Hybrid decision support system disaster management: application of lattice ordered q-rung linear Diophantine fuzzy hypersoft sets

Introduction

The frequent occurrence of uncertainty-related issues in multi-attribute decision-making (MADM) makes them difficult to foresee and manage due to the extensive modeling of these uncertainties. The fuzzy set (FS) theory introduced by Zadeh (1965) is very useful for handling the difficulties brought on by uncertainty. However, FS only has a limited ability to reflect impartial situations. To overcome these restrictions, Atanassov (1986) devised the notion of intuitionistic fuzzy sets (IFS). The IFS’s two indices are membership degree (MD) and non-membership degree (NMD), and their sum value should fall within [0,1]. To solve problems smoothly, Yager (2013) developed the Pythagorean fuzzy set (PFS) in which the total of the MD² and NMD² should fall within [0,1]. Yager (2016) also proposed the q-rung orthopair fuzzy sets (q-ROFS), where the MD^q and the NMD^q are summed together and fall inside the range [0,1]. Later, various information measures (Peng & Liu, 2019) were proposed for q-ROFS. However, each of these ideas has drawbacks of its own. To overcome these drawbacks, Riaz & Hashmi (2019) formulated the theory of the linear Diophantine fuzzy set (LDFS), which contains the notion of reference parameters (RPs). Owing to the usefulness of LDFS, several researchers from various scientific fields were interested in them, and numerous significant studies were produced as a result (Mahmood et al., 2021a, 2021b). Subsequently, the idea of quadratic diophantine fuzzy set was proposed by Zia et al. (2023). Later, Almagrabi et al. (2022) created the q-rung linear Diophantine fuzzy set (q-RLDFS), a particular extension of the IFS, q-ROFS, and LDFS. Further, many real-world decision-making studies such as company selection problem (Ali, 2025), urban planning (Petchimuthu et al., 2025), logistics (Kannan, Jayakumar & Pethaperumal, 2025) and emerging technologies (Kumar & Pamucar, 2025). However, because they are not parametrized, each theory has drawbacks. To overcome the limitations brought on by parametrization, Molodtsov (1999) developed the idea of soft set (SS) theory, which handles vagueness in a parametric manner. Later, by incorporating FS and SS, Roy & Maji (2007) provided the idea of the fuzzy soft set (FSS), which helps present fuzzy data with parametric information. Similar to this, SS theory was incorporated with other extensions of FS theory such as IFS, PFS, q-ROFS, and LDFS (Çağman & Karataş, 2013; Peng et al., 2015; Hussain et al., 2020; Riaz et al., 2020) respectively, to exhibit these fuzzy extension data with parametric information and obtained intuitionistic fuzzy soft set (IFSS), Pythagorean fuzzy soft set (PFSS), q-rung orthopair fuzzy soft set (q-ROFSS) and linear Diophantine fuzzy soft set (LDFSS). Smarandache (2018) then transformed the function into a multi-attributed function to establish the idea of the hypersoft set (HSS) as an extension of SS. By incorporating HSS with FS and IFS, Smarandache (2018) also proposed the ideas of the fuzzy hypersoft set (FHSS) and intuitionistic fuzzy hypersoft set (IFHSS), which expresses FS and IFS data with multi-sub-parameter. Similarly, by incorporating q-ROFS with HSS, Khan, Gulistan & Wahab (2022) presented the q-rung orthopair fuzzy hypersoft set (q-ROFHSS), and by incorporating q-RLDFS with HSS, Surya et al. (2024) presented the q-rung linear Diophantine fuzzy hypersoft set (q-RLDFHSS). In many real-life problems, there is a ranking among the parameters to deal with such problems very effectively. Ali et al. (2015) proposed a lattice-ordered soft set (LOSS). Later, Aslam et al. (2019) discussed the notion of lattice-ordered fuzzy soft set (LOFSS), and Mahmood et al. (2018) discussed the notion of lattice-ordered intuitionistic fuzzy soft set (LOIFSS). Further, many researchers (Rajareega & Vimala, 2021; Pandipriya, Vimala & Begam, 2018; Mahmood, Rehman & Sezgin, 2018; Begam et al., 2020; Khan, Bakhat & Iftikhar, 2019; Sabeena Begam & Vimala, 2019) developed the concepts of lattice-ordered structure to various areas of FS theory and their extensions. Likewise, to discuss real-life q-RLDFHS problems when there is a ranking among the multi-sub-parameters the notion of lattice ordered q-rung linear Diophantine fuzzy hypersoft set (LOq-RLDFHSS) is essential.

Background

This section provides the requisite notations and definitions for this article.

A binary relation $\le$ on a non-empty set $\mathfrak{A}$ is said to be a partial order on $\mathfrak{A}$ if it is antisymmetric, reflexive and transitive. Also, $\le$ is said to be a total order on $\mathfrak{A}$ if $\mathfrak{a}\ne \mathfrak{b}$, either $\mathfrak{a}\le \mathfrak{b}$ or $\mathfrak{b}\le \mathfrak{a}$ $\mathrm{\forall }\mathfrak{a}\mathfrak{,}\mathfrak{b}\in \mathfrak{A}$.

A partial order set L is said to be a lattice if the set $\{\mathfrak{a}\mathfrak{,}\mathfrak{b}\}$ has a greatest lower bound and least upper bound $\mathrm{\forall }\mathfrak{a}\mathfrak{,}\mathfrak{b}\in$ L. If L contains 1 and 0 such that $\mathrm{\forall }\mathfrak{x}\in$ L, 0 $\le \mathfrak{x}\le$ 1, then L is called a bounded lattice.

