based on I-IVIFHIOWA operator

Let $MU=\{{MU_{1},MU_{2},\ldots,MU_{m}}\}$ be alternatives, $NU=\{{NU_{1},NU_{2},\ldots,NU_{n}}\}$ be attributes with weight $mw=({mw_{1},mw_{2},\ldots,mw_{n}})$ , where $mw_{j}\in[{0,1}]$ , $\sum\limits_{j=1}^{n}{mw_{j}}=1$ . Assume that the IVIFN-matrix $MN=({MN_{ij}})_{m\times n}$ , $MN_{ij}=({[{ML_{ij},MR_{ij}}],[{NL_{ij},NR_{ij}}]})_{m\times n}$ . The decision steps of I-IVIFHIOWA method for MADM are established as follows:

Step 1: Produce the IVIF-matrix $MN=({MN_{ij}})_{m\times n}$ , $MN_{ij}=({[{ML_{ij},MR_{ij}}],[{NL_{ij},NR_{ij}}]})_{m\times n}\linebreak({i=1% ,2,\ldots,m,j=1,2,\ldots,n})$ .

Step 2: Normalize the overall IVIF-matrix $MN=({MN_{ij}})_{m\times n}$ to $\textit{NMN}=({\textit{NMN}_{ij}})_{m\times n}$ .

$\displaystyle\textit{NMN}_{ij}=({[{\textit{NML}_{ij},\textit{NMR}_{ij}}],[{% \textit{NNL}_{ij},\textit{NNR}_{ij}}]})=\left\{{{\begin{array}[]{l}{({[{ML_{ij% },MR_{ij}}],[{NL_{ij},NR_{ij}}]}),NU_{j}\mbox{ is a benefit criterion}}\\ {({[{NL_{ij},NR_{ij}}],[{ML_{ij},MR_{ij}}]}),NU_{j}\mbox{ is a cost criterion % }}\\ \end{array}}}\right.$ (22)

Step 3. Utilize the $\textit{NMN}_{ij}=({[{\textit{NML}_{ij},\textit{NMR}_{ij}}],[{\textit{NNL}_{ij% },\textit{NNR}_{ij}}]})$ and I-IVIFHIOWA operator

$\displaystyle\textit{NMN}_{i}=({[{\textit{NML}_{i},\textit{NMR}_{i}}],[{% \textit{NNL}_{i},\textit{NNR}_{i}}]})=\text{I-IVIFHIOWA}_{mw}({\langle{mu_{i1}% ,\textit{NMN}_{i1}}\rangle,\langle{mu_{i2},\textit{NMN}_{i2}}\rangle,\ldots,% \langle{mu_{in},\textit{NMN}_{in}}\rangle})$ (23)

to obtain the overall values $\textit{NMN}_{i}({i=1,2,\ldots,m})$ , where $mw=({mw_{1},mw_{2},\ldots,mw_{v}})^{T}$ is the associated weight information of the I-IVIFHIOWA operator $mw_{j}>0$ and $\sum\limits_{j=1}^{n}{mw_{j}}=1$ .

Step 4. Obtain the scores $SV({\textit{NMN}_{i}})$ , $AV({\textit{NMN}_{i}})({i=1,2,\ldots,m})$ .

Step 5. Rank the alternatives $MU_{i}({i=1,2,\ldots,m})$ and obtain the best one(s) through $SV({\textit{NMN}_{i}})$ , $AV({\textit{NMN}_{i}})({i=1,2,\ldots,m})$ .

5. Numerical example and comparative analysis

5.1 Numerical example

The construction industry is a fundamental industry of the country, and China’s construction industry has low competitiveness due to issues such as low technological content and low production efficiency. Introducing supply chain management concepts into the construction industry is one of the important ways to improve the competitiveness of construction enterprises, and supplier evaluation is the main problem to be solved in implementing supply chain management. Given the extremely important position of material cost in engineering costs, this article focuses its research on material supplier evaluation. Material supplier evaluation has the characteristics of multiple indicators, large data volume, high information correlation, and low transparency. It has highly unstructured characteristics, complex decision-making processes, and is difficult to solve with a single method. One of the effective methods to solve this problem is to apply modern evaluation theory and new algorithms to the practice of building material supplier evaluation, and optimize the efficiency of evaluation. The second is to adopt advanced information processing technology and establish an intelligent decision support system for evaluating building material suppliers. The supplier selection of building materials is a classical MADM. A point in case about the supplier selection of building materials under IVIFNs is utilized to illustrate the above methods. We shall give five building materials suppliers $MU_{i}({i=1,2,3,4,5})$ to choose. The experts consider four attributes to evaluate these five building materials suppliers: ⟀ NU ${}_{1}$ represents the product quality; ⟁ NU ${}_{2}$ means the transport cost; ⟂ NU ${}_{3}$ represents the service level; ⟃ NU ${}_{4}$ means the enterprise performance. The supplier selection of building materials is depicted with IVIFNs. The five building materials suppliers $MU_{i}({i=1,2,3,4,5})$ will be evaluated by using four criteria in the context of IVIFNs. The transport cost (NU ${}_{2}$ ) is cost attribute, others are beneficial one.

