TOPSIS Approach for MAGDM Based on Interval-Valued Hesitant Fuzzy N-Soft Environment

Abstract

Multi-attribute group decision-making (MAGDM) phenomenon is used to handle the information, when a number of attributes of alternatives are evaluated by a group of decision-makers. In this research article, we develop a novel approach called interval-valued hesitant fuzzy N-soft sets (IVHFNSSs) and extend the method of TOPSIS for solving MAGDM problems. In our proposed model, the evaluation information provided by the decision-makers is expressed in the form of ratings, grades and IVHFEs, whereas the information about attribute weights is arbitrary. The model provides us information about the occurrence of ratings or grades in hesitant situation. Moreover, we investigate some useful properties of IVHFNSSs and construct fundamental operations on them. By using the extended TOPSIS method, we describe potential applications of IVHFNSSs in MAGDM. We present our method in an algorithm and a graphical structure.

1 Introduction

In modern decision science, multi-attribute decision-making (MADM) phenomenon plays an important role in resolving the problems in our daily life. It is widely utilized in several areas, such as economics and management, society, military and engineering technology including management evaluation, project evaluation, investment decision-making and many more. In classical MADM, the evaluation of alternatives is absolutely known [9, 24] and based on truth logic. However, the attributes associated in various problems are not always pointed in real numbers, because the naturally existing objects are uncertain and fuzzy; even human thinking and preferences are ambiguous. Thus, some of these attributes involved in decision-making problems are well appropriate to be identified by fuzzy values, including linguistic variables [10], interval values [6, 34] and hesitant fuzzy elements (HFEs) [25, 26]. Nowadays, fuzzy MADM is earning more and more attention due to fuzzy decision-making model initiated by Zadeh [36] in 1965, actually this model is based on the fuzzy mathematics theory. Later on, many methods for MADM were introduced, including the maximizing deviation method [33], the TOPSIS method [14, 34] and the ELECTRE method [15]. Further, these methods were extended to set various characteristics of attribute values into bulletin, such as linguistic variables, intuitionistic fuzzy values and interval values. However, the above-mentioned methods have not been considered well for the hesitant fuzzy computation given by the decision-makers (DMs).

Hesitant fuzzy sets (HFSs) [25, 26] proposed by Torra and Narukawa as an expansion of fuzzy set illustrate the conditions that allow the membership of an element to the required set containing various values, which is beneficial to handle the uncertain data in the procedure of MADM. For instance to take a suitable decision, a decision management, which accommodates a number of decision-makers enables to assess the degree that an alternative should satisfy a criteria. The use of hesitant fuzzy computations makes the DMs assessments more feasible and significant in decision-making. Xia and Xu [30] gave applications to handle MADM problems under hesitant fuzzy environment and developed some aggregation operators. Xia et al. [31] also proposed some other hesitant fuzzy aggregation techniques and showed its importance in group decision-making. Chen et al. [7] generalized the concept of HFS and proposed the idea of interval-valued hesitant fuzzy sets (IVHFSs) in which the degree of membership of each element to the required set is not exactly defined, but indicated by different possible interval values. Simultaneously, they established an approach to group decision-making based on interval-valued hesitant preference relations, for the purpose to take the differences of opinions between each DMs. Xu and Zhang [32] developed a novel approach based on the maximizing deviation and TOPSIS method for the explanation of MADM problems, in which the evaluation information provided by the decision-maker is expressed in hesitant fuzzy elements and the information about attribute weights is incomplete. Wang et al. [28] proposed the concept of hesitant fuzzy soft sets (HFSSs) by the hybridization of soft sets and hesitant fuzzy sets. Recently, Peng and Yang [23] extended the concept of HFSSs and introduced interval-valued hesitant fuzzy soft sets (IVHFSSs). For other decision-making techniques and applications, the readers are referred to [5, 13, 16, 21, 22, 27, 29, 37].

From latest surveys of hybrid soft set models, most of the researchers in soft-set-inspired models worked on binary evaluation (either 0 or 1) or else, real numbers between 0 and 1 (see [19, 38]). But in our daily life problems, we often find data with a non-binary but discrete structure. For example, in social judgement systems, Alcantud and Laruelle [3] specified ternary voting system. Non-binary evaluations are also expected in rating or ranking positions. Examples from real-life rating systems show that rankings often take the form of number of dots and stars, which can also be displayed in the form of natural numbers. N-soft sets introduced by Fatimah et al. [11] expand the range of applications of the theories that can be used to deal with vagueness and uncertainty, which introduced the parameterized characterizations of the universe of options that depend on a finite number of grades. But the concept of N-soft sets is insufficient to provide the information about the occurrence of ratings or grades, and it is also unable to describe the occurrence of uncertainty and vagueness specially, when hesitancy occurs in decision-making problems. For this purpose, Akram et al. [1, 2] introduced the novel models with applications called Fuzzy N-soft sets and hesitant N-soft sets as the generalization of N-soft sets. Similarly, the concept of IVHFSSs is insufficient to provide the information about the opinion of experts depending upon the properties of alternatives. So, by the hybridization of two well-known concepts called N-soft sets and IVHFSSs, we introduce a new hybrid model called interval-valued hesitant fuzzy N-soft sets (IVHFNSSs). This model provides more accuracy and flexibility as compared to previously existing approaches, because it contains more information and it is more comprehensive and reasonable. Proposed model provides complete information about occurrence of ratings, uncertainty and hesitancy. It is also valuable because we extend the method of TOPSIS for MAGDM and compile the final decision by using extended TOPSIS method because of the complex structure of proposed model. For other notations and terminologies not mentioned in this paper, the readers are referred to [4, 8, 12, 17, 18, 20].

The organization of this research article is as follows:

In Sect. 2, we introduce a new hybrid model and present some of its basic operations. We illustrate this concept with real-life examples. Moreover, we develop its relationships with TOPSIS. In Sect. 3, we introduce extended TOPSIS method based on IVHFNSSs and describe its potential applications. We also present our proposed method as an algorithm and a graphical structure. In Sect. 4, we describe sensitivity analysis with related models. In Sect. 5, we present conclusion.

2 Interval-Valued Hesitant Fuzzy N-Soft Sets

In this section, we first review the basic concepts related to newly introduced model, then introduce our novel model and illustrate the properties with examples. Our model is based on the following concepts, N-soft sets from [11] and IVHFSSs from [23].