Definition 2.1. Atanassov (1986): Let $\mathfrak{G}$ be the set of alternatives. A IFS $\mathrm{I}$ is defined as

$$\mathrm{I}=\{(\mathfrak{g},{\mathrm{\Omega }}_{\mathrm{I}}(\mathfrak{g}),{\mho }_{\mathrm{I}}(\mathfrak{g}))|\mathfrak{g}\in \mathfrak{G}\}$$where ${\mathrm{\Omega }}_{\mathrm{I}}(\mathfrak{g})$ and ${\mho }_{\mathrm{I}}(\mathfrak{g})\in$ [0,1] are MD and NMD fulfilling 0 $\le {\mathrm{\Omega }}_{\mathrm{I}}(\mathfrak{g})+{\mho }_{\mathrm{I}}(\mathfrak{g})\le$ 1.

Definition 2.2. Almagrabi et al. (2022): Let $\mathfrak{G}$ be the set of alternatives. A q-RLDFS $\mathrm{Q}$ is defined as

$$\mathrm{Q}=\{(\mathfrak{g},⟨{\mathrm{\Omega }}_{\mathrm{Q}}(\mathfrak{g}),{\mho }_{\mathrm{Q}}(\mathfrak{g})⟩,⟨{\mathrm{\Delta }}_{\mathrm{Q}}(\mathfrak{g}),{▽}_{\mathrm{Q}}(\mathfrak{g})⟩)|\mathfrak{g}\in \mathfrak{G}\}$$where ${\mathrm{\Omega }}_{\mathrm{Q}}(\mathfrak{g}),{\mho }_{\mathrm{Q}}(\mathfrak{g}),{\mathrm{\Delta }}_{\mathrm{Q}}(\mathfrak{g})$ and ${▽}_{\mathrm{Q}}(\mathfrak{g})\in$ [0,1] are MD, NMD and their corresponding RPs respectively, fulfilling 0 $\le {\mathrm{\Delta }}_{\mathrm{Q}}^q(\mathfrak{g})+{▽}_{\mathrm{Q}}^q(\mathfrak{g})\le$ 1 and 0 $\le {\mathrm{\Delta }}_{\mathrm{Q}}^q(\mathfrak{g}){\mathrm{\Omega }}_{\mathrm{Q}}(\mathfrak{g})+{▽}_{\mathrm{Q}}^q(\mathfrak{g}){\mho }_{\mathrm{Q}}(\mathfrak{g})\le$ 1 $\mathrm{\forall }\mathfrak{g}\in \mathfrak{G}$, q $\ge$ 1.

Definition 2.3. Molodtsov (1999): Let $\mathfrak{G}$ be the set of alternatives, $\mathcal{E}$ be the set of attributes, and $\mathcal{A}\subseteq \mathcal{E}$. Then SS is a pair $(\mathrm{\Theta },\mathcal{A})$ defined by the mapping

$$\mathrm{\Theta }:\mathcal{A}\to P(\mathfrak{G}\mathfrak{)}$$where P $(\mathfrak{G}\mathfrak{)}$ is the power set of $\mathfrak{G}$.

Definition 2.4. Ali et al. (2015): Let $(\mathrm{\Theta },\mathcal{A})$ be a SS over $\mathfrak{G}$, where

$$\mathrm{\Theta }:\mathcal{A}\to P(\mathfrak{G}\mathfrak{)}$$Then $(\mathrm{\Theta },\mathcal{A})$ is said to be a LOSS if ${\mathfrak{a}}_1{\le }_{\mathcal{A}}{\mathfrak{a}}_2\Rightarrow \mathrm{\Theta }({\mathfrak{a}}_1)\subseteq \mathrm{\Theta }({\mathfrak{a}}_2)\mathrm{\forall }{\mathfrak{a}}_1,{\mathfrak{a}}_2\in \mathcal{A}$.

Definition 2.5. Çağman & Karataş (2013): Let $\mathfrak{G}$ be the set of alternatives, $\mathcal{E}$ be the set of attributes, and $\mathcal{A}\subseteq \mathcal{E}$. Then IFSS is a pair $(\mathrm{\Theta },\mathcal{A})$ defined by the mapping

$$\mathrm{\Theta }:\mathcal{A}\to IFP(\mathfrak{G}\mathfrak{)}$$where IFP $(\mathfrak{G}\mathfrak{)}$ is the IF power set of $\mathfrak{G}$.

Definition 2.6. Mahmood et al. (2018): Let $(\mathrm{\Theta },\mathcal{A})$ be a IFSS over $\mathfrak{G}$, where

$$\mathrm{\Theta }:\mathcal{A}\to IFP(\mathfrak{G}\mathfrak{)}$$Then $(\mathrm{\Theta },\mathcal{A})$ is said to be a LOIFSS if ${\mathfrak{a}}_1{\le }_{\mathcal{A}}{\mathfrak{a}}_2\Rightarrow \mathrm{\Theta }({\mathfrak{a}}_1)\subseteq \mathrm{\Theta }({\mathfrak{a}}_2)\mathrm{\forall }{\mathfrak{a}}_1,{\mathfrak{a}}_2\in \mathcal{A}$.

