An enhanced MADM with L q * q-Rung orthopair multi-fuzzy soft set in healthcare supplier selection

Abstract

The global healthcare systems have encountered unparalleled difficulties due to the COVID-19 pandemic, underscoring the crucial significance of effective management within healthcare supply chains. This research contributes to the field of healthcare supply chain management by presenting a robust MADM methodology called lattice ordered(L_{q
^*}) q-rung orthopair multi-fuzzy soft set(L_{q
^*} q-RO_MFS-MADM) for supplier evaluation and ranking amidst the challenges posed by the COVID-19 pandemic. Taking inspiration from multi-fuzzy soft set and q-rung orthopair fuzzy set, the present research article proposes a novel framework known as L_{q
^*} q-rung orthopair multi-fuzzy soft set (L_{q
^*} qRO_MFSS), which incorporates lattice ordering in q-rung orthopair multi-fuzzy soft set. The effectiveness of the proposed model is confirmed through successful experimentation on various important operations, including union, intersection, complement, restricted union and intersection. Moreover, the verification of De Morgan’s laws for L_{q
^*} qRO_MFSS is carried out specifically for these operations mentioned above. To highlight the significance of the proposed L_{q
^*} qRO_MFSS, a multi-attribute decision-making (MADM) problem is presented, showcasing its application in the domain of healthcare supply chain management. Furthermore, a comparative analysis is conducted to elucidate the advantages of this model in comparison to existing models.

Keywords

Lattice ordered multi-fuzzy soft set q-rung orthopair multi-fuzzy soft set supplier selection multi-attribute decision-making

1 Introduction

In recent years, healthcare suppliers played a pivotal role in maintaining healthcare services and responding to the challenges posed by the COVID-19 pandemic. Their timely and efficient provision of essential medical supplies was crucial for patient care, testing, research, and overall healthcare system functionality. The pandemic highlighted the need for robust and adaptable supply chains to ensure that healthcare systems can effectively respond to crises. Managing healthcare supply chains effectively has become crucial to ensure the smooth availability of medical items, services, and equipment, especially during difficult times. As healthcare systems grapple with increasing complexities, evolving patient needs, and dynamic market conditions, the role of supply chain management in guaranteeing patient care and operational efficiency has grown. Recent scholarly inquiries [15, 34, 35] into healthcare supply chain management have been directed towards unraveling the intricate interplay among diverse variables, encompassing elements like demand projection, supplier affiliations, inventory fine-tuning, and the incorporation of technology.

Multi-Attribute Decision Making (MADM) approaches have evolved as critical tools for assessing and selecting suppliers based on a variety of criteria in response to these problems. By carefully studying various factors such as supplier reliability, cost-effectiveness, and quality, these approaches help decision-makers to make educated decisions. The need for $MADMP$ arises in various fields such as Social Sciences, Medical Sciences, Environmental Sciences, Engineering, Ecology, Economics, and many other domains. These fields heavily rely on modeling uncertainties to understand complex systems and make informed decisions. Numerous models have been offered to address decision-making issues involving ambiguity, imprecision, and uncertainty. These include classical set theory, fuzzy set(FS) theory [32], probability theory, rough sets, and lattice theory [5].

Table 1
Advantages and Limitations of the Proposed L _q* q - RO _MFSS-MADM with Existing MADM Approaches

MADM -approaches Advantages Limitations

Lattice ordered fuzzy soft set-MADM [3] It is capable of managing elements with uncertain or imprecise parameters It is unable to manage non-membership values associated with parametrized elements

Lattice ordered intuitionistic fuzzy soft set-MADM [14] It has the capability to manage both membership values and non-non-membership values of parametrized elements This theory fails to be applicable in cases where the combined membership values and non-membership values of parametrized elements surpass unity. Additionally, these concepts are inadequate in addressing grades like abstinence

Lattice ordered multi-fuzzy soft set-MADM [18-20] It is capable of managing elements with uncertainty and flexibility using multi-dimensional memberships values It is unable to manage multi-non-membership values associated with parametrized elements

Proposed L _q* q - RO _MFSS-MADM It has the capability to manage both multi-membership values and multi-non-membership values of parametrized elements This theory fails to be applicable in cases where the sum of q^th power of multi-membership values and multi-non-membership values of parametrized elements surpass unity.

MADM -approaches	Advantages	Limitations
Lattice ordered fuzzy soft set-MADM [3]	It is capable of managing elements with uncertain or imprecise parameters	It is unable to manage non-membership values associated with parametrized elements
Lattice ordered intuitionistic fuzzy soft set-MADM [14]	It has the capability to manage both membership values and non-non-membership values of parametrized elements	This theory fails to be applicable in cases where the combined membership values and non-membership values of parametrized elements surpass unity. Additionally, these concepts are inadequate in addressing grades like abstinence
Lattice ordered multi-fuzzy soft set-MADM [18-20]	It is capable of managing elements with uncertainty and flexibility using multi-dimensional memberships values	It is unable to manage multi-non-membership values associated with parametrized elements
Proposed L _q* q - RO _MFSS-MADM	It has the capability to manage both multi-membership values and multi-non-membership values of parametrized elements	This theory fails to be applicable in cases where the sum of q^th power of multi-membership values and multi-non-membership values of parametrized elements surpass unity.

Multi-fuzzy sets [22, 23, 28] are a fresh application of fuzzy set theory that tackles some issues that are challenging to explain in other extensions, like pixel color, image recognition, etc. Additionally, yager introduced the term q-rung orthopair FS ( $q - ROFS$ ) [30] in 2017 as an improvement to the terms Pythagorean FS (PFS) [29] and intuitionistic FS (IFS) [4]. When the values q=1 and q=2, Intuitionistic FS and Pythagorean FS can be regarded as specific instances of q-ROFS. Extensive research has been conducted on IFS, PFS, Multi-Fuzzy Sets (MFS), q - ROFS.

Molodtsov [13] introduced the idea of a soft set (SS), which involves using different values for choices. In a similar vein, Maji et al. [12] explored how soft sets can be applied to decision-making, defining key concepts and their characteristics. The notions of fuzzy soft sets (FSS) and intuitionistic fuzzy soft sets (IFSS) were initially introduced by Maji et al. [11]. Building upon this, Peng et al. extended IFSS to Pythagorean fuzzy soft sets (PFSS) in 2015 [16]. By hybridizing the MFS and SS models, Yang et al. [31] introduced the idea of multi-fuzzy soft sets and applied them to decision-making. The idea of an intuitionistic multi-fuzzy soft set (IMFSS) was formulated by Das and Kar [7] as a way to tackle decision-making problems.

Hussain et al. [8] presented the q - ROFSS using the concepts of SS and qROFS. Following that, a lot of researchers looked into SS, MFSS, IMFSS, hesitant multi-fuzzy soft set and their applications [2, 9, 17]. In a recent study, the concept of q-rung orthopair multi-fuzzy soft set (q-ROMFSS) was offered by Vimala et al. [26] as an extension of IMFSS, which serves several advantageous enhancements to the decision-making process. Ali et al. [1] is the first to discuss the idea of lattice ordered soft sets and some of their characteristics. Subsequently, the concept of Lattice Ordered Fuzzy Soft Set (LOFSS) was introduced in [3], providing a means of establishing rankings among parameters in decision-making scenarios, proving to be highly advantageous.

The concept of lattice ordered soft groups was initially proposed by Vimala et al. [24, 25]. Their work also involved enhancing the lattice ordered theory with neutrosophic soft sets [27]. Building on this foundation, Mahmood et al. [10] and Muhammad Bilal Khan et al. [14] introduced the concept of lattice ordered IFSS and lattice (anti-lattice) ordered double framed soft sets (LODFrSS). As a result, Lattice ordered MFSS was later introduced by Sabeena et al. [18-20] as a broader version of successful MFSS concepts aimed at solving $MADMP$ s, but their $MADMP$ has some limitations. This paved the way for the development of L _q* q - RO _MFS set, which combines the strengths of both LOMFSS and q - ROFS to enhance the decision-making process in group-DMPs.

Here we plan to create a hybrid structure known as L _q* qRO _MFSS based on combined properties of q - ROFS and MFSS and these combinations allow us to maximize the handling of uncertain data. The proposed L _q* qRO _MFSS-MADM builds upon the foundation of the Lattice ordered MFSS, thereby extending and refining the existing framework. To assess the credibility of the presented L _q* qRO _MFSS-MADM, a case study is presented in which the five suppliers are evaluated based on five distinct criteria. The following summary encapsulates the content of this article: -

The notion of lattice ordered q-rung orthopair multi-fuzzy soft sets (LOMFSS) presents a hybrid mathematical model that combines the qualities of LOMFSS and q - ROFS. This novel technique is significant because it provides an improved mathematical tool capable of addressing a greater range of uncertainty than both LOMFSS and q - ROFS. This hybrid model is useful in a variety of decision-making contexts when there is a specific ordering of factors, allowing for a more thorough depiction of uncertainty.

The L _q* qRO _MFSS-MADM methodology allows healthcare organizations to assess suppliers based on factors like reliability, lead times, pricing, quality, and flexibility, while robust L _q* q - RO _MFSS manage uncertainty in supply chain data.

An empirical application to the discipline of supply chain management validates the rationality and accountability of the proposed techniques.