Definition 2.1

[7] Let O be a set of alternatives and I[0, 1] be the set of all closed subintervals of [0, 1]. An interval-valued hesitant fuzzy set (IVHFS) over O is defined as $\widetilde{I}=\{\langle o,\widetilde{h}(o)\rangle | o\in O\},$ where $\widetilde{h}(o)$ denotes all possible interval-valued membership degrees of element $o\in O$ to the set $\widetilde{I}.$

For convenience, Chen et al. [7] called $\widetilde{h}(o)$ an interval-valued hesitant fuzzy element (IVHFE), which is written as $\widetilde{h}(o)=\{\widetilde{r}|\widetilde{r}\in \widetilde{h}(o)\},$ denoted by h where $\widetilde{r}=[\widetilde{r}^l,\widetilde{r}^u]$ is an interval number and $\widetilde{r}^l,~\widetilde{r}^u$ represent the lower and upper boundaries, respectively. The set of all interval-valued hesitant fuzzy set over O is denoted by $\widetilde{H}(O).$

Definition 2.2

[23] A pair $(\widetilde{I},T)$ is called an interval-valued hesitant fuzzy soft set (IVHFSS) over O. If $\widetilde{I}$ is a mapping given by $\widetilde{I}:T\rightarrow \widetilde{H}(o),$ where $\widetilde{H}(o)$ is a parameterized family of interval-valued hesitant fuzzy subset of O,  for each $t\in T.$

Definition 2.3

[11] Let O be a universe of alternatives and P be the set of attributes, $T\subseteq P.$ Let $G=\{0,1,2,\ldots , N-1\}$ be the set of ordered grades where $N\in \{2,3,\ldots \}.$ A triple (F, T, N) is called an N-soft set on O if F is mapping from T to $2^{O\times G},$ with the property that for each $t\in T$ and $o\in O$ there exists a unique $(o,g_t)\in O\times G$ such that $(o,g_t)\in F(t),$$g_t\in G.$

Definition 2.4

Let O be a universe of alternatives and P be the set of attributes, $T\subseteq P.$ A pair $(\mu ,K)$ is called fuzzyN-soft set ((F, N)-soft set), when $K=(F,T,N)$ is an N-soft set on O with $N\in \{2,3,\ldots \}$, and $\mu$ is a mapping $\mu :T\rightarrow \bigcup _{t\in T}\mathcal {F}(F(t))$ such that $\mu (t)\in \mathcal {F}(F(t))$ for each $t\in T$.

According to Definition 2.4, with each attribute the mapping $\mu$ assigns a fuzzy set on the image of that attribute by the mapping F. Therefore, for each $t\in T$ and $o\in O$ there exists a unique $(o,g_t)\in O\times G$ such that $g_t\in G$ and $\langle (o,g_t), \mu _t(o)\rangle \in \mu (t)$, which is a notation that boils down to $\mu _t(o)=\mu (t)(o,g_t)$.

Definition 2.5

Let O be a universe of alternatives and P be the set of attributes, $T\subseteq P.$ A triple $(\hbar _f, T, N)$ is called hesitant fuzzyN-soft set (HFNSS), when $\hbar _f$ is a mapping $\hbar _f :O\times T\rightarrow G\times \mathcal {H}^*([0,1]).$

When $\hbar _f (o,t)=(g,\hbar _{f_t}(o)),$ we interpret that $\hbar _{f_t}(o)$ is a non-empty set formed by values in [0, 1], which denotes the possible membership degrees of the elements $o\in O$ to the subset of options approximated by t with the grade g.

In Example 2.6, we show the comparison of N-soft sets with fuzzy N-soft sets and hesitant fuzzy N-soft sets by describing their importance and feasibility.

Example 2.6

Let $O=\{o_1,o_2,o_3\}$ be the universe of alternatives and T be the set of attributes “evaluation of objects by standard parameters.” The subset $S\subseteq T$, such that $S=\{t_1,t_2,t_3\}$ is used. A 7-soft set can be obtained from Table 1, where

• Six check marks represent “excellent,”

• Five check marks represent “exceptional,”

• Four check marks represent “very good,”

• Three check marks represent “good,”

• Two check marks represent “normal,”

• One check mark represents “ordinary,”

• Box represents ‘“poor.”

This graded evaluation by check marks can easily identified with numbers as $G=\{0,1,2,3,4,5,6\},$ where

• 0 serves as “$\Box$”,

• 1 serves as “$\checkmark$”,

• 2 serves as “$\checkmark \checkmark$”,

• 3 serves as “$\checkmark \checkmark \checkmark$”,

• 4 serves as “$\checkmark \checkmark \checkmark \checkmark$”,

• 5 serves as “$\checkmark \checkmark \checkmark \checkmark \checkmark$”,

• 6 serves as “$\checkmark \checkmark \checkmark \checkmark \checkmark \checkmark$”.

The information extracted from related data is described in Table 1, and then the tabular representation of its associated 7-soft set is given in Table 2.

The information is enough when it is extracted from real data, but when the data are vague and uncertain, we need (F, N)-soft sets, which provide us information how these grades are given to alternatives. Therefore, the following (F, 7)-soft set (cf., Definition 2.4) is given in Table 3.

The case arises, when there is a confusion or hesitation for giving the membership values to evaluate the alternative by experts or decision-makers, we use HFNSS. Therefore, the following HF7SS (cf., Definition 2.5) is given in Table 4.

Table 1 Information extracted from related data

Table 2 Tabular representation of the corresponding 7-soft set

Table 3 Tabular representation of fuzzy N-soft set

Table 4 Tabular representation of hesitant fuzzy N-soft set

Our next definition introduces the novel model that emerges from the hybridization of N-soft sets and interval-valued hesitant fuzzy sets.

Definition 2.7

A triple $(\widetilde{I}_h, T, N)$ is called an interval-valued hesitant fuzzyN-soft set (IVHFNSS) over the universe of alternatives O, when $\widetilde{I}_h$ is a mapping given by $\widetilde{I}_h :O\times T\rightarrow G\times \mathcal {\widetilde{I}}^*([0,1]),$ and T is the set of evaluation attributes.

When $\widetilde{I}_h (o,t) = (g,\widetilde{h}_{{f}}(t)(o)),$ we interpret that $\widetilde{h}_{{f}}(t)(o)$ is a non-empty set of parameterized family of IVHFNS subsets of O,  with the grade g. Note that

$$\begin{aligned} g= & {} g_{jk},\hbox { where }j=1,2,\ldots ,p\hbox { and }k=1,2,\ldots ,q. \\ \widetilde{h}_{{f}}(t)(o)= & {} \bigcup \limits _{\lambda \in h_f(t)}[\lambda ^l,\lambda ^u],~\mathrm{for~all}~ t\in T. \end{aligned}$$

Example 2.8

In the situation of Example 2.6, the following IVHF7SS (cf., Definition 2.7) is defined as:

$$\begin{aligned} \widetilde{I}_h ({t_1})&=\bigg \{\bigg \langle \frac{(o_1,5)}{\{[0.3,0.5],[0.4,0.7],[0.5,0.7]\}}\bigg \rangle ,\bigg \langle \frac{(o_2,6)}{\{[0.4,0.7],[0.6,0.8]\}}\bigg \rangle , \bigg \langle \frac{(o_3,4)}{\{[0.1,0.2],[0.2,0.4]\}}\bigg \rangle \bigg \},\\ \widetilde{I}_h ({t_2})&=\bigg \{\bigg \langle \frac{(o_1,3)}{\{[0.1,0.5],[0.4,0.5]\}}\bigg \rangle ,\bigg \langle \frac{(o_2,4)}{\{[0.4,0.6],[0.6,0.8],[0.7,0.8]\}}\bigg \rangle , \bigg \langle \frac{(o_3,0)}{\{[0,0]\}}\bigg \rangle \bigg \},\\ \widetilde{I}_h ({t_3})&=\bigg \{\bigg \langle \frac{(o_1,0)}{\{[0,0]\}}\bigg \rangle ,\bigg \langle \frac{(o_2,2)}{\{[0.2,0.3]\}}\bigg \rangle , \bigg \langle \frac{(o_3,3)}{\{[0.4,0.5],[0.5,0.7],[0.6,0.8]\}}\bigg \rangle \bigg \}.\\ \end{aligned}$$

Clearly, we can display this information in tabular form as shown in Table 5.