Definition 2.7. Smarandache (2018): Let $\mathfrak{G}$ be the set of alternatives and $P(\mathfrak{G}\mathfrak{)}$ denote the Power set of $\mathfrak{G}$. Let ${\mathcal{E}}_1,{\mathcal{E}}_2,...,{\mathcal{E}}_n$ with ${\mathcal{E}}_i\cap {\mathcal{E}}_j=\mathrm{\varnothing }$ for $i,j\in \{1,2,\dots n\}$ and $i\ne j$ be the attribute values of n distinct attributes $e_1,e_2,\dots ,e_n$ respectively and for each $\mathrm{i}=1,2,\dots \mathrm{n}$, ${\mathcal{A}}_i$ be non empty subset of ${\mathcal{E}}_i$ and ${\mathrm{\aleph }}_1={\mathcal{A}}_1\times {\mathcal{A}}_2\times \dots \times {\mathcal{A}}_n\subseteq {\mathcal{E}}_1\times {\mathcal{E}}_2\times \dots \times {\mathcal{E}}_n$. Then HSS over $\mathfrak{G}$ is the pair $(\mathrm{\Theta },{\mathrm{\aleph }}_1)$ defined by the map

$$\mathrm{\Theta }:{\mathrm{\aleph }}_1\to P(\mathfrak{G}\mathfrak{)}$$This can be represented as $(\mathrm{\Theta },{\mathrm{\aleph }}_1)=\{(\eta ,\mathrm{\Theta }(\eta )):\eta \in {\mathrm{\aleph }}_1,\mathrm{\Theta }(\eta )\in P(\mathfrak{G}\mathfrak{)}\}$.

Definition 2.8. Surya et al. (2024): Let $\mathfrak{G}$ be the set of alternatives and q-RLDFP ( $\mathfrak{G}$) denote the q-RLDF Power set of $\mathfrak{G}$. Let ${\mathcal{E}}_1,{\mathcal{E}}_2,\dots ,{\mathcal{E}}_n$ with ${\mathcal{E}}_i\cap {\mathcal{E}}_j=\mathrm{\varnothing }$ for $i,j\in \{1,2,\dots n\}$ and $i\ne j$ be the attribute values of n distinct attributes $e_1,e_2,\dots ,e_n$ respectively and for each $\mathrm{i}=1,2,\dots \mathrm{n}$, ${\mathcal{A}}_i$ be non empty subset of ${\mathcal{E}}_i$ and ${\mathrm{\aleph }}_1={\mathcal{A}}_1\times {\mathcal{A}}_2\times \dots \times {\mathcal{A}}_n\subseteq {\mathcal{E}}_1\times {\mathcal{E}}_2\times \dots \times {\mathcal{E}}_n$. Then, the q-Rung Linear Diophantine Fuzzy Hypersoft Set over $\mathfrak{G}$ (q-RLDFHSS ( $\mathfrak{G}$)) is the pair $(\mathrm{\Theta },{\mathrm{\aleph }}_1)$ defined by the map

$$\mathrm{\Theta }:{\mathrm{\aleph }}_1\to q-RLDFP(\mathfrak{G}\mathfrak{)}$$This can be represented as $(\mathrm{\Theta },{\mathrm{\aleph }}_1)=\{(\eta ,\mathrm{\Theta }(\eta )):\eta \in {\mathrm{\aleph }}_1,\mathrm{\Theta }(\eta )\in q-RLDFP(\mathfrak{G}\mathfrak{)}\}$ and the q-RLDFHS Number (q-RLDFHSN)

${\mathrm{\Theta }}_{{\mathfrak{g}}_a}({\eta }_c)$ = $\{⟨{\mathrm{\Omega }}_{\mathrm{\Theta }({\eta }_c)}({\mathfrak{g}}_a),{\mho }_{\mathrm{\Theta }({\eta }_c)}({\mathfrak{g}}_a)⟩,⟨{\mathrm{\Delta }}_{\mathrm{\Theta }({\eta }_c)}({\mathfrak{g}}_a),{▽}_{\mathrm{\Theta }({\eta }_c)}({\mathfrak{g}}_a)⟩|{\mathfrak{g}}_a\in \mathfrak{G}and{\eta }_c\in {\mathrm{\aleph }}_1\}$ can be express as ${\mathfrak{J}}_{{\eta }_{ac}}$ = $\{⟨{\mathrm{\Omega }}_{{\eta }_{ac}},{\mho }_{{\eta }_{ac}}⟩,⟨{\mathrm{\Delta }}_{{\eta }_{ac}},{▽}_{{\eta }_{ac}}⟩\}$.

Definition 2.9. Surya et al. (2024): Let $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1),({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ q-RLDFHSS ( $\mathfrak{G}$), then $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)$ is said to be q-RLDFHS subset of $({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)$, if

(i) ${\mathrm{\aleph }}_1\subseteq {\mathrm{\aleph }}_2$

(ii) $\mathrm{\forall }\eta \in {\mathrm{\aleph }}_1,{\mathrm{\Theta }}_1(\eta )\subseteq {\mathrm{\Theta }}_2(\eta )$

(i.e.,) ${\mathrm{\Omega }}_{{\mathrm{\Theta }}_1(\eta )}({\mathfrak{g}}_a)\le {\mathrm{\Omega }}_{{\mathrm{\Theta }}_2(\eta )}({\mathfrak{g}}_a),{\mho }_{{\mathrm{\Theta }}_2(\eta )}({\mathfrak{g}}_a)\le {\mho }_{{\mathrm{\Theta }}_1(\eta )}({\mathfrak{g}}_a),{\mathrm{\Delta }}_{{\mathrm{\Theta }}_1(\eta )}({\mathfrak{g}}_a)\le {\mathrm{\Delta }}_{{\mathrm{\Theta }}_2(\eta )}({\mathfrak{g}}_a)$ and ${▽}_{{\mathrm{\Theta }}_2(\eta )}({\mathfrak{g}}_a)\le {▽}_{{\mathrm{\Theta }}_1(\eta )}({\mathfrak{g}}_a)\mathrm{\forall }{\mathfrak{g}}_a\in \mathfrak{G}$.