We compare our approach to existing L _q* qRO _MFSS-MADM methods using models from the LOIFSS, LOMFSS, LODFSS, and LONSS.

The organization of the article is as outlined below: Section 2 presents the preliminary concepts of MFS, MFSS, IMFSS, q - ROFS, q - RO _MFS and q - RO _MFSS. Section 3 defines the novel concept known as L _q* q - RO _MFSS with its related properties and also define the generalized De Morgan’s laws for L _q* q - RO _MFS set. Section 4, provides an illustrative example on healthcare supply chain management. Section 5, gives comparative analyses, and a conclusion to demonstrate the applicability of the suggested methodology.

2 Preliminaries

In this section, we present fundamental definitions and properties associated with L _q* q - RO _MFS set. In the course of this manuscript, we use $\ddot{\ddot{Y}}$ to denote the universe set and $E$ to represent a set of parameters. $P (\ddot{\ddot{Y}})$ corresponds to the power set of $\ddot{\ddot{Y}}$ , and $\ddot{\hat{A}}$ is a subset of $E$ .

Definition 2.1. [1] A pair $(L_{y}, \leq)$ is said to be a lattice if ≤ is a partial order on $L_{y}$ , satisfying the following conditions:

1.
For any $\ddot{l_{1}}, \ddot{l_{2}} \in L_{y}$ , the set ${\ddot{l_{1}}, \ddot{l_{2}}}$ has a supremum and an infimum in $L_{y}$ .
2.
If there exist elements 0 and 1 in $L_{u}$ such that 0 is less than or equal to every element $j \in L_{y}$ and every element $j \in L_{y}$ is less than or equal to 1, then $L_{y}$ is a bounded lattice.

Definition 2.2. [22, 23] A $MF$ set ϒ in $\ddot{\ddot{Y}}$ of dimension m is defined as a set of ordered strings: $ϒ = {\ddot{\ddot{y}} / (σ_{1} (\ddot{\ddot{y}}), σ_{2} (\ddot{\ddot{y}}), \dots, σ_{m} (\ddot{\ddot{y}})) : \ddot{\ddot{y}} \in \ddot{\ddot{Y}}},$ where $σ_{k} (\ddot{\ddot{y}}) \in P (\ddot{\ddot{Y}})$ for k = 1, 2, …, m. The function $σ_{m} (\ddot{\ddot{y}}) = (σ_{1} (\ddot{\ddot{y}}), σ_{2} (\ddot{\ddot{y}}), \dots, σ_{k} (\ddot{\ddot{y}}))$ is referred to as the multi-membership values (MV) of the $MF$ set on $\ddot{\ddot{Y}}$ .

Definition 2.3. [31] Let $(\ddot{\hat{F}}, \ddot{\hat{A}})$ be a $MFS$ set of dimension m over $\ddot{\ddot{Y}}$ , where $\ddot{\hat{F}}$ is a mapping given by $\ddot{\hat{F}}$ : $\ddot{\hat{A}}$ → $MF (\ddot{\ddot{Y}})$ .

Definition 2.4. [19] An $MFS$ set of dimension m over $\ddot{\ddot{Y}}$ , denoted as ( $\ddot{\hat{F}}$ , $\ddot{\hat{A}}$ ), is a lattice-ordered $MFS$ over $\ddot{\ddot{Y}}$ if the following condition holds: For any $e_{1}$ and $e_{2}$ in $\ddot{\hat{A}}$ such that $e_{1} \leq e_{2}$ , it follows that $\ddot{\hat{F}} (e_{1}) \subseteq \ddot{\hat{F}} (e_{2})$ . In other words, $σ_{\ddot{\hat{F}} (e_{1})} (\ddot{\ddot{y}}) \leq σ_{\ddot{\hat{F}} (e_{2})} (\ddot{\ddot{y}})$ and $τ_{\ddot{\hat{F}} (e_{1})} (\ddot{\ddot{y}}) \leq τ_{\ddot{\hat{F}} (e_{2})} (\ddot{\ddot{y}})$ for all $\ddot{\ddot{y}} \in \ddot{\ddot{Y}}$ .

Definition 2.5. [30] A q- $ROFS$ set in $\ddot{\ddot{Y}}$ is defined as $ϒ = {〈 \ddot{\ddot{y}}, σ (\ddot{\ddot{y}}), τ (\ddot{\ddot{y}}) 〉 : \ddot{\ddot{y}} \in \ddot{\ddot{Y}}}$ , where σ and τ are mappings from $\ddot{\ddot{Y}}$ to the interval [0,1]. They represent the degree of membership values (MVs) and non-membership values(NMVs) of the element $\ddot{\ddot{y}}$ in the set ϒ, respectively. The condition $0 \leq σ (\ddot{\ddot{y}}) + τ (\ddot{\ddot{y}}) \leq 1$ is satisfied. The degree of indeterminacy $π (\ddot{\ddot{y}})$ is defined as $1 - (σ (\ddot{\ddot{y}}) + τ (\ddot{\ddot{y}}))$ . Simply a q-rung orthopair fuzzy number (q- $ROFN$ ) can be expressed as $\hat{F}$ = (μ, ν).

Definition 2.6. [26] Let $\ddot{\overset{˘}{R}}$ be a $q - {RO}_{MF}$ set of dimension m defined on $\ddot{\ddot{Y}}$ , where $μ_{m} (\ddot{\ddot{Y}})$ and $ν_{m} (\ddot{\ddot{Y}})$ are real-valued functions representing multi-MVs and multi-NMVs of $\ddot{\ddot{Y}}$ in $\ddot{\overset{˘}{R}}$ . These functions satisfy $0 \leq (μ_{m} (\ddot{\ddot{Y}}))^{q} + (ν_{m} (\ddot{\ddot{Y}}))^{q} \leq 1$ for all $\ddot{\ddot{Y}} \in \ddot{\ddot{Y}}$ , where q ≥ 1 and k = 1,2,..,m. The set $\ddot{\overset{˘}{R}}$ can be represented as $\begin{matrix} \ddot{\overset{˘}{R}} = {\ddot{\ddot{Y}} / (μ_{m} (\ddot{\ddot{Y}}), ν_{m} (\ddot{\ddot{Y}})) : u \in \ddot{\ddot{Y}}} \end{matrix}$ where $(μ_{m} (\ddot{\ddot{Y}}), ν_{m} (\ddot{\ddot{Y}}))$ = ${(μ_{1} (\ddot{\ddot{Y}}), ν_{1} (\ddot{\ddot{Y}})), (μ_{2} (\ddot{\ddot{Y}}), ν_{2} (\ddot{\ddot{Y}})), . . ., (μ_{m} (\ddot{\ddot{Y}}), ν_{m} (\ddot{\ddot{Y}})) : u \in \ddot{\ddot{Y}}}$ .

The set of all $q - {RO}_{MF}$ set of dimension m in $\ddot{\ddot{Y}}$ is denoted as $q - {RO}_{MF} S$ ^m. Simply a q - ROMFS number (q- ${RO}_{MF} N$ ) can be expressed as $\hat{F}$ = (μ^m, ν ^m).

Example 2.7. Assume that a resource and development department of a company is planning to interview four candidates. Let $\ddot{\ddot{Y}}$ = ${\ddot{{\ddot{\ddot{Y}}}_{1}}, \ddot{{\ddot{\ddot{Y}}}_{2}}, \ddot{{\ddot{\ddot{Y}}}_{3}}, \ddot{{\ddot{\ddot{Y}}}_{4}}}$ be the set of four candidates. Each candidate is evaluated by their academic performance, working experience, age, and health in four-dimensional MV and NMVs. $\begin{matrix} \ddot{\overset{˘}{R}} = {〈 \ddot{{\ddot{\ddot{Y}}}_{1}}, (0.8, 0.5), (0.3, 0.5), (0.4, 0.7) 〉, \\ 〈 \ddot{{\ddot{\ddot{Y}}}_{2}} (0.7, 0.3), (0.5, 0.4), (0.6, 0.2) 〉, \\ 〈 \ddot{{\ddot{\ddot{Y}}}_{3}}, (0.5, 0.3), (0.9, 0.5), (0.7, 0.3) 〉, \\ 〈 \ddot{{\ddot{\ddot{Y}}}_{4}}, (0.4, 0.3), (0.5, 0.2), (0.9, 0.4) 〉} . \end{matrix}$

Definition 2.8. [26] A pair ( $\ddot{\hat{F}}$ , $\ddot{\hat{A}}$ ) is known to be a $q - {RO}_{MFS}$ set over $\ddot{\ddot{Y}}$ of dimension m, where $\ddot{\hat{A}}$ ⊆ $\ddot{\hat{e}}$ , $F$ : $\ddot{\hat{A}}$ → $q - {RO}_{MF} S^{m} (\ddot{\ddot{Y}})$ , which is defined as $(\ddot{\hat{F}}, \ddot{\hat{A}})$ = ${\ddot{\hat{e}}, \ddot{\hat{F (e)}}), \ddot{\hat{e}} \in \ddot{\hat{A}}, \ddot{\hat{F (e)}} \in q - {RO}_{MF} S^{m} (\ddot{\ddot{Y}})}$ , where $\ddot{\hat{F (e)}}$ = ${〈 \ddot{\ddot{y}}, μ_{{\ddot{\hat{F (e)}}}^{m}} (\ddot{\ddot{y}}), ν_{{\ddot{\hat{F (e)}}}^{m}} (\ddot{\ddot{y}}) 〉_{q} : \ddot{\ddot{y}} \in \ddot{\ddot{y}}, q \geq 1}$ and ( $μ_{{\ddot{\hat{F (e)}}}^{m}} (\ddot{\ddot{y}})$ , $ν_{{\ddot{\hat{F (e)}}}^{m}} (\ddot{\ddot{y}})$ ) ∈ [0,1] represents the degree of support for multi-MVs and degree of support for multi-NMVs of the attributes of dimension m and $0 \leq (μ_{{\ddot{\hat{F (e)}}}^{m}} (\ddot{\ddot{y}}))^{q} + (ν_{{\ddot{\hat{F (e)}}}^{m}} (\ddot{\ddot{y}}))^{q} \leq 1$ .