Table 5 Tabular representation of the IVHF7SS

Definition 2.9

Let T be a set of attribute and $R,S\subseteq T,$ where $(\widetilde{I}_h,R,N')$ and $(\widetilde{J}_h,S,N)$ be two IVHFNSSs over O. $(\widetilde{I}_h,R,N')$ is said to be an IVHFNS subset of $(\widetilde{J}_h,S,N)$ if rest of the following conditions are satisfied.

i.:

$N'\le N,$

ii.:

$R\subseteq S,$

iii.:

$S(\widetilde{h}_{\widetilde{I}_h (t)}(o))\le S(\widetilde{h}_{\widetilde{J}_h (t)}(o)).$

So, we say $(\widetilde{I}_h,R,N')\subseteq (\widetilde{J}_h,S,N)$.

Example 2.10

Consider a set of alternatives $O=\{o_1,o_2,o_3\}$ and $R, S\subseteq T$ the set of attributes, where $R=\{t_1,t_3\}$, $S=\{t_1,t_2,t_3\}$ and $(\widetilde{I}_h,R,5)$, $(\widetilde{J}_h,S,6)$ are IVHF5SS and IVHF6SS over O.

$$\begin{aligned} \widetilde{I}_h ({t_1})&=\bigg \{\bigg \langle \frac{(o_1,3)}{\{[0.4,0.5],[0.5,0.7]\}}\bigg \rangle ,\bigg \langle \frac{(o_2,3)}{\{[0.2,0.3]\}}\bigg \rangle , \bigg \langle \frac{(o_3,2)}{\{[0.1,0.2],[0.2,0.3]\}}\bigg \rangle \bigg \},\\ \widetilde{I}_h ({t_3})&=\bigg \{\bigg \langle \frac{(o_1,1)}{\{[0.3,0.5]\}}\bigg \rangle ,\bigg \langle \frac{(o_2,4)}{\{[0.5,0.6],[0.7,0.9]\}}\bigg \rangle , \bigg \langle \frac{(o_3,3)}{\{[0.3,0.4],[0.5,0.6]\}}\bigg \rangle \bigg \}. \\ \widetilde{J}_h ({t_1})&=\bigg \{\bigg \langle \frac{(o_1,5)}{\{[0.5,0.6],[0.6,0.8]\}}\bigg \rangle ,\bigg \langle \frac{(o_2,4)}{\{[0.3,0.5]\}}\bigg \rangle , \bigg \langle \frac{(o_3,3)}{\{[0.1,0.2],[0.2,0.3],[0.3,0.5]\}}\bigg \rangle \bigg \},\\ \widetilde{J}_h ({t_2})&=\bigg \{\bigg \langle \frac{(o_1,2)}{\{[0.3,0.5],[0.5,0.6]\}}\bigg \rangle ,\bigg \langle \frac{(o_2,3)}{\{[0.5,0.7],[0.6,0.8]\}}\bigg \rangle , \bigg \langle \frac{(o_3,1)}{\{[0,0.1],[0.1,0.2],[0.2,0.3]\}}\bigg \rangle \bigg \},\\ \widetilde{J}_h ({t_3})&=\bigg \{\bigg \langle \frac{(o_1,3)}{\{[0.4,0.7]\}}\bigg \rangle ,\bigg \langle \frac{(o_2,5)}{\{[0.6,0.8],[0.8,0.9]\}}\bigg \rangle , \bigg \langle \frac{(o_3,4)}{\{[0.4,0.5],[0.6,0.7]\}}\bigg \rangle \bigg \}. \end{aligned}$$

Each condition given in Definition 2.9 is satisfied. So, Example 2.10 shows that $(\widetilde{I}_h,R,5)\subseteq (\widetilde{J}_h,S,6)$.

Definition 2.11

Two IVHFNSSs $(\widetilde{I}_h,R,N')$ and $(\widetilde{J}_h,S,N)$ over universe O are said to be equal. If $\widetilde{I}_h=\widetilde{J}_h$, $R=S$, $N'=N.$ In other words, one can say each of them is IVHFNS subset of other.

Definition 2.12

Two IVHFNSSs $(\widetilde{I}_h,R,N')$ and $(\widetilde{I}_h,S,N)$ over universe O are said to be equivalent. If $R=S$, $N'=N,$ but the corresponding grades and intervals are not same.

Example 2.13

Let $\{o_1,o_2\}$ be a set of alternatives and $R=S=\{t_1\}$ be the sets of attributes, where $N'=N=4.$

$$\begin{aligned} \widetilde{I}_h ({t_1})&=\bigg \{\bigg \langle \frac{(o_1,3)}{\{[0.2,0.4],[0.3,0.4]\}}\bigg \rangle ,\bigg \langle \frac{(o_2,2)}{\{[0.1,0.3]\}}\bigg \rangle \bigg \},\\ \widetilde{J}_h ({t_1})&=\bigg \{\bigg \langle \frac{(o_1,2)}{\{[0.3,0.5],[0.4,0.6]\}}\bigg \rangle ,\bigg \langle \frac{(o_2,2)}{\{[0.3,0.4]\}}\bigg \rangle \bigg \}.\\ \end{aligned}$$

Example 2.13 shows, $(\widetilde{J}_h,R,4')\cong (\widetilde{I}_h,S,4)$ but $(\widetilde{J}_h,R,4')\ne (\widetilde{I}_h,S,4).$

Definition 2.14

An IVHFNSS $(\widetilde{I}_h,T,N)$ over O is said to be empty IVHFNSS, if $\widetilde{I}_h(t)=\langle {0,\{[0,0]\}}\rangle$ for all $t\in T,$ and denoted by $\widetilde{\Phi }_T.$ An IVHFNSS $(\widetilde{I}_h,T,N)$ over O is said to be full IVHFNSS, if $\widetilde{I}_h(t)=\langle {N-1,\{[1,1]\}}\rangle$ for all $t\in T,$ and denoted by $\widetilde{O}_T.$

Definition 2.15

An IVHFNSS $(\widetilde{I}_h^c,T,N)$ over O is said to be weak complement of an IVHFNSS $(\widetilde{I}_h,T,N)$ over O if rest of the following conditions are satisfied.

i.:

$F^c(t)\cap F(t)=\Phi ,$

ii.:

$\widetilde{h}_f^c(t)=\bigcup \limits _{\lambda \in h_f(t)}[1-\lambda ^u,1-\lambda ^l]$, for all $t\in T.$

Remark 1

More than one weak complement can be defined, because the choice of grades is arbitrary, and can be changed with same interval values.

Example 2.16

In the situation of Example 2.6, the weak complements of the IVHFNSS are, respectively, given in Tables 6 and 7, respectively.