Algebraic operations of loq-rldfhss

In this section, the fundamental algebraic operations of LOq-RLDFHSS are presented.

Definition 3.1. A $q\text{-}RLDFHSS(\mathfrak{G}\mathfrak{)}$ $(\mathrm{\Theta },{\mathcal{A}}_1\times {\mathcal{A}}_2\times ...\times {\mathcal{A}}_n={\mathrm{\aleph }}_1)$ is said to be lattice ordered q-RLDFHSS over $\mathfrak{G}$ (LOq-RLDFHSS ( $\mathfrak{G}$)) if for mapping $\mathrm{\Theta }:{\mathrm{\aleph }}_1\to q\text{-}RLDFP(\mathfrak{G}\mathfrak{)}$,

${\eta }_1{\le }_{{\mathrm{\aleph }}_1}{\eta }_2\Rightarrow \mathrm{\Theta }({\eta }_1)\subseteq \mathrm{\Theta }({\eta }_2)$ $\mathrm{\forall }$ ${\eta }_1,{\eta }_2\in {\mathrm{\aleph }}_1$

(i.e.,) ${\eta }_1{\le }_{{\mathrm{\aleph }}_1}{\eta }_2$

   $\Rightarrow {\mathrm{\Omega }}_{\mathrm{\Theta }({\eta }_1)}({\mathfrak{g}}_a)\le {\mathrm{\Omega }}_{\mathrm{\Theta }({\eta }_2)}({\mathfrak{g}}_a)$, ${\mho }_{\mathrm{\Theta }({\eta }_2)}({\mathfrak{g}}_a)\le {\mho }_{\mathrm{\Theta }({\eta }_1)}({\mathfrak{g}}_a)$,

    ${\mathrm{\Delta }}_{\mathrm{\Theta }({\eta }_1)}({\mathfrak{g}}_a)\le {\mathrm{\Delta }}_{\mathrm{\Theta }({\eta }_2)}({\mathfrak{g}}_a)$ and ${▽}_{\mathrm{\Theta }({\eta }_2)}({\mathfrak{g}}_a)\le {▽}_{\mathrm{\Theta }({\eta }_1)}({\mathfrak{g}}_a)\mathrm{\forall }$ ${\mathfrak{g}}_a\in \mathfrak{G}$

where ${\eta }_1=({\eta }_{1_1},{\eta }_{1_2},\dots ,{\eta }_{1_n}),{\eta }_2=({\eta }_{2_1},{\eta }_{2_2},\dots ,{\eta }_{2_n})$ and ${\eta }_{1_i},{\eta }_{2_i}\in {\mathcal{A}}_i$ for $i\in \{1,2,\dots ,n\}$.

Also, each ${\mathcal{A}}_i$ is defined by its corresponding binary relation ${\le }_{{\mathcal{A}}_i}$ and ${\mathrm{\aleph }}_1$ forms a relation defined by $({\eta }_{1_1},{\eta }_{1_2},\dots ,{\eta }_{1_n}){\le }_{{\mathrm{\aleph }}_1}({\eta }_{2_1},{\eta }_{2_2},\dots ,{\eta }_{2_n})\iff {\eta }_{1_i}{\le }_{{\mathcal{A}}_i}{\eta }_{2_i}$ in ${\mathcal{A}}_i$ for $i\in \{1,2,\dots ,n\}$.

The following example clarifies the definition above.

EXAMPLE 1. Let $\mathfrak{G}\mathfrak{=}\{{\mathfrak{g}}_{\mathfrak{1}}\mathfrak{,}{\mathfrak{g}}_{\mathfrak{2}}\mathfrak{,}{\mathfrak{g}}_{\mathfrak{3}}\}$ be the set of hotels for accommodation, consider the attributes $e_1$ = {charges}, $e_2=\{$food}, $e_3=\{$service} and ${\mathcal{E}}_1$ = {extra charges $(e_{11})$, room rent $(e_{12})\}$, ${\mathcal{E}}_2$ = {taste $(e_{21})$, hygiene $(e_{22})\}$, ${\mathcal{E}}_3$ = {customer service $(e_{31})\}$ be their corresponding attribute values respectively.

Suppose that,

For each i = 1, 2, 3, ${\mathcal{A}}_i$ = ${\mathcal{E}}_i$

The elements in each set ${\mathcal{A}}_1,{\mathcal{A}}_{2}and{\mathcal{A}}_3$ have an order among them, they are

The elements in set ${\mathcal{A}}_1$ are in the order $e_{11}{\le }_{{\mathcal{A}}_1}{e}_{12}$

The elements in set ${\mathcal{A}}_2$ are in the order $e_{21}{\le }_{{\mathcal{A}}_2}{e}_{22}$

${\mathcal{A}}_3$ has only one element $e_{31}$ and

$\mathrm{\aleph }={\mathcal{A}}_1\times {\mathcal{A}}_2\times {\mathcal{A}}_3=\{{\eta }_1=(e_{11},e_{21},e_{31}),{\eta }_2=(e_{11},e_{22},e_{31}),{\eta }_3=(e_{12},e_{21},e_{31}),$

${\eta }_4=(e_{12},e_{22},e_{31})\}$

Then the order of elements in set $\mathrm{\aleph }$ is shown in Fig. 1.