Definition 2.9. [10] The comparison matrix is a matrix that organizes information regarding the relationships between objects in a universe. The columns of the matrix represent the names of the universe objects, denoted as ${\ddot{\ddot{y_{1}}}, \ddot{\ddot{y_{2}}}, . . ., \ddot{\ddot{y_{n}}}}$ , while the rows are labeled by criteria $\ddot{\ddot{{e_{1}}}, \ddot{\ddot{e_{2}}}, . . ., \ddot{\ddot{e_{n}}}}$ . The entries of the matrix, denoted as q _ij, indicate the number of criteria for which the value of $y_{i}$ is greater than or equal to the value of $y_{j}$ .

The entries q _ij are computed as follows: q _ij = c - d, where “c” represents the count of how many times the MV $σ_{\ddot{\ddot{y_{i}}} (\ddot{\ddot{e_{j})}}}$ is greater than or equal to $σ_{\ddot{\ddot{y_{j}}} (\ddot{\ddot{e_{j})}}}$ for $\ddot{\ddot{y_{i}}} \neq \ddot{\ddot{y_{j}}}$ , considering all $y_{k}$ in the universe $\ddot{\ddot{Y}}$ . Similarly, “d” represents the count of how many times the NMV $τ_{\ddot{\ddot{y_{i}}} (\ddot{\ddot{e_{j})}}}$ is greater than or equal to $τ_{\ddot{\ddot{y_{j}}} (\ddot{\ddot{e_{j})}}}$ for $\ddot{\ddot{y_{i}}} \neq \ddot{\ddot{y_{j}}}$ , considering all $\ddot{\ddot{y_{k}}}$ in the universe $\ddot{\ddot{Y}}$ .

Definition 2.10. [10] The score of an element $y_{i}$ in S _i is determined by calculating the sum of q _ij for all $s_{k}$ : D _i = ∑q _ij for $s_{k}$ , for all k.
3 L _{q
^*} q-Rung orthopair multi-fuzzy soft set

In this section, we introduce lattice-ordered (L _{q
^*}) q-rung orthopair multi-fuzzy soft sets, which is the extension of the LOMFSS. We also provide the basic definitions and fundamental operations such as arbitrary union and arbitrary intersection for the L _{q
^*} q - RO _MFSS.

Fig. 1

Graphical representation of proposed methodology.

Definition 3.1. A $q - {RO}_{MFS}$ set $(\ddot{\hat{F}}, \ddot{\hat{A}})$ over $\ddot{\ddot{Y}}$ is defined as L _{q
^*} $q - {RO}_{MFS}$ set if for $\ddot{\hat{e_{1}}}$ , $\ddot{\hat{e_{2}}}$ ∈ $\ddot{\hat{A}}$ , such that $\ddot{\hat{e_{1}}}$ ≤ $\ddot{\hat{e_{2}}}$ implies $\ddot{\hat{F (e_{1})}}$ ⊆ $\ddot{\hat{F (e_{2})}}$ . i.e., $μ_{{\ddot{\hat{F (e_{1})}}}^{m}} (\ddot{\ddot{y}})$ ≤ $μ_{{\ddot{\hat{F (e_{2})}}}^{m}} (\ddot{\ddot{y}})$ and $ν_{{\ddot{\hat{F (e_{1})}}}^{m}} (\ddot{\ddot{y}})$ ≥ $ν_{{\ddot{\hat{F (e_{2})}}}^{m}} (\ddot{\ddot{y}})$ , for all $\ddot{\ddot{y}}$ ∈ $\ddot{\ddot{Y}}$

Example 3.2. In a rapidly evolving world, three visionary international companies, XyloBaby, YumiKids, and ZenInfants, have revolutionized the manufacturing of eco-friendly baby products for the global market. Germany is the top buyer, and the companies primarily advertise in G20 and European countries. However, since the groups G20 and Europe have some overlap but do not fully encompass each other, it is important to assess sales performance in Germany, G20, Europe, and globally. The evaluation is based solely on revenue generated, ranging from no revenue (0) to the highest revenue (1).

Consequently, the score increases as the region under evaluation expands, as more sales are expected to take place in larger regions. However, it is important to note that the committee has decided to assess the two products, baby moisturizer and baby wash, separately, despite both being produced by these companies. In this example, the results for each of the four regions are summarised in the tables below Table 2.

Fig. 2

Order of parameters.

Table 2

Tabular Representation of L_{q
^*} $q - {RO}_{MFS} S$

Suppliers	Attributes
	Germany	G20	Europe	Globally
$\ddot{\ddot{x}}$	[(0.4,0.8),	[(0.5,0.6),	[(0.4,0.4),	[(0.7,0.2),
	(0.5,0.9)]	(0.6,0.7)]	(0.7,0.6)]	(0.8,0.3)]
$\ddot{\ddot{Y}}$	[(0.6,0.8),	[(0.8,0.7),	[(0.8,0.4),	[(0.9,0.3),
	(0.4,0.7)]	(0.4,0.6)]	(0.6,0.5)]	(0.7,0.4)]
$\ddot{\ddot{z}}$	[(0.4,0.9),	[(0.4,0.8),	[(0.6,0.6),	[(0.7,0.5),
	(0.3,0.7)]	(0.5,0.7)]	(0.7,0.6)]	(0.8, 0.4)]

With all of these inputs:

(i) Use the non-empty set to identify the three companies $\ddot{\ddot{Y}}$ = ${\ddot{\ddot{x}}, \ddot{\ddot{y}}, \ddot{\ddot{z}}}$

(ii)Indicate the four regions with the lattice of parameters $\ddot{\hat{e}}$ = ${\ddot{\hat{e_{1}}} = Japan, \ddot{\hat{e_{2}}} = G 20, \ddot{\hat{e_{3}}} = Europe, \ddot{\hat{e_{4}}} = globally}$ for which $\ddot{\hat{e_{1}}} \leq \ddot{\hat{e_{2}}} \leq \ddot{\hat{e_{3}}}$ and $\ddot{\hat{e_{1}}} \leq \ddot{\hat{e_{2}}} \leq \ddot{\hat{e_{4}}}$ , but no such relationship established between $\ddot{\hat{e_{3}}}$ and $\ddot{\hat{e_{4}}}$ .

(iii) Use the set of indexes to denote the two assessment categories. $\ddot{\hat{A}}$ = {b ₁ = baby moisturizer, b ₂ = baby wash}. Then the representation of L_{q
^*} $q - {RO}_{MFS}$ set is shown below: $(\ddot{\hat{F}}, \ddot{\hat{e}})$ = ${\ddot{\hat{F (e_{1})}}$ = ${\frac{\ddot{\hat{x}}}{[(0.4, 0.8), (0.5, 0.9)]}, \frac{\ddot{\ddot{y}}}{[(0.6, 0.8), (0.4, 0.7)]}, \frac{\ddot{\hat{z}}}{[(0.4, 0.9), (0.3, 0.7)]}}$ $\ddot{\hat{F (e_{2})}}$ = ${\frac{\ddot{\hat{x}}}{[(0.5, 0.6), (0.6, 0.7)]}, \frac{\ddot{\ddot{y}}}{[(0.8, 0.7), (0.4, 0.6)]}, \frac{\ddot{\hat{z}}}{[(0.4, 0.8), (0.5, 0.7)]}}$ $\ddot{\hat{F (e_{3})}}$ = ${\frac{\ddot{\hat{x}}}{[(0.4, 0.4), (0.7, 0.6)]}, \frac{\ddot{\ddot{y}}}{[(0.8, 0.4), (0.6, 0.5)]}, \frac{\ddot{\hat{z}}}{[(0.6, 0.6), (0.7, 0.6)]}}$ $\ddot{\hat{F (e_{4})}}$ = ${\frac{\ddot{\hat{x}}}{[(0.7, 0.2), (0.8, 0.3)]}, \frac{\ddot{\ddot{y}}}{[(0.9, 0.3), (0.7, 0.4)]}, \frac{z}{[(0.7, 0.5), (0.8, 0.4)]}}}$ ⊆ L_{q
^*} $q - {RO}_{MFS} S$ $(\ddot{\ddot{Y}})$

Definition 3.3. Let $(\ddot{\hat{F}}, \ddot{\hat{A}})$ , $(\ddot{\hat{G}}, \ddot{\hat{B}})$ be two L _{q
^*} q - RO _MFSSs of dimension m over $\ddot{\ddot{Y}}$ . (i)