Table 6 Tabular representation of weak complement of IVHF7SS

Table 7 Tabular representation of an other weak complement of IVHF7SS

Definition 2.17

Let $(\widetilde{I}_h,R,N')$ and $(\widetilde{J}_h,S,N)$ be two IVHFNSSs. The AND operation on these two IVHFNSSs over the same universe O denoted by $(\widetilde{I}_h,R,N')\wedge (\widetilde{J}_h,S,N)$ is defined as

$$\begin{aligned}&(\widetilde{K}_h,R\times S,\min (N',N)), \forall ~(r,s)\in R\times S\, \mathrm{and}\;o_j\in O, \\&(g_{jk}, \widetilde{h}_f(r,s)(o)) \in \widetilde{K}_h(r,s)(o)\Leftrightarrow g_{jk}=\min (g^1_{jk},g^2_{jk}), \\&\mathrm{and} \;\widetilde{h}_f(r,s)(o)=\widetilde{h}_{f1}(r)(o)\cap \widetilde{h}_{f2}(s)(o), \\&\mathrm{where}\;\widetilde{h}_{f1}(r)(o)\cap \widetilde{h}_{f2}(s)(o)=\{\min (\lambda ^l_{\widetilde{h}_{f1}(r)}(o),\lambda ^l_{\widetilde{h}_{f2}(s)}(o)),\min (\lambda ^u_{\widetilde{h}_{f1}(r)}(o),\lambda ^u_{\widetilde{h}_{f2}(s)}(o))\}, \\&\mathrm{such\;that}\;\widetilde{h}_{f1}(r)(o)\in \widetilde{I}_{h}(r)(o) ~\;\mathrm{and}\;~~\widetilde{h}_{f2}(s)(o)\in \widetilde{J}_{h}(s)(o). \end{aligned}$$

Definition 2.18

Let $(\widetilde{I}_h,R,N')$ and $(\widetilde{J}_h,S,N)$ be two IVHFNSSs. The OR operation on these two IVHFNSSs over the same universe O denoted by $(\widetilde{I}_h,R,N')\vee (\widetilde{J}_h,S,N)$ is defined as

$$\begin{aligned}&(\widetilde{L}_h,R\times S,\max (N',N)), \forall ~(r,s)\in R\times S\;\;\mathrm{and}\;\;o_j\in O, \\&(g_{jk},\widetilde{h}_f(r,s)(o)) \in \widetilde{L}_h(r,s)(o)\Leftrightarrow g_{jk}=\max (g^1_{jk},g^2_{jk}), \\&\mathrm{and}\;~\widetilde{h}_f(r,s)(o)=\widetilde{h}_{f1}(r)(o)\cap \widetilde{h}_{f2}(s)(o), \\&\mathrm{where}\;\;\widetilde{h}_{f1}(r)(o)\cap \widetilde{h}_{f2}(s)(o)=\{\max (\lambda ^l_{\widetilde{h}_{f1}(r)}(o),\lambda ^l_{\widetilde{h}_{f2}(s)}(o)),\max (\lambda ^u_{\widetilde{h}_{f1}(r)}(o),\lambda ^u_{\widetilde{h}_{f2}(s)}(o))\}, \\&\mathrm{such\;that}\;\;\widetilde{h}_{f1}(r)(o)\in \widetilde{I}_{h}(r)(o) ~\;\mathrm{and}\;~~\;\widetilde{h}_{f2}(s)(o)\in \widetilde{J}_{h}(s)(o). \end{aligned}$$

Example 2.19

Consider a set of universe of objects $O=\{o_1,o_2\}$ and $R, S\subseteq T$ the set of attributes, where $R=\{t_1,t_2,t_3\}$ and $S=\{t_1,t_2\},$ whereas $(\widetilde{I}_h,R,4)$ and $(\widetilde{J}_h,S,5)$ are IVHF4SS and IVHF5SS over O given as follows:

$$\begin{aligned} \widetilde{I}_h ({t_1})&=\bigg \{\bigg \langle \frac{(o_1,3)}{\{[0.5,0.6],[0.6,0.8]\}}\bigg \rangle ,\bigg \langle \frac{(o_2,2)}{\{[0.3,0.4],[0.4,0.5],[0.5,0.6]\}}\bigg \rangle \bigg \},\\ \widetilde{I}_h ({t_2})&=\bigg \{\bigg \langle \frac{(o_1,2)}{\{[0.2,0.3],[0.3,0.5]\}}\bigg \rangle ,\bigg \langle \frac{(o_2,1)}{\{[0,0.1],[0.1,0.2]\}}\bigg \rangle \bigg \},\\ \widetilde{I}_h ({t_3})&=\bigg \{\bigg \langle \frac{(o_1,1)}{\{[0.1,0.2]\}}\bigg \rangle ,\bigg \langle \frac{(o_2,0)}{\{[0,0]\}}\bigg \rangle \bigg \},\\ \widetilde{J}_h ({t_1})&=\bigg \{\bigg \langle \frac{(o_1,4)}{\{[0.6,0.7],[0.8,0.9]\}}\bigg \rangle ,\bigg \langle \frac{(o_2,2)}{\{[0.3,0.5],[0.4,0.5]\}}\bigg \rangle \bigg \},\\ \widetilde{J}_h ({t_2})&=\bigg \{\bigg \langle \frac{(o_1,3)}{\{[0.4,0.6]\}}\bigg \rangle ,\bigg \langle \frac{(o_2,2)}{\{[0.1,0.2],[0.2,0.3],[0.3,0.4]\}}\bigg \rangle \bigg \}.\\ \end{aligned}$$

The result of AND operation $(\widetilde{I}_h,R,N')\wedge (\widetilde{J}_h,S,N)=(\widetilde{K}_h,R\times S,\min (N',N))$ is given in Table 8.

Similarly, the result of OR operation $(\widetilde{I}_h,R,N')\vee (\widetilde{J}_h,S,N)=(\widetilde{L}_h,R\times S,\max (N',N))$ is given in Table 9.

Table 8 Tabular representation of AND operation

Table 9 Tabular representation of OR operation

Definition 2.20

Let $(\widetilde{I}_h,R,N')$ and $(\widetilde{J}_h,S,N)$ be two IVHFNSSs. The intersection operation on these two IVHFNSSs over the same universe O denoted by $(\widetilde{I}_h,R,N')\cap (\widetilde{J}_h,S,N)$ is defined as $(\widetilde{\xi }_h,R\cap S,\min (N',N)),$ where

$$\begin{aligned} \widetilde{\xi }_h(t_k)=\widetilde{I}_h(t_k)\cap \widetilde{J}_h(t_k),~\forall ~t_k\in R\cap S,~o_j\in O. \end{aligned}$$

Definition 2.21

Let $(\widetilde{I}_h,R,N')$ and $(\widetilde{J}_h,S,N)$ be two IVHFNSSs. The union operation on these two IVHFNSSs over the same universe O denoted by $(\widetilde{I}_h,R,N')\cup (\widetilde{J}_h,S,N)$ is defined as $(\widetilde{\rho }_h,R\cup S,\max (N',N)),$ where

$$\begin{aligned} \widetilde{\rho }_h(t_k)=\left\{ \begin{array}{ll} \widetilde{I}_h(t_k), &{} ~if~~ t_k\in R-S, \\ \widetilde{J}_h(t_k), &{} ~if~~ t_k\in S-R , \\ \widetilde{I}_h(t_k)\cup \widetilde{J}_h(t_k), &{} ~if~~ t_k\in S\cup R . \\ \end{array} \right. \end{aligned}$$

Example 2.22

In the situation of Example 2.19, the intersection and union of the IVHF4SS and IVHF5SS are, respectively, given in Tables 10 and 11.

