Selection of healthcare waste disposal strategies with L q * q-rung orthopair multi-fuzzy soft set similarity measures

Abstract

The global challenges associated with urbanization and the escalating waste production have been magnified in recent times, particularly in the context of the COVID-19 pandemic. In response to these challenges, municipal authorities, especially in developing nations, are confronted with the imperative task of discerning the most suitable healthcare waste (HCW) disposal methods. These methods are crucial for the effective management of medical waste, both during and after the COVID-19 era. This study introduces a novel similarity measure designed for lattice ordered q-rung orthopair multi-fuzzy soft sets (L_{q
^*} q-ROMⁿFSSs) and exploring some of their essential characteristics. Currently, no established methods are available for gauging the similarity of L_{q
^*} q-ROMⁿFSSs sets. Therefore, this paper takes a pioneering step by presenting similarity measures tailored for L_{q
^*} q-ROMⁿFSSs sets. Moreover, we propose an evaluation methodology that leverages the lattice ordered q-rung orthopair multi-fuzzy soft information to determine the optimal health care waste (HCW) disposal approach. This approach seeks to enhance decision-making within the realm of waste management, facilitating more informed and effective choices in handling healthcare waste.

Keywords

Multi-fuzzy soft set healthcare waste disposal technique

1 Introduction

In the last five decades, global population growth, especially in developing nations, has raised significant concerns for both human and animal well-being. Managing healthcare waste (HCW) in these nations is challenging due to resource limitations, inadequate infrastructure, inappropriate disposal methods, insufficient training and awareness, restricted regulatory frameworks, and inadequate transportation systems. Addressing these challenges requires a comprehensive strategy involving infrastructure improvement, regulatory framework fortification, education, training, and sufficient financial resources. The World Health Organization (WHO) defines HCW to include materials like used syringes, blood-soaked items, bandages, scalpels, body parts, chemicals, cytotoxic substances, and radioactive components [17, 36]. WHO reports that around 85% of HCW is nonhazardous, while the remaining 15% poses risks such as contagion, toxicity, or radioactivity [35, 39]. Inadequate handling of this 15% can lead to various ecological and health risks. Establishing an environmentally friendly HCW management system is a primary concern for healthcare organizations [40]. Therefore, selecting the most suitable HCW disposal technique from a set of alternatives is a complex multi-criteria decision-making (MCDM) process due to multiple qualitative and quantitative criteria from sustainable perspectives [41].

The decision-making process faces numerous challenges when dealing with unclear, ambiguous, imprecise, and vague criteria in multi-criteria decision-making. In such situations, crisp sets are rendered ineffective in addressing uncertainty and imprecision in information, while fuzzy information can adeptly manage these issues. Zadeh [48] proposed the idea of fuzzy set (FS), debates on MAGDM with fuzzy information have never ended. However, the fuzzy set cannot adequately represent all decision-making difficulties. For example, during the site selection process, some assessors agree with a scheme, while others oppose or even reject it. As a result, how to successfully employ them to cope with the fuzzy decision-making problem is directly tied to people’s productivity, living, and societal advancement. Atanassov [2] introduced the intuitionistic fuzzy set (IFS), which is an extension of FS, for this purpose. The IFS depicts the fuzziness of objects from three perspectives: membership values(MVs), nonmembership values (NMVs), and hesitation degree; hence, it has greater fuzziness and uncertainty flexibility.

Yager [43, 44] showed that the existing frameworks of FS and IFS were inadequate for capturing human opinions in a more practical manner. He introduced Pythagorean Fuzzy Sets (PyFSs), which extended the scope of information by introducing a novel conditional constraint (i.e.,) an element is assigned a membership values(MVs) of 0.65 and a non-membership values(NMVs) degree of 0.85. When you square these values, (0.65) ² + (0.85) ² = 1.145, which is greater than 1. This means that PFSs (Pythagorean Fuzzy Sets) can’t handle this situation. To address such cases, Senapati and Yager [34] introduced Fermatean fuzzy sets (FFSs) with a condition: 0 ≤ (δ) ³ + (λ) ³ ≤ 1. This expanded range of uncertainties allows us to handle the problems discussed earlier because (0.65) ³ + (0.85) ³ = 0.888, which is less than or equal to 1. Later on, Yager [45] contributed by introducing q-rung orthopair fuzzy sets (q-ROFSs) or generalized orthopair fuzzy sets, where the sum of the qth power of belongingness and non-belongingness values of elements doesn’t exceed one. These q-ROFSs can be simplified to IFSs, PFSs, and FFSs for different values of q: q = 1, 2, and q = 3, respectively.