Subset: ( $\ddot{\hat{F}}$ , $\ddot{\hat{A}}$ ) is a L _q
^* q-rung orthopair multi-fuzzy soft subset of ( $\ddot{\hat{G}}$ , $\ddot{\hat{B}}$ ) if $\ddot{\hat{A}}$ ⊆ $\ddot{\hat{B}}$ and for any $\ddot{\hat{e}}$ ∈ $\ddot{\hat{A}}$ , $\ddot{\hat{F (e)}}$ ⊆ $\ddot{\hat{G (e)}}$ and it is denoted by ( $\ddot{\hat{F}}$ , $\ddot{\hat{A}}$ ) $\hat{\subseteq}$ ( $\ddot{\hat{G}}$ , $\ddot{\hat{B}}$ ). i.e., $μ_{{\ddot{\hat{F (e)}}}^{m}} (\ddot{\ddot{y}})$ ≤ $μ_{{\ddot{\hat{G (e)}}}^{m}} (\ddot{\ddot{y}})$ and $ν_{{\ddot{\hat{F (e)}}}^{m}} (\ddot{\ddot{y}})$ ≥ $ν_{{\ddot{\hat{G (e)}}}^{m}} (\ddot{\ddot{y}})$ , for all $\ddot{\hat{e}}$ ∈ $\ddot{\hat{A}}$ , $\ddot{\ddot{y}}$ ∈ $\ddot{\ddot{Y}}$ .

(ii)

Equal: If ( $\ddot{\hat{F}}$ , $\ddot{\hat{A}}$ ) = ( $\ddot{\hat{G}}$ , $\ddot{\hat{B}})$ , then $μ_{{\ddot{\hat{F (e)}}}^{m}} (\ddot{\ddot{y}})$ = $μ_{{\ddot{\hat{G (e)}}}^{m}} (\ddot{\ddot{y}})$ and $ν_{{\ddot{\hat{F (e)}}}^{m}} (\ddot{\ddot{y}})$ = $ν_{{\ddot{\hat{G (e)}}}^{m}} (\ddot{\ddot{y}})$ , for all $\ddot{\hat{e}}$ ∈ $\ddot{\hat{A}}$ , $\ddot{\ddot{y}}$ ∈ $\ddot{\ddot{Y}}$ .

(iii)

Null: L _q
^* q - RO _MFSS is said to be null L _q
^* q - RO _MFSS set if $μ_{{\ddot{\hat{F (e)}}}^{m}} (\ddot{\ddot{y}})$ = 0 and $ν_{{\ddot{\hat{F (e)}}}^{m}} (\ddot{\ddot{y}})$ = 1 and is denoted by $\emptyset_{\ddot{\hat{A}}}^{m}$ , for all $\ddot{\hat{e}}$ ∈ $\ddot{\hat{A}}$ , $\ddot{\ddot{y}}$ ∈ $\ddot{\ddot{Y}}$ .

(iv)

Absolute: L _q
^* q - RO _MFSS is said to be universal L _q
^* q - RO _MFSS if $μ_{{\ddot{\hat{F (e)}}}^{m}} (\ddot{\ddot{y}})$ = 1 and $ν_{{\ddot{\hat{F (e)}}}^{m}} (\ddot{\ddot{y}})$ = 0 and is denoted by ${\ddot{\ddot{y}}}_{\ddot{\hat{A}}}^{m}$ , for all $\ddot{\hat{e}}$ ∈ $\ddot{\hat{A}}$ , $\ddot{\ddot{y}}$ ∈ $\ddot{\ddot{Y}}$ .

(v)

Complement: The complement of L _q
^* $q - {RO}_{MFSS} (F, \ddot{\hat{A}})$ over $\ddot{\ddot{y}}$ is denoted by $(\ddot{\hat{F}}, \ddot{\hat{A}})^{c}$ and defined as $\begin{matrix} {\ddot{\hat{F}}}^{c} (\ddot{\hat{e}}) = (\ddot{\hat{F}} (\ddot{\hat{e}}))^{c} = {〈 \ddot{\ddot{Y}}, ν_{\ddot{\hat{F}} (e)}^{m} (\ddot{\ddot{y}}), μ_{\ddot{\hat{F}} (\ddot{\hat{e}})}^{m} (\ddot{\ddot{y}}) 〉 : \\ \ddot{\ddot{y}} \in \ddot{\ddot{Y}}} \end{matrix}$ where ${\ddot{\hat{F}}}^{c}$ : $\ddot{\hat{A}}$ → $q - {RO}_{MFS}^{(\ddot{\ddot{Y}})}$ and for all $\ddot{\hat{e}}$ ∈ $\ddot{\hat{A}}$ .

Proposition 3.4. Let ( $\ddot{\hat{F}}$ , $\ddot{\hat{A}}$ ), ( $\ddot{\hat{G}}$ , $\ddot{\hat{B}}$ ) be two L _{q
^*} q - RO _MFSSs of dimension m over $\ddot{\ddot{Y}}$ . Then (i)

$(\ddot{\hat{F}}$ , $\ddot{\hat{A}})$ $\hat{\subseteq}$ $(\ddot{\hat{F}}, \ddot{\hat{A}})$ .

(ii)

$\emptyset_{\ddot{\hat{A}}}^{m}$ $\hat{\subseteq}$ $(\ddot{\hat{F}}, \ddot{\hat{A}})$ .

(iii)

$(\ddot{\hat{F}}$ , $\ddot{\hat{A}})$ $\hat{\subseteq}$ $(\ddot{\hat{G}}, \ddot{\hat{B}})$ , $(\ddot{\hat{G}}$ , $\ddot{\hat{B}})$ $\hat{\subseteq}$ $(\ddot{\hat{H}}, \ddot{\hat{C}})$ this implies that $(\ddot{\hat{F}}$ , $\ddot{\hat{A}})$ $\hat{\subseteq}$ $(\ddot{\hat{H}}, \ddot{\hat{C}})$ .

(iv)

$(\ddot{\hat{F}}, \ddot{\hat{A}})$ $\hat{\subseteq}$ ${\ddot{\ddot{Y}}}_{\ddot{\hat{A}}}^{m}$ .

Proof. Straightforward□

Definition 3.5. Let $(\ddot{\hat{F}}, \ddot{\hat{A}})$ , $(\ddot{\hat{G}}, \ddot{\hat{B}})$ be two L_{q
^*} $q - {RO}_{MFS}$ sets of dimension m over the common universe $\ddot{\ddot{Y}}$ . (i)

Union of two L_q
^* $q - {RO}_{MFS} S$ ( $\ddot{\ddot{Y}}$ ) is defined as ( $\ddot{\hat{H}}$ , $\ddot{\hat{C}}$ ) = ( $\ddot{\hat{F}}$ , $\ddot{\hat{A}}$ ) $\hat{\cup}$ ( $\ddot{\hat{G}}$ , $\ddot{\hat{B}}$ ), where $\ddot{\hat{C}}$ = $\ddot{\hat{A}} \cup \ddot{\hat{B}}$ , and ∀ $\ddot{\hat{e}}$ ∈ $\ddot{\hat{C}}$ , $\ddot{\ddot{y}} \in \ddot{\ddot{Y}}$ $\begin{matrix} μ_{\ddot{\hat{H (e)}}}^{m} = {\begin{matrix} μ_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ μ_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ \max (μ_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}), μ_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y})}) \\ if \ddot{\hat{e}} \in \ddot{\hat{A}} \cap \ddot{\hat{B}} \end{matrix} \end{matrix}$ $\begin{matrix} ν_{\ddot{\hat{H (e)}}}^{m} = {\begin{matrix} ν_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ ν_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{Y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ \min (ν_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y})}), ν_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y})}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} \cap \ddot{\hat{B}} . \end{matrix} \end{matrix}$

(ii)

Intersection of two L_q
^* $q - {RO}_{MFS} S$ ( $\ddot{\ddot{Y}}$ ) is defined as ( $\ddot{\hat{H}}$ , $\ddot{\hat{C}}$ ) = ( $\ddot{\hat{F}}$ , $\ddot{\hat{A}}$ ) $\hat{\cap}$ ( $\ddot{\hat{G}}$ , $\ddot{\hat{B}}$ ), where $\ddot{\hat{C}}$ = $\ddot{\hat{A}} \cup \ddot{\hat{B}}$ , and ∀ $\ddot{\hat{e}}$ ∈ $\ddot{\hat{C}}$ , $\ddot{\ddot{y}} \in \ddot{\ddot{Y}}$ $\begin{matrix} μ_{\ddot{\hat{H (e)}}}^{m} = {\begin{matrix} μ_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ μ_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ \min (μ_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}), μ_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y})}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} \cap \ddot{\hat{B}} \end{matrix} \end{matrix}$ $\begin{matrix} ν_{\ddot{\hat{H (e)}}}^{m} = {\begin{matrix} ν_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ ν_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ \max (ν_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}), ν_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y})}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} \cap \ddot{\hat{B}} \end{matrix} \end{matrix}$

(iii)

Restricted union of two L_q
^* $q - {RO}_{MFS} S$ ( $\ddot{\ddot{Y}}$ ) is defined as ( $\ddot{\hat{H}}$ , $\ddot{\hat{C}}$ ) = ( $\ddot{\hat{F}}$ , $\ddot{\hat{A}}$ ) ${\hat{\cup}}_{R}$ ( $\ddot{\hat{G}}$ , $\ddot{\hat{B}}$ ), where $\ddot{\hat{C}}$ = $\ddot{\hat{A}} \cap \ddot{\hat{B}}$ ≠ φ, and $\ddot{\hat{H (e)}}$ = $\ddot{\hat{F (e)}}$ ∪ $\ddot{\hat{G (e)}}$ . (i.e.,) $μ_{\ddot{\hat{H (e)}}}^{m}$ = max ${\ddot{\hat{F (e)}}, \ddot{\hat{G (e)}}}$ , $ν_{\ddot{\hat{H (e)}}}^{m}$ = min ${\ddot{\hat{F (e)}}, \ddot{\hat{G (e)}}}$ , ∀ $\ddot{\hat{e}}$ ∈ $\ddot{\hat{C}}$ .