Table 10 Tabular representation of intersection operation

Table 11 Tabular representation of union operation

Theorem 2.23

If$(\widetilde{I}_h,R,N')$and$(\widetilde{J}_h,S,N)$are two IVHFNSSs over the universeO,  then

• 1. $(\widetilde{I}_h,R,N')\cap (\widetilde{I}_h,R,N')=(\widetilde{I}_h,R,N')$

• 2. $(\widetilde{I}_h,R,N')\cap \widetilde{\Phi }_R= \widetilde{\Phi }_R$

• 3. $(\widetilde{I}_h,R,N')\cap \widetilde{O}_R=(\widetilde{I}_h,R,N')$

• 4. $(\widetilde{I}_h,R,N')\cap \widetilde{O}_S=(\widetilde{I}_h,R\cap S,\min (N',N))$

• 5. $(\widetilde{I}_h,R,N')\cap \widetilde{\Phi }_S=\widetilde{\Phi }_{R\cap S}$

• 6. $(\widetilde{I}_h,R,N')\cap (\widetilde{J}_h,S,N)=(\widetilde{J}_h,S,N)\cap (\widetilde{I}_h,R,N')$

Theorem 2.24

If$(\widetilde{I}_h,R,N')$and$(\widetilde{J}_h,S,N)$are two IVHFNSSs over the universeO,  then

• 1. $(\widetilde{I}_h,R,N')\cup (\widetilde{I}_h,R,N')=(\widetilde{I}_h,R,N')$

• 2. $(\widetilde{I}_h,R,N')\cup \widetilde{\Phi }_R=(\widetilde{I}_h,R,N')$

• 3. $(\widetilde{I}_h,R,N')\cup \widetilde{O}_R=\widetilde{O}_R$

• 4. $(\widetilde{I}_h,R,N')\cup \widetilde{O}_S=\widetilde{O}_S$ if and only if $R\subseteq S$

• 5. $(\widetilde{I}_h,R,N')\cup \widetilde{\Phi }_S=(\widetilde{I}_h,R,N')$ if and only if $S\subseteq R$

• 6. $(\widetilde{I}_h,R,N')\cup (\widetilde{J}_h,S,N)=(\widetilde{J}_h,S,N)\cup (\widetilde{I}_h,R,N')$

Remark 2

In Theorems 2.23 and 2.24 trivial cases are discussed, both can be proved easily by using Definitions 2.20 and 2.21.

3 Applications

In this section, we explain the MAGDM mechanism that operates on the model that we have described in previous sections. MAGDM is a process in which a group of experts or decision-makers work together to decide the best alternative from a set of feasible alternatives that are characterized in terms of their attributes under particular situation. Therefore, we define respective algorithm for problems that are characterized by IVHFNSSs. In order to prove its importance and feasibility, we apply the method of extended TOPSIS to real situations that are fully developed.

3.1 Extended TOPSIS method based on IVHFNSSs

(TOPSIS) “Technique for Order Preference by Similarity to an Ideal Solution” is a method to choose the alternative that should have precise distance from the positive ideal solution (PIS) and the long distance from the negative ideal solution (NIS). This method is basically used for the ranking of objects and to get the best performance in MADM. We extend this method to deal with complex and IVHFNS information.

Let $O=\{o_1,o_2,\ldots ,o_p\}$ be the set of alternatives and $T=\{t_1,t_2,\ldots ,t_q\}$ be the set of attributes “evaluation of alternatives by standard parameters” specially the criteria determined by experts, respectively. Experts have an authority to assign weights to each alternative according to their choice. We suppose that the weights $(w_k\in (0,1])$ assigned by the experts satisfy the normalized condition. i.e., $\sum \nolimits _{k=1}^qw_k=1.$

The ratings of alternatives $o_j\in O$ given by evaluation attributes $t_k$ are based on IVHFNS, $\widetilde{I}_{h_j}=\{(g_{jk},\widetilde{h}_{{f}}(t)(o))|~~ k=1,2,\ldots q \},$ where $\widetilde{h}_{{f}}(t)(o)$ is a non-empty set of parameterized family of IVHFNS subsets of O. Positive ideal solution $\widetilde{I}^+_h$ and negative ideal solution $\widetilde{I}^-_h$ under the IVHFNS environment are defined as follows:

$$\begin{aligned} \widetilde{I}_h^+=\bigg \{&\bigg \langle \max g_{j1},\{[(\lambda _{1j}^{1l})^+,(\lambda _{1j}^{1u})^+],[(\lambda _{1j}^{2l})^+,(\lambda _{1j}^{2u})^+],\ldots ,[(\lambda _{1j}^{rl})^+,(\lambda _{1j}^{ru})^+]\}\bigg \rangle ,\nonumber \\&\bigg \langle \max g_{j2},\{[(\lambda _{2j}^{1l})^+,(\lambda _{2j}^{1u})^+],[(\lambda _{2j}^{2l})^+,(\lambda _{2j}^{2u})^+],\ldots ,[(\lambda _{2j}^{rl})^+,(\lambda _{2j}^{ru})^+]\}\bigg \rangle ,\nonumber \\&\quad \vdots \nonumber \\&\bigg \langle \max g_{jr},\{[(\lambda _{rj}^{1l})^+,(\lambda _{rj}^{1u})^+],[(\lambda _{rj}^{2l})^+,(\lambda _{rj}^{2u})^+],\cdots ,[(\lambda _{rj}^{rl})^+,(\lambda _{rj}^{ru})^+]\}\bigg \rangle \bigg \}. \end{aligned}$$

(1)

$$\begin{aligned} \widetilde{I}_h^-=\bigg \{&\bigg \langle \min g_{j1},\{[(\lambda _{1j}^{1l})^-,(\lambda _{1j}^{1u})^-],[(\lambda _{1j}^{2l})^-,(\lambda _{1j}^{2u})^-],\ldots ,[(\lambda _{1j}^{rl})^-,(\lambda _{1j}^{ru})^-]\}\bigg \rangle ,\nonumber \\&\bigg \langle \max g_{j2},\{[(\lambda _{2j}^{1l})^-,(\lambda _{2j}^{1u})^-],[(\lambda _{2j}^{2l})^-,(\lambda _{2j}^{2u})^-],\ldots ,[(\lambda _{2j}^{rl})^-,(\lambda _{2j}^{ru})^-]\}\bigg \rangle ,\nonumber \\&\quad \vdots \nonumber \\&\bigg \langle \max g_{jr},\{[(\lambda _{rj}^{1l})^-,(\lambda _{rj}^{1u})^-],[(\lambda _{rj}^{2l})^-,(\lambda _{rj}^{2u})^-],\ldots ,[(\lambda _{rj}^{rl})^-,(\lambda _{rj}^{ru})^-]\}\bigg \rangle \bigg \}. \end{aligned}$$

(2)

For calculating the separation measures $\widetilde{D}_j^+$ and $\widetilde{D}_j^-$ of each alternative, we take the difference between grades and calculate the distance of each alternative from IVHFNSPIS and IVHFNSNIS and we use IVHF Euclidean distance. The separation measures $\widetilde{D}_j^+$ and $\widetilde{D}_j^-$ of alternatives are defined as follows:

$$\begin{aligned} \widetilde{D}_j^+&=\langle \widetilde{g}_j^+,\widetilde{d}_j^+\rangle ,~~\mathrm{where}~j=1,2,\ldots ,p.\nonumber \\ \widetilde{g}_j^+&=\sum \limits _{k=1}^q w_k|(g_{jk}^{\gamma l})^+-(g_{k}^{\gamma l})^+|,\nonumber \\ \widetilde{d}_j^+&=\sum \limits _{k=1}^q w_k d(\widetilde{\lambda }_{jk},\widetilde{\lambda }_j^+),\nonumber \\&=\sum \limits _{k=1}^qw_k\sqrt{\frac{1}{2r}\sum \limits _{\gamma =1}^r\bigg (|\lambda _{jk}^{\gamma l}-(\lambda _{k}^{\gamma l})^+|^2+|\lambda _{jk}^{\gamma u}-(\lambda _{k}^{\gamma u})^+|^2\bigg )}. \end{aligned}$$