Molodtsov [23] introduced a completely novel theory known as the theory of soft sets (S _t S), which provides a parameterized mathematical framework that overcomes the aforementioned limitations. Yager and Sebastian and Ramakrishnan [32, 33, 42] quantify the concept of multi-fuzzy sets for duplicate elements in a set with similar or distinct membership values. Yang et al. [46] introduced the concept of an MFSS by merging MFS and SS and applied it in the context of MCDM. Meanwhile, Das and Kar [8] and Asit and Dey [4] introduced the Intuitionistic Multi-Fuzzy Soft Set (IMFS_tS) model and hesitant multi-fuzzy soft set model as a extension of MFS_tS. Recently, Vimala et al. and Jeevitha et al. successfully extended the IMFS_tS model by introducing the q-rung orthopair multi-fuzzy soft set (q-ROMFS_tS) and linear diophantine multi-fuzzy soft set (LDMFS_tS) models.

Lattice theory holds significant significance in various everyday life domains. Birkhoff [6] made crucial contributions to lattice theory, exploring its practical applications. However, the absence of lattice-ordered frameworks in hybrid sets prompted further developments to extend their applicability across various domains. Vimala et al. proposed several theories on lattices [1, 18, 19, 38], broadening their reach into diverse fields. Additionally, Sabeena Begam et al. introduced the concept of lattice ordered MFS_tS and applied it in various contexts [5, 28, 29]. Recently, the extension of lattice ordered MFS_tS is developed by incorporating multi-NMVS [22]. Vimala et al. recently [39] proposed lattice-ordered linear diophantine MFS_tS and their application in Prognostication of Myocardial Infarction.

Similarity measures(SMs) are essential for evaluating associations between items. Initially SMs defined for vague stes [14], soft sets [11], soft matrices [12], fuzzy S_tS [9, 16, 21], further they were extended to various hybrid fuzzy models such as interval fuzzy sets (IFS) [24], intuitionistic fuzzy sets (IFS) [7, 13, 15], and PyFSs [20, 27, 31]. In recent years, researchers have explored different types of similarity measures (SMs) [27, 30, 31, 47], applying them to decision-making, pattern recognition, and image processing across various soft set models, including FSS [9, 16], intuitionistic FSS [25], and Py FSS [27]. The aforementioned similarity measures are not well-suited for managing multi-dimensional data. In response to this challenge, Al-Qudah and Hassan [3] devised a novel similarity measure tailored for managing multi-dimensional attributes within the framework of complex MⁿFSS. This hybridized approach extends the range of values available, effectively addressing the intricacies of handling uncertainty associated with periodic data.

Subsequently, Sabeena et al. [5] introduced a similarity measure within the framework of lattice-ordered MⁿFSS, specifically designed to manage multi- dimensional data by incorporating multi-membership values. Recently, Saeed et al. [30] introduced the theory of complex multi-fuzzy hypersoft similarity measure. This measure extends the membership function by translating it onto a unit circle, incorporating phase and amplitude components within the hypersoft set domain. However, it is essential to recognize that all the existing similarity measures developed for multi-fuzzy sets have limitations when it comes to handling multi-non-membership values (multi-NMVs). This limitation prompted the development of the L_{q
^*} q-ROMⁿFS set similarity measure. This innovative approach is adept at addressing both multi-membership values (multi-MVs) and multi-non-membership values (multi-NMVs), providing a comprehensive solution for effectively handling complex multi-dimensional data sets. Consequently, the significant contributions are outlined as follows:

-
The innovative L_q
^* q-ROMⁿFS set model has been specially designed to tackle the complex challenges presented by lattice-ordered q-ROMⁿFS information. L_q
^* q-ROMⁿFS set not only elevates computational efficiency but also enables the seamless management of increasing uncertainties, as it accommodates the order of uncertainty ranging from 1 to the qth power of multi-membership degree and multi-non-membership degree within datasets.
-
An robust similarity measure has been formulated specifically for Lq ^* q-ROMⁿFSSs, with its essential properties, and subsequently utilized to evaluate criteria weights.
-
The suggested similarity measure was utilized in the context of healthcare waste disposal method selection, demonstrating its practicality and efficiency. We addressed the healthcare waste disposal management challenge and demonstrated the superiority of the proposed similarity measures over the currently employed ones.
Paper Outline: In Section 2, we review some basic terms and explain the main idea of a q-ROMⁿFS similarity measures, along with important concepts. Section 3 introduces the idea of q-ROMⁿFS sets, q-ROMⁿFS matrices, measures of similarity, and weighted similarity measures for q-ROMⁿFS sets. In Section 4, we examine a real-life case about choosing the best method for disposing of healthcare waste. Finally, Section 5 wraps up the paper with conclusions and suggests potential future research directions.
2 Preliminaries

This section discusses the fundamental concepts, definitions, and properties of MFS, MFSS, q-ROFS, and q-ROMFSS.

Definition 2.1. [45] Let $Z$ be the universe of discourse with $δ_{Q}$ : $Z$ ⟶ [0,1] and $λ_{Q}$ : $Z$ ⟶ [0,1] represents the membership values and non-membership values respectively. The set $Q$ = ${〈 z, δ_{Q} (z), λ_{Q} (z) 〉 / z \in Z}$ is known as q-rung orthopair fuzzy set $Q$ in $Z$ if it satisfies the condition $0 \leq δ_{Q} (z) + λ_{Q} (z) \leq 1$ with the degree of hesitancy given by $Π_{Q} (z)$ = $1 - δ_{Q} (z) + λ_{Q} (z)$ .

Definition 2.2. [32] Let n be a non-zero non-negative integer. A multi-fuzzy set, $T$ = ${〈 z, δ_{T}^{1} (z), . . ., δ_{T}^{n} (z) 〉 / z \in Z}$ such that $\begin{matrix} δ_{T}^{k} : Z ⟶ [0, 1], k = 1, 2, . . ., n \end{matrix}$ A function $δ_{T}^{k} (z)$ = ${δ_{T}^{1} (z), δ_{T}^{2} (z), . . ., δ_{T}^{n} (z)$ is said to be multi-membership values(M-MVs) of MF-sets of $T$ , $n = Dim T$ and their collection in $Z$ is $M^{n} FS (Z)$ .

Definition 2.3. [46] A pair $(H, T)$ is said to be MⁿFSS with dimension n over $Z$ if $H$ : $T$ ⟶ $M^{n} FS (Z)$ and $H (t)$ , $t$ ∈ $T$ is its collection of e-approximate members.

Definition 2.4. [28] A MⁿFSS over $Z$ is said to be lattice ordered MⁿFSS over $Z$ if for $t_{1}$ , $t_{2}$ ∈ $T_{1}$ , such that $t_{1}$ ≤ $t_{2}$ ⇒ $H (t_{1})$ ⊆ $H (t_{2})$ , i.e., $δ_{H (t)}^{k} (z) \leq δ_{H (t)}^{k} (z)$ and $λ_{H (t)}^{k} (z) \geq λ_{H (t)}^{k} (z)$ , ∀ $z \in Z$

Definition 2.5. [5] Let $(H, T)$ , $(L, T)$ ∈ LOMⁿFSS. The similarity measure of $(H, T)$ and $(L, T)$ over $Z$ is defined as $\begin{matrix} {Sim}_{L_{q^{*}}} ((H, T), (L, T)) \\ = \frac{\sum_{k = 1}^{n} {Sim}_{L_{q^{*}}}^{k} ((H, T), (L, T))}{| k |} \\ where {Sim}_{L_{q^{*}}}^{k} ((H, T), (L, T)) \\ = \frac{\sum_{j = 1}^{m} \lor_{z \in Z} ((H_{δ^{k} (t_{j})} (z))^{q} \land ((L_{δ^{k} (t_{j})} (z))^{q})}{\sum_{j = 1}^{m} \lor_{z \in Z} ((H_{δ^{k} (t_{j})} (z))^{q} \lor ((L_{δ^{k} (t_{j})} (z))^{q})} \end{matrix}$