(iv)

Restricted intersection of L_q
^* $q - {RO}_{MFS} S$ ( $\ddot{\ddot{Y}}$ ) is defined as ( $\ddot{\hat{H}}$ , $\ddot{\hat{C}}$ ) = ( $\ddot{\hat{F}}$ , $\ddot{\hat{A}}$ ) ${\hat{\cap}}_{R}$ ( $\ddot{\hat{G}}$ , $\ddot{\hat{B}}$ ), where $\ddot{\hat{C}}$ = $\ddot{\hat{A}} \cap \ddot{\hat{B}}$ ≠ φ, and $\ddot{\hat{H (e)}}$ = $\ddot{\hat{F (e)}}$ ∩ $\ddot{\hat{G (e)}}$ . (i.e.,) $μ_{\ddot{\hat{H (e)}}}^{m}$ = min ${\ddot{\hat{F (e)}}, \ddot{\hat{G (e)}}}$ , $ν_{\ddot{\hat{H (e)}}}^{m}$ = max ${\ddot{\hat{F (e)}}, \ddot{\hat{G (e)}}}$ , ∀ $\ddot{\hat{e}}$ ∈ $\ddot{\hat{C}}$ .

Theorem 3.6. De Morgan’s Law For two L_{q
^*} $q - {RO}_{MFS}$ sets $(\ddot{\hat{F}}, \ddot{\hat{A}})$ and $(\ddot{\hat{G}}, \ddot{\hat{B}})$ of dimension m over $\ddot{\ddot{Y}}$ . Then (i)

$[(\ddot{\hat{F}}, \ddot{\hat{A}}) \hat{\cup} (\ddot{\hat{G}}, \ddot{\hat{B}})]^{c}$ = $(\ddot{\hat{F}}, \ddot{\hat{A}})^{c}$ $\hat{\cap}$ $(\ddot{\hat{G}}, \ddot{\hat{B}})^{c}$ .

(ii)

$[(\ddot{\hat{F}}, \ddot{\hat{A}}) \hat{\cap} (\ddot{\hat{G}}, \ddot{\hat{B}})]^{c}$ = $(\ddot{\hat{F}}, \ddot{\hat{A}})^{c}$ $\hat{\cup}$ $(\ddot{\hat{G}}, \ddot{\hat{B}})^{c}$ .

Proof. (i) Suppose that $(\ddot{\hat{F}}, \ddot{\hat{A}}) \hat{\cup} (\ddot{\hat{G}}, \ddot{\hat{B}})$ = $(\ddot{\hat{H}}, \ddot{\hat{C}})$ where $\ddot{\hat{C}}$ = $\ddot{\hat{A}} \cup \ddot{\hat{B}}$ and for all $\ddot{\hat{e}} \in \ddot{\hat{C}}$ , $\ddot{\ddot{y}} \in \ddot{\ddot{Y}}$ . $\begin{matrix} μ_{\ddot{\hat{H}} (\ddot{\hat{e}})}^{m} (\ddot{\ddot{y}}) \\ = {\begin{matrix} μ_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ μ_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ \max (μ_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}), μ_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y}})) & if \ddot{\hat{e}} \in \ddot{\hat{A}} \cap \ddot{\hat{B}} \end{matrix} \\ ν_{\ddot{\hat{H (e)}}}^{m} (\ddot{\ddot{y}}) \\ = {\begin{matrix} ν_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ ν_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ \min (ν_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}), ν_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y}})) & if \ddot{\hat{e}} \in \ddot{\hat{A}} \cap \ddot{\hat{B}} \end{matrix} \end{matrix}$ Thus $(\ddot{\hat{F}}, \ddot{\hat{A}}) \hat{\cup} (\ddot{\hat{G}}, \ddot{\hat{B}})^{c}$ = $(\ddot{\hat{H}}, \ddot{\hat{C}})^{c}$ = $({\ddot{\hat{H}}}^{c}, \ddot{\hat{C}})$ and for all $\ddot{\hat{e}} \in \ddot{\hat{C}}$ $\begin{matrix} μ_{{\ddot{\hat{H}}}^{c} (\ddot{\hat{e}})}^{m} (\ddot{\ddot{y}}) \\ = {\begin{matrix} ν_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ ν_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ \max (ν_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}), ν_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y}})) & if \ddot{\hat{e}} \in \ddot{\hat{A}} \cap \ddot{\hat{B}} \end{matrix} \\ ν_{H^{c} (e)}^{m} (\ddot{\ddot{y}}) = \\ {\begin{matrix} μ_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ μ_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ \min (μ_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}), μ_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y}})) & if \ddot{\hat{e}} \in \ddot{\hat{A}} \cap \ddot{\hat{B}} \end{matrix} \end{matrix}$ Again, Suppose that $(\ddot{\hat{F}}, \ddot{\hat{A}})^{c}$ $\hat{\cap}$ $(\ddot{\hat{G}}, \ddot{\hat{B}})^{c}$ = $({\ddot{\hat{F}}}^{c}, \ddot{\hat{A}})$ $\hat{\cap}$ $({\ddot{\hat{G}}}^{c}, \ddot{\hat{B}})$ = $(O, D)$ where $D$ = $\ddot{\hat{A}}$ ∪ $\ddot{\hat{B}}$ and for all $\ddot{\hat{e}}$ ∈ $D$ . $\begin{matrix} μ_{O^{c} (\ddot{\hat{e}})}^{m} (\ddot{\ddot{y}}) = \\ {\begin{matrix} ν_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ ν_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ \max (ν_{\ddot{\hat{F (e)}} (\ddot{\ddot{y}}}^{m}), ν_{\ddot{\hat{G (e)}} (\ddot{\ddot{y}}}^{m})) & if \ddot{\hat{e}} \in \ddot{\hat{A}} \cap \ddot{\hat{B}} \end{matrix} \\ ν_{O^{c} (e)}^{m} (\ddot{\ddot{y}}) = \\ {\begin{matrix} μ_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ μ_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y}}) & if \ddot{\hat{e}} \in \ddot{\hat{A}} - \ddot{\hat{B}} \\ \min (μ_{\ddot{\hat{F (e)}}}^{m} (\ddot{\ddot{y}}), μ_{\ddot{\hat{G (e)}}}^{m} (\ddot{\ddot{y}})) & if \ddot{\hat{e}} \in \ddot{\hat{A}} \cap \ddot{\hat{B}} \end{matrix} \end{matrix}$ we see that $\ddot{\hat{C}}$ = $D$ and for all $\ddot{\hat{e}}$ ∈ $\ddot{\hat{C}}$ ( $D$ ), ${\ddot{\hat{H}}}^{c} (\ddot{\hat{e}})$ = $O (e)$ .□

Proof. (ii). can be proved analogously.□

Theorem 3.7. De Morgan’s Law For two L_{q
^*} $q - {RO}_{MFS}$ sets $(\ddot{\hat{F}}, \ddot{\hat{A}})$ and $(\ddot{\hat{G}}, \ddot{\hat{B}})$ of dimension m over $\ddot{\ddot{Y}}$ . Then (i)

$[(\ddot{\hat{F}}, \ddot{\hat{A}}) {\hat{\cup}}_{R} (\ddot{\hat{G}}, \ddot{\hat{B}})]^{c}$ = $(\ddot{\hat{F}}, \ddot{\hat{A}})^{c}$ $\hat{\cap}$ $(\ddot{\hat{G}}, \ddot{\hat{B}})^{c}$ .

(ii)

$[(\ddot{\hat{F}}, \ddot{\hat{A}}) \hat{\cap} (\ddot{\hat{G}}, \ddot{\hat{B}})]^{c}$ = $(\ddot{\hat{F}}, \ddot{\hat{A}})^{c}$ ${\hat{\cup}}_{R}$ $(\ddot{\hat{G}}, \ddot{\hat{B}})^{c}$ .