Further, the following is how the attributes are categorized

• The attribute “charges” and its attribute values indicates whether the alternative is cheap or not cheap

• The attribute “food” and its attribute values indicates whether the alternative is good or not good

• The attribute “service” and its attribute values indicates whether the alternative satisfies or dissatisfies

Then, the Cartesian product of attribute values exemplifies that the alternative is (cheap, good, satisfies) altogether or (not cheap, not good, dissatisfies) altogether.

Then, q-RLDFHSS ( $\mathrm{\Theta },\mathrm{\aleph }$) may be expressed as

$$\begin{array}{r}(\mathrm{\Theta },\mathrm{\aleph })=\{⟨{\eta }_1,\left(\frac{{\mathfrak{g}}_1}{⟨(0.4,0.8),(0.3,0.9)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.3,0.7),(0.4,0.9)⟩},\frac{{\mathfrak{g}}_3}{⟨(0.4,0.7),(0.2,0.7)⟩}\right)⟩,\\ ⟨{\eta }_2,\left(\frac{{\mathfrak{g}}_1}{⟨(0.4,0.7),(0.4,0.8)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.4,0.6),(0.5,0.7)⟩},\frac{{\mathfrak{g}}_3}{⟨(0.5,0.6),(0.4,0.6)⟩}\right)⟩,\\ ⟨{\eta }_3,\left(\frac{{\mathfrak{g}}_1}{⟨(0.5,0.7),(0.5,0.8)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.4,0.6),(0.5,0.8)⟩},\frac{{\mathfrak{g}}_3}{⟨(0.4,0.6),(0.3,0.6)⟩}\right)⟩,\\ ⟨{\eta }_4,\left(\frac{{\mathfrak{g}}_1}{⟨(0.6,0.6),(0.5,0.6)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.7,0.6),(0.7,0.5)⟩},\frac{{\mathfrak{g}}_3}{⟨(0.8,0.4),(0.7,0.3)⟩}\right)⟩\}\end{array}$$

We will assume that q = 3.

The characteristic of this q-RLDFHSS $(\mathrm{\Theta },\mathrm{\aleph })$ is $(⟨$MD, NMD $⟩,⟨$(cheap, good, satisfies), (not cheap, not good, dissatisfies) $⟩)$ $\mathrm{\forall }{\eta }_c\in \mathrm{\aleph }$.

Clearly $\mathrm{\Theta }({\eta }_1)\subseteq \mathrm{\Theta }({\eta }_2)\subseteq \mathrm{\Theta }({\eta }_4)$ and $\mathrm{\Theta }({\eta }_1)\subseteq \mathrm{\Theta }({\eta }_3)\subseteq \mathrm{\Theta }({\eta }_4)$, therefore, $(\mathrm{\Theta },{\mathrm{\aleph }}_1)$ is a LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$.

Definition 3.2. Let $\mathfrak{G}$ be the set of alternatives and $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1),({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$. Their Restricted union is defined by $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1){\cup }_{RES}({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)=({\mathrm{\Theta }}_3,{\mathrm{\aleph }}_3)$ where ${\mathrm{\aleph }}_3={\mathrm{\aleph }}_1\cap {\mathrm{\aleph }}_2$ and $\mathrm{\forall }$ $\eta \in {\mathrm{\aleph }}_3,\mathfrak{g}\in \mathfrak{G}$ we have ${\mathrm{\Theta }}_1(\eta )\cup {\mathrm{\Theta }}_2(\eta )={\mathrm{\Theta }}_3(\eta )$.

${\mathrm{\Omega }}_{{\mathrm{\Theta }}_3(\eta )}(\mathfrak{g})=Max\{{\mathrm{\Omega }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mathrm{\Omega }}_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g})\}$,

${\mho }_{{\mathrm{\Theta }}_3(\eta )}(\mathfrak{g})=Min\{{\mho }_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mho }_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g})\}$,

${\mathrm{\Delta }}_{{\mathrm{\Theta }}_3(\eta )}(\mathfrak{g})=Max\{{\mathrm{\Delta }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mathrm{\Delta }}_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g})\}$ and

${▽}_{{\mathrm{\Theta }}_3(\eta )}(\mathfrak{g})=Min\{{▽}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{▽}_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g})\}$.

Proposition 3.3. Let $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1),({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$. Then $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1){\cup }_{RES}({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$.

Proof. See “Proof of Proposition 3.2”. □

Definition 3.4. Let $\mathfrak{G}$ be the set of alternatives and $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1),({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$. Their Restricted intersection is defined by $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1){\cap }_{RES}({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)=({\mathrm{\Theta }}_3,{\mathrm{\aleph }}_3)$ where ${\mathrm{\aleph }}_3={\mathrm{\aleph }}_1\cap {\mathrm{\aleph }}_2$ and $\mathrm{\forall }$ $\eta \in {\mathrm{\aleph }}_3,\mathfrak{g}\in \mathfrak{G}$ we have ${\mathrm{\Theta }}_1(\eta )\cap {\mathrm{\Theta }}_2(\eta )={\mathrm{\Theta }}_3(\eta )$.