(3)

$$\begin{aligned} \widetilde{D}_j^-&=\langle \widetilde{g}_j^-,\widetilde{d}_j^-\rangle ,~~\mathrm{where}~j=1,2,\cdots ,p.\nonumber \\ \widetilde{g}_j^-&=\sum \limits _{k=1}^q w_k|(g_{jk}^{\gamma l})^--(g_{k}^{\gamma l})^-|,\nonumber \\ \widetilde{d}_j^-&=\sum \limits _{k=1}^q w_k d(\widetilde{\lambda }_{jk},\widetilde{\lambda }_j^-),\nonumber \\&=\sum \limits _{k=1}^qw_k\sqrt{\frac{1}{2r}\sum \limits _{\gamma =1}^r\bigg (|\lambda _{jk}^{\gamma l}-(\lambda _{k}^{\gamma l})^-|^2+|\lambda _{jk}^{\gamma u}-(\lambda _{k}^{\gamma u})^-|^2\bigg )}. \end{aligned}$$

(4)

The relative closeness coefficient of each alternative $O_j$ can be computed by using the following formula:

$$\begin{aligned} \widetilde{E}_j=\bigg \langle \frac{\widetilde{g}_j^-}{\widetilde{g}_j^++\widetilde{g}_j^-},\frac{\widetilde{d}_j^-}{\widetilde{d}_j^++\widetilde{d}_j^-}\bigg \rangle ,~j=1,2,\ldots ,p. \end{aligned}$$

(5)

An alternative $O_j$ is nearer to the IVHFNSPIS $(\widetilde{I}_h^+)$ and farther from the IVHFNSNIS $(\widetilde{I}_h^-)$ as $\widetilde{E}_j$ approaches 1. So, according to the closeness coefficient $\widetilde{E}_j,$ we can determine the ranking order of each alternative and choose the best one.

Based on above technique, we develop practical approaches for MAGDM with IVHFNS information. In such a practical approaches, the information about evaluation attribute weights is completely unknown. The graphical structure of TOPSIS is described in Fig.1.

Fig. 1

Graphical structure of TOPSIS

1. Selection of a synthetic polymer

Synthetic polymers are the polymers made by human. From the utility point of view, they are commonly classified into various categories. They are found in a variety of consumer products; specially, they are used in manufacturing of different households products. In the age of science, a variety of synthetic polymers is available in market. In such a variety of synthetic polymers with different charming properties, plastic industries face the problem to choose a synthetic polymer for the manufacturing of its products. In the age of competition, every plastic industry wants to choose the best one, from synthetic organic polymers, which are commonly found in households that are described as

• Low-density polyethylene (LDPE),

• High-density polyethylene (HDPE),

• Polyvinyl chloride (PVC),

• Polytetrafluoroethylene (PTFE).

The products made by these polymers are mostly recycled. These polymers have different properties and are used to manufacture different products that are described as

• Low-density polyethylene is flexible, chemically inert and insulator, and it is used to manufacture toys, squeeze bottles, insulation cover (electric wires), flexible pipes and six pack rings, etc.

• High-density polyethylene is thermally stable, inert, high tensile strength and tough, and it is used to manufacture bottles, inner insulation (dielectric) of coax cable, plastic bags and pipes, etc.

• Polyvinyl chloride is flame retardant, insulator and chemically inert, and it is used to manufacture fencing, pipe (mainly draining), handbags, lawn chairs, non-food bottles, toys, vinyl flooring and electrical installation insulations, etc.

• Polytetrafluoroethylene is excellent dielectric, very low coefficient of friction and chemically inert, and it is used to manufacture non-stick pans, low-friction bearings, inner insulation (dielectric) of coax cable and coating against chemical attack, etc.

Industries are unable to take a decision about the selection of best one from these polymers, for such a MADM process, we use the technique of extended TOPSIS based on IVHFNSSs. We take the set of four polymers P mentioned above as alternatives and E the set of attributes “evaluation of polymers by the experts” based on the above-mentioned properties, where $E=\{e_1,e_2,e_3,e_4\}$ is a set of four different experts of polymers and experts have different opinions about polymers. All the experts are hesitant about other experts’ decision, because all have their own opinion about the properties of polymers. According to their opinion, each expert assigns different ratings to the polymers in the form of check marks that are based on properties of polymers. First of all, experts fix criteria with their mutual understanding; what these check marks represent is described as

• Five check marks represent “special” (extra properties from required criteria),

• Four check marks represent “best” (better properties from required criteria),

• Three check marks represent “better” (properties that fulfill the required criteria),

• Two check marks represent “good” (properties near to required criteria),

• One check mark represents “normal” (slightly less properties from required criteria),

• Box represents “minor” (only one property from required criteria).

This graded evaluation by check marks can be determined by numbers such as $G=\{0,1,2,3,4,5\},$ and what these evaluations in the form of numbers serve is given in Example 2.6. The structure for selection of best polymer by experts (that assign ratings to each synthetic polymer) is shown in Fig. 2.

Fig. 2

Structure for selection of best polymer

The ratings compiled from the four experts on related data are described in Table 12, which provide the information about the individual ratings of experts. The tabular representation of the 6-soft set that results from their fusion is given in Table 13.

Table 12 Ratings given by experts

Table 13 Corresponding 6-soft set

Table 13, provides the information about the opinion of experts in the form of 6-soft set. Further, experts (DMs) express their information in the form of IVHF6SS as described in the IVHFNS decision matrix (see Table 14).

Table 14 Tabular representation of IVHF6S decision matrix for synthetic polymers

Obviously, the numbers of IVHFEs are different in IVHFSs. In order to gain more accuracy, we prolong the larger one as far as, the length of both becomes equal, when we compare them. According to the regulations as describe, we consider that the experts show optimistic response in such a case, and change the IVHF data by adding the maximal values as mentioned in Table 15.

Table 15 Tabular representation of IVHF6S decision matrix for synthetic polymers by adding maximal values

Suppose the data about the weights of attributes completely depend upon the experts. Weights assigned to synthetic polymers by each expert are as follows:

$$\begin{aligned} w=(0.267,0.259,0.227,0.247). \end{aligned}$$

Use Eqs. (1) and (2) to determine the IVHFNSPIS $(\widetilde{I}_h^+)$ and IVHFNSNIS ($\widetilde{I}_h^-)$, respectively.

$$\begin{aligned} \widetilde{I}_h^+&=\{\langle 4,\{[0.6,0.7],[0.7,0.9],[0.7,0.9]\}\rangle ,\langle 5,\{[0.6,0.7],[0.7,0.8],[0.7,0.8]\}\rangle ,\\&\quad \quad \langle 5,\{[0.7,0.8],[0.7,0.9],[0.9,1]\}\rangle ,\langle 4,\{[0.6,0.8],[0.6,0.8],[0.6,0.8]\}\rangle \}.\\ \widetilde{I}_h^-&=\{\langle 1,\{[0.1,0.2],[0.2,0.3],[0.2,0.3]\}\rangle ,\langle 2,\{[0.2,0.4],[0.3,0.5],[0.4,0.5]\}\rangle ,\\&\quad \quad \langle 2,\{[0.3,0.4],[0.4,0.5],[0.4,0.5]\}\rangle ,\langle 3,\{[0.2,0.4],[0.4,0.5],[0.5,0.6]\}\rangle \}. \end{aligned}$$