Proof. Suppose that $(\ddot{\hat{F}}, \ddot{\hat{A}}) {\hat{\cup}}_{R} (\ddot{\hat{G}}, \ddot{\hat{B}})$ = $(\ddot{\hat{H}}, \ddot{\hat{C}})$ where $\ddot{\hat{C}}$ = $\ddot{\hat{A}}$ ∩ $\ddot{\hat{B}}$ and for all $\ddot{\hat{e}}$ ∈ $\ddot{\hat{C}}$ , $\ddot{\hat{H (e)}}$ = $\ddot{\hat{F (e)}} \cup \ddot{\hat{G (e)}}$ . Thus $[(\ddot{\hat{F}}, \ddot{\hat{A}}) {\hat{\cup}}_{R} (\ddot{\hat{G}}, \ddot{\hat{B}})]^{c}$ = $(\ddot{\hat{H}}, \ddot{\hat{C}})^{c}$ = $({\ddot{\hat{H}}}^{c}, \ddot{\hat{C}})$ and for all $\ddot{\hat{e}}$ ∈ $\ddot{\hat{C}}$ , ${\ddot{\hat{H}}}^{c} (\ddot{\hat{e}})$ = $F^{c} (e) \cap G^{c} (e)$ . Again, Suppose that $(\ddot{\hat{F}}, \ddot{\hat{A}})^{c}$ $\hat{\cap}$ $(\ddot{\hat{G}}, \ddot{\hat{B}})^{c}$ = $({\ddot{\hat{F}}}^{c}, \ddot{\hat{A}})$ $\hat{\cap}$ $({\ddot{\hat{G}}}^{c}, \ddot{\hat{B}})$ = $(O, D)$ , where $D$ = $\ddot{\hat{A}}$ ∩ $\ddot{\hat{B}}$ and for all $\ddot{\hat{e}}$ ∈ $D$ , $O (e)$ = $F^{c} (e) \cap G^{c} (e)$ . We see that $\ddot{\hat{C}}$ = $D$ and for all $\ddot{\hat{e}}$ ∈ $\ddot{\hat{C}}$ ( $D$ ), ${\ddot{\hat{H}}}^{c} (\ddot{\hat{e}})$ = $O (e)$ .□

Proof. (ii) similar to (i).□

4 Implementation of the proposed L _{q
^*} q-Rung orthopair multi-fuzzy MADM approach

The process of decision-making using L _{q
^*} q-RO _MFSS-MADM necessitates the assessment of all potential decision alternatives, a common practice in most decision-making scenarios. However, the complexity arises from the fact that a solitary criterion is often inadequate for comprehensively evaluating all available options. Consequently, the efficacy of selecting the optimal alternative from the considered set is not consistently optimal.

This section introduces a pair of innovative and tailored algorithms designed to effectively apply the information from L _{q
^*} q-RO _MFSS in addressing decision-making problems. These algorithms build upon the frameworks established in the works of [31] and [33], incorporating modifications to enhance their performance within this context.

4.1 Algorithm 1

Step A: Consider the $L_{q^{*}}$ q- ${RO}_{MFS} S$ over $U$ and write it in tabular form. $\begin{matrix} [Q] = [q_{ij}] = [(μ_{\hat{F} (e_{j})^{m}}, ν_{\hat{F} (e_{j})^{m}})]_{m \times n} = \\ {(\begin{matrix} [(μ_{11^{1}}, ν_{11^{1}}), . ., & \dots & [(μ_{1 s^{1}}, ν_{1 s^{1}}), . ., \\ (μ_{11^{m}}, ν_{11^{m}})] & (μ_{1 s^{m}}, ν_{1 s^{m}})] \\ [(μ_{21^{1}}, ν_{21^{1}}), . ., & \dots & [(μ_{2 s^{1}}, ν_{2 s^{1}}), . ., \\ (μ_{21^{m}}, ν_{21^{m}})] & (μ_{2 s^{m}}, ν_{2 s^{m}})] \\ ⋮ & ⋱ & ⋮ \\ [(μ_{t 1^{1}}, ν_{t 1^{1}}), . ., & \dots & [(μ_{{ts}^{1}}, ν_{{ts}^{1}}), . ., \\ (μ_{t 1^{k}}, ν_{t 1^{k}})] & (μ_{{ts}^{m}}, ν_{{ts}^{m}})] \end{matrix})}_{t \times s} \end{matrix}$ Step B: Convert the $L_{q^{*}} q - {RO}_{MFS} S$ ( $F, A)$ to the q- $ROFS$ by obtaining the weighted aggregation values of $(μ_{\hat{F} (e)^{m}} (u), ν_{\hat{F} (e)^{m}} (u))$ , ∀ $e$ ∈ A, ∀ $u \in U$ and k = 1,2,...,m as in the following equation:

$\begin{matrix} {(\begin{matrix} [(μ_{11^{1}}, ν_{11^{1}}), . ., & \dots & [(μ_{1 s^{1}}, ν_{1 s^{1}}), . ., \\ (μ_{11^{m}}, ν_{11^{m}})] & (μ_{1 s^{m}}, ν_{1 s^{m}})] \\ [(μ_{21^{1}}, ν_{21^{1}}), . ., & \dots & [(μ_{2 s^{1}}, ν_{2 s^{1}}), . ., \\ (μ_{21^{m}}, ν_{21^{m}})] & (μ_{2 s^{m}}, ν_{2 s^{m}})] \\ ⋮ & ⋱ & ⋮ \\ [(μ_{t 1^{1}}, ν_{t 1^{1}}), . ., & \dots & [(μ_{{ts}^{1}}, ν_{{ts}^{1}}), . ., \\ (μ_{t 1^{k}}, ν_{t 1^{k}})] & (μ_{{ts}^{m}}, ν_{{ts}^{m}})] \end{matrix})}_{t \times s} (\begin{matrix} ω_{1}^{m} \\ ω_{2}^{m} \\ ⋮ \\ ω_{s}^{m} \end{matrix}) \\ = (\begin{matrix} (\sum_{j = 1}^{s} ω_{j}^{m} μ_{j^{m}} (u_{1}), \sum_{j = 1}^{s} ω_{j}^{m} ν_{j^{m}} (u_{1})) \\ (\sum_{j = 1}^{s} ω_{j}^{m} μ_{j^{m}} (u_{2}), \sum_{j = 1}^{s} ω_{j}^{m} ν_{j^{m}} (u_{1})) \\ ⋮ \\ (\sum_{j = 1}^{s} ω_{j}^{m} μ_{j^{m}} (u_{t}), \sum_{j = 1}^{s} ω_{j}^{m} ν_{j^{m}} (u_{1})) \end{matrix}) \end{matrix}$ (1)

Step C: Compute the comparison matrix of q- $ROFN$ . The entries are q _ij where q _ij, is the number of parameters for which the value of $s_{i}$ exceeds or equal to the value $s_{j}$ . We calculate that entries by q _ij = c - d, where “c” is the integer calculated as “how many times $μ_{s_{i} (e_{j})}$ exceeds or equal to $μ_{s_{i} (e_{j})}$ ”, for $s_{i}$ ≠ $s_{j}$ , ∀ $s_{k}$ ∈ $U$ , “d” is the integer calculated as “how many times $ν_{s_{i} (e_{j})}$ exceeds or equal to $ν_{s_{i} (e_{j})}$ ”, for $s_{i}$ ≠ $s_{j}$ , ∀ $s_{k}$ ∈ $U$ .

Step D: Compute the score D _i = ∑q _ij of $s_{k}$ , ∀ k.

Step E: Find D _i = max d _k.

4.2 Algorithm 2

Step A: Same as Step A in Algorithm 1.

Step B: Same as Step B in Algorithm 1.

Step C: Insert s threshold fuzzy set λ: $A$ → [0,1] ((or) specify a threshold score (or) adopt a mid-level decision criterion (or) adopt a top-level decision criterion) for an approach to decision-making.

Step D: Evaluate the level SS L(Δ _f; λ) of Δ _f in accordance with the threshold fuzzy set λ ((or) the t-level SS L(Δ _f; t); (or) the mid-level SS L(Δ _f; mid) (or) the top-level SS L(Δ _f; max)).

Step E: Exhibit the level SS (L(Δ _f; t) or L(Δ _f; mid) or L(Δ _f; max)) in the form of table and determine the choice score c _i of u _i, ∀ i.

Step F: The ideal decision is to choose u _j if c _j = $\max_{i} C_{i}$ .

Step G: Any one of u _j can be identified, if j has multiple values.

4.3 Case scenario

In a hypothetical scenario, a regional healthcare system was grappling with supply chain management during the COVID-19 pandemic. In such a crisis, the need for sustainable suppliers becomes even more critical as hospitals strive to maintain uninterrupted operations and provide effective patient care. The suppliers are assessed based on several important attributes: flexibility, reliability, cost, customer service, and quality. These attributes collectively contribute to the effectiveness of the healthcare supply chain, ensuring that medical resources are efficiently procured and distributed. To make this complex decision, they employ a specialized MADM technique called L _{q
^*} q-RO_MFS-MADM.

In this novel technique, a lattice structure is employed to visually represent the relationships and priorities among the aforementioned attributes in a three-dimensional framework (see Fig. 3). At the top of the lattice structure lies the most fundamental attribute,quality. Quality is the overarching concern, as healthcare supply chain management demands uncompromising standards to safeguard patient well-being. Directly beneath quality, we find reliability, positioned at the next level. Reliability closely follows quality due to its crucial role in ensuring a consistent flow of essential medical supplies. Flexibility, representing a supplier’s adaptability to fluctuating demands and unforeseen situations, is positioned at a lower level in the lattice. While it is valued, it holds a lesser position in comparison to quality and reliability. Interestingly, both cost and customer services appear independently in the lattice structure, linked directly to reliability. This signifies that while cost and customer services are not directly connected, they share a relationship with reliability and are positioned just below it. This positioning highlights the experts’ recognition of cost as an important consideration that needs to be balanced with reliability, along with the emphasis on maintaining satisfactory customer service levels. This lattice structure serves as a valuable tool for understanding the relative significance and interplay of attributes from the experts’ viewpoint.