${\mathrm{\Omega }}_{{\mathrm{\Theta }}_3(\eta )}(\mathfrak{g})=Min\{{\mathrm{\Omega }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mathrm{\Omega }}_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g})\}$,

${\mho }_{{\mathrm{\Theta }}_3(\eta )}(\mathfrak{g})=Max\{{\mho }_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mho }_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g})\}$,

${\mathrm{\Delta }}_{{\mathrm{\Theta }}_3(\eta )}(\mathfrak{g})=Min\{{\mathrm{\Delta }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mathrm{\Delta }}_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g})\}$ and

${▽}_{{\mathrm{\Theta }}_3(\eta )}(\mathfrak{g})=Max\{{▽}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{▽}_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g})\}$.

Proposition 3.5. Let $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1),({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$. Then $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1){\cap }_{RES}({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$.

Proof. See “Proof of Proposition 3.4”. □

Definition 3.6. Let $\mathfrak{G}$ be the set of alternatives and $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1),({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$. Their extended union is defined by $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1){\cup }_{EXT}({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)=({\mathrm{\Theta }}_3,{\mathrm{\aleph }}_3)$ where ${\mathrm{\aleph }}_3={\mathrm{\aleph }}_1\cup {\mathrm{\aleph }}_2$

$$({\mathrm{\Theta }}_3,{\mathrm{\aleph }}_3)=\{\begin{array}{cc}\{⟨{\mathrm{\Omega }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mho }_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g})⟩,⟨{\mathrm{\Delta }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{▽}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g})⟩\}\hfill & if\eta \in {\mathrm{\aleph }}_1-{\mathrm{\aleph }}_2\hfill \\ \{⟨{\mathrm{\Omega }}_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g}),{\mho }_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g})⟩,⟨{\mathrm{\Delta }}_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g}),{▽}_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g})⟩\}\hfill & if\eta \in {\mathrm{\aleph }}_2-{\mathrm{\aleph }}_1\hfill \\ \{⟨Max\{{\mathrm{\Omega }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mathrm{\Omega }}_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g})\},Min\{{\mho }_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mho }_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g})\}⟩,\hfill & if\eta \in {\mathrm{\aleph }}_1\cap {\mathrm{\aleph }}_2\hfill \\ ⟨Max\{{\mathrm{\Delta }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mathrm{\Delta }}_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g})\},Min\{{▽}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{▽}_{{\mathrm{\Theta }}_2(\eta )}(\mathfrak{g})\}⟩\}\hfill & \hfill \end{array}$$

Proposition 3.7. Let $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1),({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$. Then $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1){\cup }_{EXT}({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$, if one of them is a LOq-RLDFHSS subset of other.

Proof. See “Proof of Proposition 3.6”. □

Definition 3.8. Let ${\mathrm{\aleph }}_1,{\mathrm{\aleph }}_2\subseteq {\mathcal{E}}_1\times {\mathcal{E}}_2\times \dots \times {\mathcal{E}}_n$. Then partial order ${\le }_{{\mathrm{\aleph }}_1\times {\mathrm{\aleph }}_2}$ on ${\mathrm{\aleph }}_1\times {\mathrm{\aleph }}_2$ is defined as for any $({\eta }_1,{\varsigma }_1),({\eta }_2,{\varsigma }_2)\in {\mathrm{\aleph }}_1\times {\mathrm{\aleph }}_2,({\eta }_1,{\varsigma }_1){\le }_{{\mathrm{\aleph }}_1\times {\mathrm{\aleph }}_2}({\eta }_2,{\varsigma }_2)\iff {\eta }_1{\le }_{{\mathrm{\aleph }}_1}{\eta }_2$ and ${\varsigma }_1{\le }_{{\mathrm{\aleph }}_2}{\varsigma }_2$.

Definition 3.9. Let $\mathfrak{G}$ be the set of alternatives and $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1),({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$. Their “AND” operation is defined by $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)\wedge ({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)=(\mathrm{\Xi },{\mathrm{\aleph }}_1\times {\mathrm{\aleph }}_2)$

where

$$\mathrm{\Xi }({\mathrm{\aleph }}_1\times {\mathrm{\aleph }}_2)=\{(\eta ,\varsigma ),(\mathfrak{g},\mathrm{\Xi }(\eta ,\varsigma )(\mathfrak{g})):\mathfrak{g}\in \mathfrak{G},(\eta ,\varsigma )\in {\mathrm{\aleph }}_1\times {\mathrm{\aleph }}_2\}$$and $\mathrm{\Xi }(\eta ,\varsigma )(\mathfrak{g})=\{⟨Min\{{\mathrm{\Omega }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mathrm{\Omega }}_{{\mathrm{\Theta }}_2(\varsigma )}(\mathfrak{g})\},Max\{{\mho }_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mho }_{{\mathrm{\Theta }}_2(\varsigma )}(\mathfrak{g})\}⟩,$

$⟨Min\{{\mathrm{\Delta }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mathrm{\Delta }}_{{\mathrm{\Theta }}_2(\varsigma )}(\mathfrak{g})\},Max\{{▽}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{▽}_{{\mathrm{\Theta }}_2(\varsigma )}(\mathfrak{g})\}⟩\}$.

Proposition 3.10. Let $\mathfrak{G}$ be the set of alternatives and $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1),({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$. Then $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)\wedge ({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$.