Use Eqs. (3) and (4) to calculate the separation measures $\widetilde{D}_j^+$ and $\widetilde{D}_j^-$ of each synthetic polymer from IVHFNSPIS and IVHFNSNIS, and we use interval-valued hesitant fuzzy Euclidean distance. The separation measures $\widetilde{D}_j^+$ and $\widetilde{D}_j^-$ of synthetic polymers are calculated as follows:

$$\begin{aligned}&\widetilde{D}_1^+=\langle 0.5,0.1311\rangle ,~~~~~~\widetilde{D}_1^-=\langle 2.0,0.2822\rangle , \\&\widetilde{D}_2^+=\langle 1.4,0.2307\rangle ,~~~~~~\widetilde{D}_2^-=\langle 1.1,0.1845\rangle , \\&\widetilde{D}_3^+=\langle 0.6,0.0345\rangle ,~~~~~~\widetilde{D}_3^-=\langle 1.9,0.3708\rangle , \\&\widetilde{D}_4^+=\langle 2.5,0.4169\rangle ,~~~~~~\widetilde{D}_4^-=\langle 0.0,0.0000\rangle . \end{aligned}$$

Use Eq. (5) to calculate the relative closeness coefficients $\widetilde{E}_j$ of each alternative.

$$\begin{aligned}&\widetilde{E}_1=\langle 0.8,0.6828\rangle ,~~~~~~\widetilde{E}_2=\langle 0.4,0.4444\rangle , \\&\widetilde{E}_3=\langle 0.8,0.9149\rangle ,~~~~~~\widetilde{E}_4=\langle 0.0,0.0000\rangle . \end{aligned}$$

Finally, arrange the synthetic polymers $p_j,~(j=1,2,3,4)$ according to their ranks and grades obtained from relative closeness coefficients. It is easy to see

$$\begin{aligned} p_3>p_1>p_2>p_4. \end{aligned}$$

So, $p_3$ is the most suitable synthetic polymer according to the opinion of all experts. Finally, we say polyvinyl chloride (PVC) is the best synthetic polymer for plastic industry to utilize.

2. Analysis of minerals in the sand

Sand is a spontaneously impure material made up of elegantly divided mineral particles and rocks. It is defined by volume and considered refined than rocks and raw than residue. It can also be considered as a structural class of soil (a soil composed of more than $85\%$ sand-sized particles by mass). Most of the minerals may exist as sand grains somewhere. The configuration of sand may vary depending upon the sources of local rock and situations, but the most fundamental ingredient of sand is silica (silicon dioxide, or SiO$_2$), commonly exists in the form of quartz, and is specially found in inland continental and non-tropical coastal settings. Other minerals have limited quantity in sand or are found only in peculiar locations. Sand that contains the most quantity of silica is called silica sand, and normally it contains $95\%$ of silica in structure of quartz, extensively very low level of detrimental impurities specifically iron oxides, clay and refractory minerals including chromite, hematite, magnetite, limonite and ilmenite. The list of minerals in silica sand that describes their properties is subsequently presented.

• The most important mineral found in sand is quartz as compared to others. In most cases, the massiveness of sand is actually due to quartz and it is mostly found every where. It can have almost every color, but purely it is transparent. Usually, its grains are rounded and their rust-colored appearance is due to the fine covering of hematite pigment. It is excessively resistant to weathering, as well as universal rock-forming mineral and has no split behavior.

• Clay minerals are not sand-forming minerals, and due to enough small size, they are not considered as sand. Mostly, they exist in sand in the form of mud when it is wet or in the form of dust when it is dry. Beach sands are commonly free from dust and behave like a plain deposits. Dust is found especially in inland sand samples and in many rivers. Most of the minerals of clay are the weathering product of feldspars, but many other natural minerals also have a frequency to convert into clay.

• Chromite is basically an oxide mineral and shows the crystal structure similar to magnetite, but it has weakly magnetic behavior. Mostly, it is found in black to brownish black color and brown to brownish black color on thin edges in transmitted light. Chromite is an ore of chromium and is found in ultra-mafic igneous rocks.

• Hematite naturally exists as a common component of many sandstones and sand, but mostly its size is small as compared to sand-sized grains. Actually, it is very refind pigment on the covering of another grain. More generally, it is found in metallic gray and dull to bright red colors. It also exist in reddish rust-colored hue to many sandstones. A large number of hematite grains exist in the huge mineral portion of sand.

• Magnetite is a rock mineral and one of the main iron ores, and its identification is easy because of its magnetic behavior. Its color is black, gray with brownish tint in reflected sun. It has opaque crystal structure and some of its grains may found in octahedral crystal structure. The size of its grains is mostly small, and it actually comes from metamorphic and igneous sources.

• Limonite is one of the iron ores which is basically a mixture of hydrated iron, in varying composition. It is actually the hydration of hematite and magnetite and is relatively dense with a specific gravity. It has crystal habit with fine grained aggregate and powdery coating. Generally, it is found in various shades of brown and yellow.

• In most of igneous and metamorphic rocks, ilmenite is a widespread adornment mineral. Its color is metallic black and has opaque nature. Due to intergrown magnetite, It shows a weak magnetic behavior. For its titanium content, ilmenite is mined.

Sand minerals are considered as the most increasingly important components in products that are part of our everyday life. Different companies are working in such a way, to check the amount of minerals in sand with the help of their geological experts team. The purpose of these companies is to analyze the minerals present in sand. Sometimes companies are unable to take a decision about the analysis of sand, because it is composed of a lot of minerals that are not easy to analyze. For such a multi-attribute decision-making process, we use the method of extended TOPSIS based on IVHFNSS. In such a case, we take the sample of silica sand, and normally it contains $95\%$ of silica in the form of quartz, as mentioned above. The purpose of our method is to analyze the detrimental impurities found in silica sand other than quartz. According to it, we consider the set of minerals $M=\{clay, chromite, hematite, magnetite, limonite, ilmenite\}$ as alternatives which are found in the sample of silica sand other than quartz and G the set of attributes “evaluation of minerals by the geology experts team” based on the above-mentioned properties of each mineral, where $G=\{g_1,g_2,g_3\}$ is a set of geology experts team and they have different opinions about minerals present in silica sand. All the geology experts in team are hesitant about other experts’ decision, because all have their own opinion about the properties of minerals. According to their opinion, each geology expert assigns different ratings to the minerals in the form of check marks that are based on properties of minerals. First of all, geology experts team set a criteria with their mutual understanding; what these check marks represent is described as

• Five check marks represent “extra ordinary pure,”

• Four check marks represent “pure structure,”

• Three check marks represent “slightly pure structure,”

• Two check marks represent “impure structure,”

• One check mark represents “totally impure structure,”

• Box represents “strange structure.”

This graded evaluation by check marks can be determined by numbers such that $G=\{0,1,2,3,4,5\},$ and what these evaluations in the form of numbers serve given in Example 2.6. The structure of analysis of sand minerals by geology experts (that assign ratings to each mineral) is shown in Fig. 3.

Fig. 3

Structure of analysis of minerals of sand

The ratings compiled from geology experts team on related data are described in Table 16, which provides the information about the individual ratings of geology experts, and the tabular representation of the 6-soft set that results from their fusion is given in Table 17.

Table 16 Ratings given by experts

Table 17 Corresponding 6-soft set

Table 17 provides the information about the opinion of experts in the form of 6-soft set. Further, experts (DMs) express their information in the form of IVHF6SS as described in the IVHFNS decision matrix (see Table 18).