Fig. 3

Lattice of parameters.

4.4 Illustartion

Let $\ddot{\hat{Y}}$ = ${\ddot{\hat{S_{1}}}, \ddot{\hat{S_{2}}}, \ddot{\hat{S_{3}}}, \ddot{\hat{S_{4}}}, \ddot{\hat{S_{5}}}}$ be the set of suppliers. Using the proposed L _{q
^*} q-RO_MFS-MADM, The hospital aims to create a list that shows which suppliers are the best. They will get advice from three experts. The first expert knows a lot about how to manage the movement of medical stuff. The second expert is good at making sure sick people get the right care and knows about how to keep people healthy in general. The third expert is an expert in making healthcare systems work well and making sure that supplies are managed correctly.

By looking at what others have written, the hospital found that there are five important things to think about when choosing suppliers. These things are: how much you can trust the supplier ( $\ddot{\hat{e_{1}}}$ = reliability), how much money the supplies will cost ( $\ddot{\hat{e_{2}}}$ = cost), how well the supplier treats the hospital and patients ( $\ddot{\hat{e_{3}}}$ = customer services), how fast the supplier can bring the supplies ( $\ddot{\hat{e_{4}}}$ = delivery time), and how good the supplies are ( $\ddot{\hat{e_{5}}}$ = quality).

4.4.1 Algorithm 1: Calculation part

Step A. Contruct a L_{q
^*} q-RO_MFSS decision matrix of dimension 3 in the form of L _{q
^*} q - RO _MFSS, shown in Tables 3 and 4.

Table 3
L_{q
^*} $q - {RO}_{MFS} S$ of dimension 3

Suppliers Attributes

$\ddot{\hat{e_{1}}}$ $\ddot{\hat{e_{2}}}$ $\ddot{\hat{e_{3}}}$

$\ddot{\hat{S_{1}}}$ [(0.3,0.7), [(0.5,0.6), [(0.6,0.5),

(0.4,0.5), (0.7,0.5), (0.7,0.4),

(0.4,0.4)] (0.8,0.4)] (0.8,0.3)]

$\ddot{\hat{S_{2}}}$ [(0.4,0.9), [(0.5,0.8), [(0.6,0.7),

(0.5,0.8), (0.6,0.7), (0.7,0.6),

(0.5,0.7)] (0.7,0.6)] (0.8,0.4)]

$\ddot{\hat{S_{3}}}$ [(0.3,0.8), [(0.4,0.7), [(0.6,0.6),

(0.5,0.7), (0.5,0.6), (0.7,0.6),

(0.7,0.6)] (0.8,0.5)] (0.8,0.4)]

$\ddot{\hat{S_{4}}}$ [(0.4,0.9), [(0.4,0.8), [(0.6,0.6),

(0.45,0.7), (0.5,0.7), (0.7,0.4),

(0.5,0.6)] (0.6,0.5)] (0.8,0.4)]

$\ddot{\hat{S_{5}}}$ [(0.4,0.9), [(0.45,0.7), [(0.5,0.6),

(0.5,0.8), (0.6,0.6), (0.7,0.5),

(0.6,0.8)] (0.7,0.5)] (0.7,0.3)]

Suppliers	Attributes
$\ddot{\hat{S_{1}}}$	[(0.3,0.7),	[(0.5,0.6),	[(0.6,0.5),
	(0.4,0.5),	(0.7,0.5),	(0.7,0.4),
	(0.4,0.4)]	(0.8,0.4)]	(0.8,0.3)]
$\ddot{\hat{S_{2}}}$	[(0.4,0.9),	[(0.5,0.8),	[(0.6,0.7),
	(0.5,0.8),	(0.6,0.7),	(0.7,0.6),
	(0.5,0.7)]	(0.7,0.6)]	(0.8,0.4)]
$\ddot{\hat{S_{3}}}$	[(0.3,0.8),	[(0.4,0.7),	[(0.6,0.6),
	(0.5,0.7),	(0.5,0.6),	(0.7,0.6),
	(0.7,0.6)]	(0.8,0.5)]	(0.8,0.4)]
$\ddot{\hat{S_{4}}}$	[(0.4,0.9),	[(0.4,0.8),	[(0.6,0.6),
	(0.45,0.7),	(0.5,0.7),	(0.7,0.4),
	(0.5,0.6)]	(0.6,0.5)]	(0.8,0.4)]
$\ddot{\hat{S_{5}}}$	[(0.4,0.9),	[(0.45,0.7),	[(0.5,0.6),
	(0.5,0.8),	(0.6,0.6),	(0.7,0.5),
	(0.6,0.8)]	(0.7,0.5)]	(0.7,0.3)]

Table 4

L_{q
^*} $q - {RO}_{MFS} S$ of dimension 3

Suppliers	Attributes
	$\ddot{\hat{e_{4}}}$	$\ddot{\hat{e_{5}}}$
$\ddot{\hat{S_{1}}}$	[(0.6,0.4),	[(0.8,0.3),
	(0.7,0.3),	(0.9,0.2),
	(0.9,0.2)]	(0.9,0.1)]
$\ddot{\hat{S_{2}}}$	[(0.55,0.7),	[(0.7,0.4),
	(0.6,0.4)]	(0.85,0.3)]
$\ddot{\hat{S_{3}}}$	[(0.6,0.4),	[(0.7,0.3),
	(0.6,0.3),	(0.8,0.2),
	(0.8,0.2)]	(0.9,0.1)]
$\ddot{\hat{S_{4}}}$	[(0.5,0.6),	[(0.6,0.5),
	(0.6,0.4),	(0.7,0.4),
	(0.7, 0.4)]	(0.8,0.2)]
$\ddot{\hat{S_{5}}}$	[(0.6,0.4),	[(0.7,0.3),
	(0.7,0.3),	(0.8,0.2),
	(0.8,0.2)]	(0.9,0.1)]

Step B. Convert L_{q
^*} q-RO_MFSS decision matrix into weighted $q - ROFSS$ by imposing the following weights for the each parameters in $\ddot{\hat{A}}$ :

$ω (\ddot{\hat{e_{1}}}) = (0.2, 0.3, 0.5)$

$ω (\ddot{\hat{e_{2}}}) = (0.1, 0.2, 0.7)$

$ω (\ddot{\hat{e_{3}}}) = (0.1, 0.3, 0.6)$

$ω (\ddot{\hat{e_{4}}}) = (0.1, 0.3, 0.6)$

$ω (\ddot{\hat{e_{5}}}) = (0.1, 0.4, 0.5)$ .

Using Equation (1) L_{q
^*} q-RO_MFSS decision matrix in Tables 3 and 4 is converted into q-ROFSS shown in Tables 5 and 6.

Table 5

Weighted aggregation of L_{q
^*} $q - {RO}_{MFS} S$ of dimension 3

Suppliers	Attributes
	$\ddot{\hat{e_{1}}}$	$\ddot{\hat{e_{2}}}$	$\ddot{\hat{e_{3}}}$
	$ω (\ddot{\hat{e_{1}}}) =$	$ω (\ddot{\hat{e_{2}}}) =$	$ω (\ddot{\hat{e_{3}}}) =$
	(0.2,0.3,0.5)	(0.1,0.2,0.7)	(0.1,0.3,0.6)
$\ddot{\hat{S_{1}}}$	(0.38,0.49)	(0.75,0.44)	(0.75,0.35)
$\ddot{\hat{S_{2}}}$	(0.48,0.77)	(0.66,0.64)	(0.75,0.49)
$\ddot{\hat{S_{3}}}$	(0.56,0.67)	(0.70,0.54)	(0.75,0.48)
$\ddot{\hat{S_{4}}}$	(0.465,0.69)	(0.56,0.57)	(0.75,0.42)
$\ddot{\hat{S_{5}}}$	(0.53,0.82)	(0.655,0.54)	(0.68,0.39)

Table 6

Weighted aggregation of L_{q
^*} $q - {RO}_{MFS} S$ of dimension 3

Suppliers	Attributes
	$\ddot{\hat{e_{4}}}$	$\ddot{\hat{e_{5}}}$
	$ω (\ddot{\hat{e_{4}}})$ =	$ω (\ddot{\hat{e_{5}}})$ =
	(0.1, 0.3, 0.6)	(0.1, 0.4, 0.5)
$\ddot{\hat{S_{1}}}$	(0.81,0.25)	(0.89,0.16)
$\ddot{\hat{S_{2}}}$	(0.595,0.46)	(0.815,0.35)
$\ddot{\hat{S_{3}}}$	(0.72, 0.25)	(0.84, 0.16)
$\ddot{\hat{S_{4}}}$	(0.65,0.42)	(0.74,0.31)
$\ddot{\hat{S_{5}}}$	(0.75, 0.25)	(0.84, 0.16)

Step C. Table 7 shows the comparison matrix for q-ROFSS.