Proof. See “Proof of Proposition 3.9”. □

Definition 3.11. Let $(\mathfrak{G}\mathfrak{)}$ be the set of alternatives and $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1),({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$. Then their “OR” operation is defined by $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)\vee ({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)=(\mathrm{\Xi },{\mathrm{\aleph }}_1\times {\mathrm{\aleph }}_2)$

where

$(\mathrm{\Xi },{\mathrm{\aleph }}_1\times {\mathrm{\aleph }}_2)=\{(\eta ,\varsigma ),(\mathfrak{g},\mathrm{\Xi }(\eta ,\varsigma )(\mathfrak{g})):\mathfrak{g}\in \mathfrak{G}\mathfrak{,}\mathfrak{(}\eta \mathfrak{,}\varsigma \mathfrak{)}\in {\mathrm{\aleph }}_{\mathfrak{1}}\times {\mathrm{\aleph }}_{\mathfrak{2}}\}$ and $\mathrm{\Xi }(\eta ,\varsigma )(\mathfrak{g})=\{⟨Max\{{\mathrm{\Omega }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mathrm{\Omega }}_{{\mathrm{\Theta }}_2(\varsigma )}(\mathfrak{g})\},Min\{{\mho }_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mho }_{{\mathrm{\Theta }}_2(\varsigma )}(\mathfrak{g})\}⟩,$

$⟨Max\{{\mathrm{\Delta }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mathrm{\Delta }}_{{\mathrm{\Theta }}_2(\varsigma )}(\mathfrak{g})\},Min\{{▽}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{▽}_{{\mathrm{\Theta }}_2(\varsigma )}(\mathfrak{g})\}⟩\}$

Proposition 3.12. Let $\mathfrak{G}$ be the set of alternatives and $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1),({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$. Then $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)\vee ({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$.

Proof. See “Proof of Proposition 3.11”. □

Definition 3.13. Let $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$.

If ${\mathrm{\Omega }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g})$ = ${\mathrm{\Delta }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g})$ = 0, ${\mho }_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g})$ = ${▽}_{{\mathrm{\Omega }}_1(\eta )}(\mathfrak{g})$ = 1 $\mathrm{\forall }\eta \in {\mathrm{\aleph }}_1$ and $\mathfrak{g}\in \mathfrak{G}$, Then, $({\mathrm{\Omega }}_1,{\mathrm{\aleph }}_1)$ is called the relative null LOq-RLDFHSS and is denoted by ${\mathrm{\varnothing }}_{{\mathrm{\aleph }}_1}$.

Definition 3.14. Let $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$.

If ${\mathrm{\Omega }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g})$ = ${\mathrm{\Delta }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g})$ = 1, ${\mho }_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g})$ = ${▽}_{{\mathrm{\Omega }}_1(\eta )}(\mathfrak{g})$ = 0 $\mathrm{\forall }\eta \in {\mathrm{\aleph }}_1$ and $\mathfrak{g}\in \mathfrak{G}$, Then, $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)$ is called the relative universal LOq-RLDFHSS and is denoted by ${\mathfrak{U}}_{{\mathrm{\aleph }}_1}$.

Proposition 3.15. Let $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$. Then

1. $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1){\cup }_{RES}({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)$ = $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)$

2. $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1){\cup }_{RES}{\mathrm{\varnothing }}_{{\mathrm{\aleph }}_1}$ = $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)$

3. $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1){\cup }_{RES}{\mathfrak{U}}_{{\mathrm{\aleph }}_1}$ = ${\mathfrak{U}}_{{\mathrm{\aleph }}_1}$

4. $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1){\cap }_{RES}({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)$ = $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)$

5. $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1){\cap }_{RES}{\mathrm{\varnothing }}_{{\mathrm{\aleph }}_1}$ = ${\mathrm{\varnothing }}_{{\mathrm{\aleph }}_1}$

6. $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1){\cap }_{RES}{\mathfrak{U}}_{{\mathrm{\aleph }}_1}$ = $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)$

Proof. Straightforward. □

Definition 3.16. Let $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$. Then complement of $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)$ denoted by $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1{)}^c$ and is defined as follows

$({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1{)}^c$ = $\{(\mathfrak{g},\{⟨{\mho }_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mathrm{\Omega }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g})⟩,⟨{▽}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mathrm{\Delta }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g})⟩\}):\eta \in {\mathrm{\aleph }}_1$ and $\mathfrak{g}\in \mathfrak{G}\}$.

Proposition 3.17. Let $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$. Then $(({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1{)}^c{)}^c=({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)$

Proof. Let $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)\in$ LOq-RLDFHSS $(\mathfrak{G}\mathfrak{)}$. Then complement of $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)$ is

$({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1{)}^c$ = $\{(\mathfrak{g},\{⟨{\mho }_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mathrm{\Omega }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g})⟩,⟨{▽}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mathrm{\Delta }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g})⟩\}):\eta \in {\mathrm{\aleph }}_1$ and $\mathfrak{g}\in \mathfrak{G}\}$,

Now complement of $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1{)}^c$ is

$(({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1{)}^c{)}^c$ = $\{(\mathfrak{g},\{⟨{\mathrm{\Omega }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{\mho }_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g})⟩,⟨{\mathrm{\Delta }}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g}),{▽}_{{\mathrm{\Theta }}_1(\eta )}(\mathfrak{g})⟩\}):\eta \in {\mathrm{\aleph }}_1$ and $\mathfrak{g}\in \mathfrak{G}\}$ = $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)$. □