Then, we employ the developed I-IVIFHIOWA operator to choose the outstanding building materials supplier under IVIFSs. The specific calculation steps are as follows.

Step 1. The evaluation IVIF-matrix $MN=({MN_{ij}})_{5\times 4}$ which is shown in Table 1.

Table 1
IVIFN-matrix

Alternatives	NU ${}_{1}$	NU ${}_{2}$
MU ${}_{1}$	([0.17, 0.25], [0.27, 0.46])	([0.27, 0.35], [0.38, 0.49])
MU ${}_{2}$	([0.39, 0.43], [0.46, 0.53])	([0.29, 0.32], [0.25, 0.37])
MU ${}_{3}$	([0.37, 0.43], [0.43, 0.49])	([0.19, 0.23], [0.29, 0.32])
MU ${}_{4}$	([0.17, 0.23], [0.36, 0.41])	([0.27, 0.35], [0.36, 0.43])
MU ${}_{5}$	([0.34, 0.38], [0.22, 0.35])	([0.23, 0.45], [0.29, 0.41])
Alternatives	NU ${}_{3}$	NU ${}_{4}$
MU ${}_{1}$	([0.26, 0.29], [0.36, 0.43])	([0.23, 0.29], [0.36, 0.41])
MU ${}_{2}$	([0.19, 0.23], [0.41, 0.52])	([0.19, 0.24], [0.35, 0.46])
MU ${}_{3}$	([0.34, 0.37], [0.39, 0.46])	([0.23, 0.42], [0.29, 0.35])
MU ${}_{4}$	([0.27, 0.34], [0.36, 0.45])	([0.39, 0.42], [0.47, 0.53])
MU ${}_{5}$	([0.34, 0.45], [0.29, 0.38])	([0.41, 0.46], [0.38, 0.42])

Step 2. Transform cost attributes into benefit attributes (See Table 2).

Table 2

The normalized decision matrix

Alternatives	NU ${}_{1}$	NU ${}_{2}$
MU ${}_{1}$	([0.17, 0.25], [0.27, 0.46])	([0.38, 0.49], [0.27, 0.35])
MU ${}_{2}$	([0.39, 0.43], [0.46, 0.53])	([0.25, 0.37], [0.29, 0.32])
MU ${}_{3}$	([0.37, 0.43], [0.43, 0.49])	([0.29, 0.32], [0.19, 0.23])
MU ${}_{4}$	([0.17, 0.23], [0.36, 0.41])	([0.36, 0.43], [0.27, 0.35])
MU ${}_{5}$	([0.34, 0.38], [0.22, 0.35])	([0.29, 0.41], [0.23, 0.45])
Alternatives	NU ${}_{3}$	NU ${}_{4}$
MU ${}_{1}$	([0.26, 0.29], [0.36, 0.43])	([0.23, 0.29], [0.36, 0.41])
MU ${}_{2}$	([0.19, 0.23], [0.41, 0.52])	([0.19, 0.24], [0.35, 0.46])
MU ${}_{3}$	([0.34, 0.37], [0.39, 0.46])	([0.23, 0.42], [0.29, 0.35])
MU ${}_{4}$	([0.27, 0.34], [0.36, 0.45])	([0.39, 0.42], [0.47, 0.53])
MU ${}_{5}$	([0.34, 0.45], [0.29, 0.38])	([0.41, 0.46], [0.38, 0.42])

Step 3. Invited experts employed induced variables to depict the attitude characteristics of the opinions of different board members. The results are depicted in Table 3.

Table 3

Inducing variables

	NU ${}_{1}$	NU ${}_{2}$	NU ${}_{3}$	NU ${}_{4}$
MU ${}_{1}$	17	10	15	11
MU ${}_{2}$	18	11	21	22
MU ${}_{3}$	19	13	16	11
MU ${}_{4}$	13	11	10	7
MU ${}_{5}$	9	18	17	20

Step 4. The uncertain information obtained in Table 2 is connected with I-IVIFHIOWA which has weight $mw=({0.20,0.30,0.30,0.20})$ , the overall values $\textit{NMN}_{i}({i=1,2,\ldots,5})$ is obtained in Table 4.