Table 7

Comaprison matrix

Attributes	Suppliers
	$\ddot{\hat{S_{1}}}$	$\ddot{\hat{S_{2}}}$	$\ddot{\hat{S_{3}}}$	$\ddot{\hat{S_{4}}}$	$\ddot{\hat{S_{5}}}$
$\ddot{\hat{e_{1}}}$	0	-1	3	-1	-1
$\ddot{\hat{e_{2}}}$	4	-2	1	-3	-1
$\ddot{\hat{e_{3}}}$	4	0	1	2	-1
$\ddot{\hat{e_{4}}}$	4	-4	0	-2	1
$\ddot{\hat{e_{5}}}$	4	-3	1	-3	1
Score(S _i)	12	-10	6	-7	-1
Rank	I	V	II	IV	III

Table 8

Tabular representation of the mid-level soft set (Λ _f ; mid) with choice values

Suppliers	Attributes
	$\ddot{\hat{e_{1}}}$	$\ddot{\hat{e_{2}}}$	$\ddot{\hat{e_{3}}}$	$\ddot{\hat{e_{4}}}$	$\ddot{\hat{e_{5}}}$	Choice Values
$\ddot{\hat{S_{1}}}$	1	1	1	1	1	5
$\ddot{\hat{S_{2}}}$	0	0	0	0	0	0
$\ddot{\hat{S_{3}}}$	0	1	0	1	1	3
$\ddot{\hat{S_{4}}}$	0	0	0	0	1	1
$\ddot{\hat{S_{5}}}$	0	0	0	1	1	2

Step D. Calculate the score value using (Definition 2.10) shown in Table 7

Step E. Rank the suppliers according to the obtained score values.

$\ddot{\hat{S_{1}}} \geq \ddot{\hat{S_{3}}} \geq \ddot{\hat{S_{5}}} \geq \ddot{\hat{S_{4}}} \geq \ddot{\hat{S_{2}}}$ .

As a result, $\ddot{\hat{S_{1}}}$ is the best supplier for purchasing equipment, shown in Table 7.

4.4.2 Algorithm 2: Calculation part

In L_{q
^*} q-RO_MFSS-MADM, one has the flexibility to employ various rules or thresholds as part of an adaptable approach. For instance, when addressing the issue using a mid-level decision rule, it becomes evident that the mid-level threshold of Δ _F takes the form of a fuzzy set. mid Δ _F = ${({\ddot{\hat{e}}}_{1}, (0.483, 0.688)), ({\ddot{\hat{e}}}_{2}, (0.665, 0.546)), ({\ddot{\hat{e}}}_{3}, (0.736, 0.418)), ({\ddot{\hat{e}}}_{4}, (0.699, 0.328)), ({\ddot{\hat{e}}}_{5}, (0.825, 0.228))}$ and we will derive the mid-level soft set L (Δ _f ; mid) of Δ _f using tabular representation, as shown in Table 9. Based on the data in Table 9, it can be observed that the maximum choice value is 5, and the optimal decision is to choose $S_{1}$ as the supplier. Hence, the hospital should select $S_{1}$ as the best supplier, taking into account the weights assigned to different parameters.

Table 9
Comaprison of proposed approach using Algorithm 1

Approach Suppliers

$\ddot{\hat{S_{1}}}$ $\ddot{\hat{S_{2}}}$ $\ddot{\hat{S_{3}}}$ $\ddot{\hat{S_{4}}}$ $\ddot{\hat{S_{5}}}$ Optimal

Proposed 12 -10 6 -7 -1 $\ddot{\hat{S_{1}}}$

LOMFSS [18] 16 9 16 6 10 $\ddot{\hat{S_{1}}}$

LODFrSS [14] - - - - - ×

LOIFSS [10] - - - - - ×

LONSS [27] - - - - - ×

Approach	Suppliers
Proposed	12	-10	6	-7	-1	$\ddot{\hat{S_{1}}}$
LOMFSS [18]	16	9	16	6	10	$\ddot{\hat{S_{1}}}$
LODFrSS [14]	-	-	-	-	-	×
LOIFSS [10]	-	-	-	-	-	×
LONSS [27]	-	-	-	-	-	×

5 Comparison and benefits of the proposed methodology

The suggested L_{q
^*} q-RO_MFSS-MADM framework establishes a comprehensive method for decision-making, specifically within the realm of healthcare supplier selection amid the COVID-19 pandemic. It is compared with some existing techniques [10, 14, 19, 27] to demonstrate the proficiency of the proposed work. The obtained results are summarised in Tables 9 and 10. It is clear from these tables that the results obtained by LOMFSS [19] are the same as the proposed approach. Here the ranking is the same because we reduce the L_{q
^*} q-RO_MFSS to LOMFSS by only taking the membership values of the objects in the L_{q
^*} q-RO_MFSS and ignoring their non-membership values. However, our presented approach is not comparable with the existing approaches used in [10, 14, 27]. Thus this sophisticated framework is expected to incorporate a range of decision criteria (attributes) that hold relevance in healthcare supply chain management. These attributes may encompass elements like supplier dependability, cost, lead time, product quality, and the ability to navigate disruptions.

Table 10
Comaprison of proposed approach using Algorithm 2

Approach Suppliers

$\ddot{\hat{S_{1}}}$ $\ddot{\hat{S_{2}}}$ $\ddot{\hat{S_{3}}}$ $\ddot{\hat{S_{4}}}$ $\ddot{\hat{S_{5}}}$ Optimal

Proposed 5 0 3 1 2 $\ddot{\hat{S_{1}}}$

LOMFSS [18] 5 1 5 1 3 $\ddot{\hat{S_{1}}}$

LODFrSS [14] - - - - - ×

LOIFSS [10] - - - - - ×

LONSS [27] - - - - - ×

Approach	Suppliers
Proposed	5	0	3	1	2	$\ddot{\hat{S_{1}}}$
LOMFSS [18]	5	1	5	1	3	$\ddot{\hat{S_{1}}}$
LODFrSS [14]	-	-	-	-	-	×
LOIFSS [10]	-	-	-	-	-	×
LONSS [27]	-	-	-	-	-	×

5.1 Benefits

The primary benefits of this paper is to articulate multi-fuzzy information through the utilization of L_{q
^*} q-rung orthopair multi-fuzzy soft information, which encompasses complex mathematical structures capable of accommodating uncertain and imprecise data within a multi-attribute decision-making framework. This concept exemplifies the endeavor to formulate an advanced framework that can offer heightened authenticity and precision in evaluations within intricate scenarios, such as the assessment of healthcare supplier rankings in times of uncertainty, such as the COVID-19 pandemic.

6 Conclusion

Recently, $q - ROFS$ s have been recognised as an effective tool for expressing fuzzy information, and they have been extensively researched and applied in MADM procedures. The proposed methodology enables healthcare organizations to systematically assess supplier performance based on multiple attribute, including reliability, lead times, pricing, quality, and flexibility. The incorporation of lattice ordering in q-rung orthopair multi-fuzzy soft set analysis aids in ranking alternatives by giving some order among parameters. The application of robust L_{q
^*} $q - {RO}_{MFS}$ -MADM methodology offers a structured and effective approach to evaluating and ranking suppliers, ultimately strengthening the healthcare system’s ability to respond to emergencies while maintaining a consistent supply of vital resources. In future, we will develop the algebraic structures of L_{q
^*} q-RO_MFSS.

Footnotes

Acknowledgments

The article has been written with the joint financial support of RUSA-Phase 2.0 grant sanctioned vide letter No.F24-51/2014-U, Policy (TN Multi-Gen), Dept. of Edn. Govt. of India, Dt. 09.10.2018, DST-PURSE 2nd Phase programme vide letter No. SR/PURSE Phase 2/38 (G) Dt. 21.02.2017 and DST (FIST - level I) 657876570 vide letter No.SR/FIST/MS-I/2018/17 Dt.20.12.2018.

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An enhanced MADM with L q * q-Rung orthopair multi-fuzzy soft set in healthcare supplier selection

Abstract

Keywords

1 Introduction

4.1 Algorithm 1

4.3 Case scenario

4.4.1 Algorithm 1: Calculation part

Table 9 Comaprison of proposed approach using Algorithm 1 Approach Suppliers S 1 ˆ ̈ S 2 ˆ ̈ S 3 ˆ ̈ S 4 ˆ ̈ S 5 ˆ ̈ Optimal Proposed 12 -10 6 -7 -1 S 1 ˆ ̈ LOMFSS [18] 16 9 16 6 10 S 1 ˆ ̈ LODFrSS [14] - - - - - × LOIFSS [10] - - - - - × LONSS [27] - - - - - ×

Table 10 Comaprison of proposed approach using Algorithm 2 Approach Suppliers S 1 ˆ ̈ S 2 ˆ ̈ S 3 ˆ ̈ S 4 ˆ ̈ S 5 ˆ ̈ Optimal Proposed 5 0 3 1 2 S 1 ˆ ̈ LOMFSS [18] 5 1 5 1 3 S 1 ˆ ̈ LODFrSS [14] - - - - - × LOIFSS [10] - - - - - × LONSS [27] - - - - - ×

6 Conclusion

Footnotes

Acknowledgments

References