Table 18 Tabular representation of IVHF6S decision matrix for minerals of sand

Similarly, as previously described, the numbers of IVHFEs are different in IVHFSs. In order to gain more accuracy, we prolong the smaller one as far as, the length of both becomes equal, when we compare them. According to the regulations as described, we consider that the experts show pessimistic response in such a case and change the IVHF data by adding the minimal values as mentioned in Table 19.

Table 19 Tabular representation of IVHF6S decision matrix for minerals of sand by adding minimal values

Suppose that the data about the weights of attributes completely depend upon the experts. Weights assigned to minerals by each expert are as follows:

$$\begin{aligned} w=(0.35,0.32,0.33). \end{aligned}$$

Use Eqs. (1) and (2) to determine the IVHFNSPIS $(\widetilde{I}_h^+)$ and IVHFNSNIS ($\widetilde{I}_h^-)$, respectively.

$$\begin{aligned} \widetilde{I}_h^+&=\{\langle 5,\{[0.8,0.9],[0.6,0.8],[0.6,0.8],[0.6,0.8]\}\rangle ,\langle 5,\{[0.7,0.8],[0.6,0.7],\\&\quad \quad [0.5,0.6],[0.5,0.6]\}\rangle ,\langle 4,\{[0.7,0.9],[0.6,0.7],[0.5,0.6],[0.4,0.5]\}\rangle \}.\\ \widetilde{I}_h^-&=\{\langle 2,\{[0.4,0.5],[0.3,0.4],[0.2,0.3],[0.1,0.2]\}\rangle ,\langle 2,\{[0.3,0.4],[0.2,0.3],\\&\quad \quad [0.1,0.2],[0.1,0.2]\}\rangle ,\langle 2,\{[0.2,0.3],[0.1,0.2],[0.1,0.2],[0.1,0.2]\}\rangle \}. \end{aligned}$$

Use Eqs. (3) and (4) to calculate the separation measures $\widetilde{D}_j^+$ and $\widetilde{D}_j^-$ of each mineral from IVHFNSPIS and IVHFNSNIS, and we use interval-valued hesitant fuzzy Euclidean distance. The separation measures $\widetilde{D}_j^+$ and $\widetilde{D}_j^-$ of minerals are calculated as follows:

$$\begin{aligned}&\widetilde{D}_1^+=\langle 0.7,0.14\rangle ,~~~~~~\widetilde{D}_1^-=\langle 2.0,0.30\rangle , \\&\widetilde{D}_2^+=\langle 1.0,0.56\rangle ,~~~~~~\widetilde{D}_2^-=\langle 0.7,0.35\rangle , \\&\widetilde{D}_3^+=\langle 2.0,0.28\rangle ,~~~~~~\widetilde{D}_3^-=\langle 0.7,0.16\rangle , \\&\widetilde{D}_4^+=\langle 0.3,0.03\rangle ,~~~~~~\widetilde{D}_4^-=\langle 2.4,0.41\rangle , \\&\widetilde{D}_5^+=\langle 1.7,0.27\rangle ,~~~~~~\widetilde{D}_5^-=\langle 0.9,0.18\rangle , \\&\widetilde{D}_6^+=\langle 2.1,0.25\rangle ,~~~~~~\widetilde{D}_6^-=\langle 0.7,0.11\rangle . \end{aligned}$$

Use Eq. (5) to calculate the relative closeness coefficients $\widetilde{E}_j$ of each mineral.

$$\begin{aligned}&\widetilde{E}_1=\langle 0.7,0.68\rangle ,~~~~~~\widetilde{E}_2=\langle 0.7,0.35\rangle , \\&\widetilde{E}_3=\langle 0.3,0.36\rangle ,~~~~~~\widetilde{E}_4=\langle 0.9,0.93\rangle , \\&\widetilde{E}_5=\langle 0.4,0.40\rangle ,~~~~~~\widetilde{E}_6=\langle 0.3,0.31\rangle . \end{aligned}$$

For the comparison, arrange the minerals $m_j~(j=1,2,3,4,5,6)$ according to their interval values obtained from relative closeness coefficients. It is easy to see

$$\begin{aligned} m_4>m_1>m_5>m_3>m_2>m_6. \end{aligned}$$

But the percentage of each mineral present in silica sand is taken from ranks or gradings calculated from relative closeness coefficients. From relative closeness coefficients, we compile:

• The quantity of “clay” is $0.7\%$ in silica sand.

• The quantity of “chromite” is $0.7\%$ in silica sand.

• The quantity of “hemitite” is $0.3\%$ in silica sand.

• The quantity of “magnetite” is $0.9\%$ in silica sand.

• The quantity of “limonite” is $0.4\%$ in silica sand.

• The quantity of “ ilmenite” is $0.3\%$ in silica sand.

Finally, we conclude that the sample of silica sand consists of $96.7\%$ of silica in the form of quartz and $3.3\%$ of other minerals as described above. The proposed method of MAGDM is described in an algorithm.

Graphical representation of the proposed approach for MAGDM is described in Fig. 4.

Fig. 4

Graphical representation of the proposed approach for MAGDM or extended TOPSIS

4 Sensitivity Analysis and Discussion

(1) Advantages of the proposed model

Generally, the real-world MADM and MAGDM problems occur in complex environment under imprecise and uncertain data, which is difficult to handle. The proposed model is very suitable for the situation when the information is complex, imprecise and uncertain. Specially, when the existing data based upon grades, ratings and hesitant decisions by decision-makers. We also extend the TOPSIS method to deal with complex and IVHFNS information. It is suitable when complete information regarding comparison and actual amount of alternatives is required.

(2) Disadvantages of the proposed model

A little bit flaws are there in the proposed model, including its complex structure, the huge data in the form of grades and hesitant information. Such a huge data are difficult to handle, because of the massive calculations, which are not so easy to perform.

(3) Comparison with existing models

The proposed model is basically the combination of N-soft sets [11] and IVHFSSs [23]. Both the previously existing models have their own worth in the literature: former one is helpful in dealing the information when data exist in the form of grades or ratings according to real situations, and the latter one is beneficial when a wide range of data are required according to the hesitant situation of decision-makers. In MAGDM, proposed model is able to provide the complete information about available data such as the occurrence of grades or ratings, to choose the best alternative by using extended TOPSIS method, to compare the alternatives and to provide the exact ratio of each alternative. Both the previously existing models are unable to define all this information at the same time.

5 Conclusion

MAGDM is considered as one of the most general and attractive phenomena in the theory of decision science. A variety of methods have been reported in the literature to handle the MAGDM problems. In this research article, we have presented novel model called IVHFNSSs, the hybridization of N-soft sets and IVHFSs, which provides more accuracy and flexibility in decision-making problems. Further, we have developed extended TOPSIS approach for MAGDM based on IVHFNS environment, in which the weights of decision-makers are considered as an arbitrary approach. Moreover, we have investigated the properties and basic operations of IVHFNSSs. Finally, we have designed an algorithm to solve practical problems for MAGDM. It is also noted that our proposed model is complex due to its massive calculations. In the future, we plan to extend our research study to reduce the massive calculations in our proposed model by (1) computer programming and construct-related hybrid models; (2) interval-valued hesitant fuzzy N-soft graphs; (3) N-soft hesitant fuzzy rough hypergraphs; (4) fuzzy N-soft rough diagraphs; and (5) TOPSIS method to multiple criteria decision-making with Pythagorean fuzzy N-soft sets.