Table 4

The overall values $\textit{NMN}_{i}({i=1,2,\ldots,5})$

Alternatives	$\textit{NMN}_{i}$
MU ${}_{1}$	([0.29, 0.32], [0.25, 0.43])
MU ${}_{2}$	([0.39, 0.41], [0.36, 0.57])
MU ${}_{3}$	([0.24, 0.36], [0.47, 0.52])
MU ${}_{4}$	([0.34, 0.38], [0.43, 0.49])
MU ${}_{5}$	([0.31, 0.34], [0.23, 0.34])

Step 5. Calculate the scores information $SV({\textit{NMN}_{i}})({i=1,2,\ldots,5})$ .

Table 5

The scores information $SV\linebreak({\textit{NMN}_{i}})({i=1,2,\ldots,5})$

Alternatives	$SV({\textit{NMN}_{i}})$
MU ${}_{1}$	0.7331
MU ${}_{2}$	0.8532
MU ${}_{3}$	0.8107
MU ${}_{4}$	0.8996
MU ${}_{5}$	0.7474

Table 6

The decision orders

Methods	Ranking
I-IVIFEOWA operator [55]	$MU_{4}\succ MU_{2}\succ MU_{3}\succ MU_{5}\succ MU_{1}$
I-IVIFEOWG operator [56]	$MU_{4}\succ MU_{2}\succ MU_{5}\succ MU_{3}\succ MU_{1}$
I-IIFHA operator [57]	$MU_{4}\succ MU_{2}\succ MU_{3}\succ MU_{5}\succ MU_{1}$
I-IIFHG operator [57]	$MU_{4}\succ MU_{2}\succ MU_{5}\succ MU_{3}\succ MU_{1}$
IG-IVIFHSA operator [58]	$MU_{4}\succ MU_{2}\succ MU_{3}\succ MU_{5}\succ MU_{1}$

Step 6. Rank all the building materials suppliers $MU_{i}({i=1,2,3,4,5})$ through scores $SV({\textit{NMN}_{i}})(u=\linebreak 1,2,\ldots,5)$ : $MU_{4}\succ MU_{2}\succ MU_{3}\succ MU_{5}\succ MU_{1}$ , and thus the most desirable building materials supplier is $MU_{4}$ .

5.2 Comparative analyses

The I-IVIFHIOWA method is compared with induced interval-valued intuitionistic fuzzy Einstein ordered weighted average (I-IVIFEOWA) operator [55], induced interval-valued intuitionistic fuzzy Einstein ordered weighted geometric (I-IVIFEOWG) operator [56], induced interval-valued intuitionistic fuzzy hybrid averaging (I-IIFHA) operator [57], induced interval-valued intuitionistic fuzzy hybrid geometric (I-IIFHG) operator [57] and induced generalized interval-valued intuitionistic fuzzy hybrid Shapley averaging (IG-IVIFHSA) operator [58]. The decision orders are produced in Table 6.

Comparing the results of the I-IVIFHIOWA operator with existing methods, the calculating results are slightly different and the best building materials supplier and the worst building materials supplier is same. Different models could effectively deal with MADM from diverse research angles.

6. Conclusion

The issue of supplier selection has always been a hot topic in supply chain research, and the evaluation and selection of suppliers is an important link in the supply chain and an important content of supply chain management. In construction projects, the cost of building materials accounts for approximately 50% to 70% of the total construction cost, and the quality of supplied materials and whether they are supplied on time will directly affect the quality and duration of the entire project. Therefore, it is necessary to adopt scientific and reasonable methods to select and evaluate building material suppliers, and select the most reasonable and competitive suppliers. Thus, the building material supplier selection is frequently regarded as the MADM problem. In this paper, some calculating laws on IVIFSs, Hamacher sum, Hamacher product are introduced, and the induced interval-valued intuitionistic fuzzy Hamacher interactive ordered weighted averaging (IVIFHIOWA) operators (I-IVIFHIOWA) operator is proposed. Meanwhile, some ideal properties of I-IVIFHIOWA operator are studied. Then, the I-IVIFHIOWA operator is employed to cope with the MADM under IVIFSs. Finally, an example for building material supplier selection is employed to test the I-IVIFHIOWA operator. The use of IVIFSs theory to study complex uncertain MADM problems is very valuable both in theory and in practice. It is worth exploring the research direction of in-depth research on information fusion methods suitable for different complex decision-making environments, such as mixed aggregation operators and automatic determination methods of induced variables [59, 60, 61, 62, 63, 64, 65].

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Optimizing building material supplier selection through integrated interval-valued intuitionistic fuzzy multi-attribute decision making

Abstract

Keywords

1. Introduction

2. Preliminaries

5.1 Numerical example

Table 1 IVIFN-matrix

6. Conclusion

References

Table 1
IVIFN-matrix