EXAMPLE 2. Let $\mathfrak{G}\mathfrak{=}\{{\mathfrak{g}}_{\mathfrak{1}}\mathfrak{,}{\mathfrak{g}}_{\mathfrak{2}}\}$ be a set of alternatives, ${\mathrm{\aleph }}_1=\{{\eta }_1,{\eta }_2\}$ be a set of parameters with an order defined by ${\eta }_1{\le }_{\mathrm{\aleph }}{\eta }_2$ and ${\mathrm{\aleph }}_2=\{{\eta }_1,{\eta }_3\}$ be another set of parameters with an order defined by ${\eta }_1{\le }_{\mathrm{\aleph }}{\eta }_3$. Then, let

$$\begin{array}{rl}({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)=& \{⟨{\eta }_1,(\frac{{\mathfrak{g}}_1}{⟨(0.3,0.8),(0.4,0.8)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.2,0.8),(0.2,0.8)⟩})⟩,\\ & ⟨{\eta }_2,(\frac{{\mathfrak{g}}_1}{⟨(0.6,0.5),(0.5,0.7)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.6,0.5),(0.6,0.4)⟩})⟩\}\end{array}$$be a q-RLDFHSS with q as 3, and since ${\mathrm{\Theta }}_1({\eta }_1)\subseteq {\mathrm{\Theta }}_1({\eta }_2)$, this implies $({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)$ is a LOq-RLDFHSS. Also, let

$$\begin{array}{rl}({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)=& \{⟨{\eta }_1,(\frac{{\mathfrak{g}}_1}{⟨(0.5,0.8),(0.4,0.7)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.3,0.7),(0.2,0.8)⟩})⟩,\\ & ⟨{\eta }_3,(\frac{{\mathfrak{g}}_1}{⟨(0.6,0.7),(0.6,0.6)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.7,0.3),(0.9,0.2)⟩})⟩\}\end{array}$$be another q-RLDFHSS with q as 3, and since ${\mathrm{\Theta }}_2({\eta }_1)\subseteq {\mathrm{\Theta }}_2({\eta }_3)$, this implies $({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)$ is a LOq-RLDFHSS.

The following operations are then derived:

• $$({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1){\cup }_{RES}({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)=\left\{⟨{\eta }_1,\left(\frac{{\mathfrak{g}}_1}{⟨(0.5,0.8),(0.4,0.7)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.3,0.7),(0.2,0.8)⟩}\right)⟩\right\}$$

• $$({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1){\cap }_{RES}({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)=\left\{⟨{\eta }_1,\left(\frac{{\mathfrak{g}}_1}{⟨(0.3,0.8),(0.4,0.8)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.2,0.8),(0.2,0.8)⟩}\right)⟩\right\}$$

• $$\begin{array}{r}({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1){\cup }_{EXT}({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)=\{⟨{\eta }_1,\left(\frac{{\mathfrak{g}}_1}{⟨(0.5,0.8),(0.4,0.7)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.3,0.7),(0.2,0.8)⟩}\right)⟩,\\ ⟨{\eta }_2,\left(\frac{{\mathfrak{g}}_1}{⟨(0.6,0.5),(0.5,0.7)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.6,0.5),(0.6,0.4)⟩}\right)⟩,\\ ⟨{\eta }_3,\left(\frac{{\mathfrak{g}}_1}{⟨(0.6,0.7),(0.6,0.6)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.7,0.3),(0.9,0.2)⟩}\right)⟩\}\end{array}$$

• $$\begin{array}{r}({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)\vee ({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)=\{⟨({\eta }_1,{\eta }_1),\left(\frac{{\mathfrak{g}}_1}{⟨(0.5,0.8),(0.4,0.7)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.3,0.7),(0.2,0.8)⟩}\right)⟩,\\ ⟨({\eta }_1,{\eta }_3),\left(\frac{{\mathfrak{g}}_1}{⟨(0.6,0.7),(0.6,0.7)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.7,0.3),(0.9,0.2)⟩}\right)⟩,\\ ⟨({\eta }_2,{\eta }_1),\left(\frac{{\mathfrak{g}}_1}{⟨(0.6,0.5),(0.5,0.7)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.6,0.5),(0.6,0.4)⟩}\right)⟩,\\ ⟨({\eta }_2,{\eta }_3),\left(\frac{{\mathfrak{g}}_1}{⟨(0.6,0.5),(0.6,0.7)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.7,0.3),(0.9,0.2)⟩}\right)⟩\}\end{array}$$

• $$\begin{array}{r}({\mathrm{\Theta }}_1,{\mathrm{\aleph }}_1)\wedge ({\mathrm{\Theta }}_2,{\mathrm{\aleph }}_2)=\{⟨({\eta }_1,{\eta }_1),\left(\frac{{\mathfrak{g}}_1}{⟨(0.3,0.8),(0.4,0.8)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.2,0.8),(0.2,0.8)⟩}\right)⟩,\\ ⟨({\eta }_1,{\eta }_3),\left(\frac{{\mathfrak{g}}_1}{⟨(0.3,0.8),(0.4,0.8)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.2,0.8),(0.2,0.8)⟩}\right)⟩,\\ ⟨({\eta }_2,{\eta }_1),\left(\frac{{\mathfrak{g}}_1}{⟨(0.5,0.8),(0.4,0.7)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.3,0.7),(0.2,0.8)⟩}\right)⟩,\\ ⟨({\eta }_2,{\eta }_3),\left(\frac{{\mathfrak{g}}_1}{⟨(0.6,0.7),(0.5,0.7)⟩},\frac{{\mathfrak{g}}_2}{⟨(0.6,0.5),(0.6,0.4)⟩}\right)⟩\}\end{array}$$