Definition 2.6. Let $T$ be a q - ROM ⁿ FS set of dimension n defined on $Z$ , where $δ_{T}^{k} (z)$ and $λ_{T}^{k} (z)$ are real-valued functions representing M - MVs and M - NMVs of $Z$ in $T$ . These functions satisfy $0 \leq (δ_{T}^{k} (z))^{q} + (λ_{T}^{k} (z))^{q} \leq 1$ for all $z \in Z$ , where q ≥ 1 and k = 1,2,..,n. The set $T$ can be represented as $\begin{matrix} T = {z / (δ_{T}^{k} (z), λ_{T}^{k} (z)) : z \in Z} \end{matrix}$ where $(δ_{T}^{k} (z), λ_{T}^{k} (z))$ = ${(δ_{T}^{1} (z), λ_{T}^{1} (z), (δ_{T}^{2} (z), λ_{T}^{2} (z), . . ., (δ_{T}^{n} (z), λ_{T}^{n} (z) : z \in Z}$ .

The collection of all q - ROM ⁿ FS set of dimension n in $Z$ is denoted as q - ROMFS ⁿ. Simply a q - ROMFN (q - ROMFN) can be expressed as (δ ^k, λ^k).

Definition 2.7. [37] A pair $(H, T)$ is said to be qROMFSS with dimension n over $Z$ if $H$ : $T$ ⟶ ${qROM}^{n} FS (Z)$ and it is characterized as $\begin{matrix} (H, T) = {(t, H (t)) : t \in T, H (t) \in {qROM}^{n} FS (z)} \end{matrix}$ where $H (t)$ = ${〈 z, δ_{H (t)}^{k} (z), λ_{H (t)}^{k} (z) 〉 / t \in T, z \in Z, k = 1, 2, . . . n}$ where $δ_{H (t)}^{k} (z)$ and $λ_{H (t)}^{k} (z)$ is a multi-membership values(M - MVs) and multi-non-membership values(M - NMVs) of $z \in Z$ with real valued functions. The collection of all q-rung orthopair multi-fuzzy soft sets is denoted as qROMⁿFS set.

3 L _{q
^*} q-Rung orthopair multi-fuzyy soft similarity measure

Definition 3.1. A q - ROMFS set $(H, T)$ over $Z$ is defined as L _{q
^*} q - ROM ⁿ FS set if for $t_{1}$ , $t_{2}$ ∈ $T$ , such that $t_{1}$ ≤ $t_{2}$ implies $H (t_{1})$ ⊆ $H (t_{1})$ . i.e., $δ_{H (t_{1})}^{k} (z)$ ≤ $δ_{H (t_{1})}^{k} (z)$ and $λ_{H (t_{1})}^{k} (z)$ ≥ $λ_{H (t_{1})}^{k} (z)$ , for all $z$ ∈ $Z$ .

Example 3.2. In the field of renewable energy, three innovative international companies, SolarX, WindEco, and HydroGen, have transformed the production of sustainable energy solutions for the global market. The United States is a major consumer, and these companies primarily market their products in G7 and North American countries. However, since the groups G7 and North America have some overlap but do not fully encompass each other, it is crucial to assess sales performance in the United States, G7, North America, and globally. The evaluation is based solely on energy production capacity, ranging from no capacity (0) to the highest capacity (1). As a result, the evaluation score increases as the region under assessment expands, as more energy solutions are expected to be installed in larger regions. It’s important to note that the evaluation committee has chosen to assess two types of renewable energy products, solar panels and wind turbines, separately, even though both are manufactured by these companies.Let the three companies be $Z$ = ${z_{1}, z_{2}, z_{3}}$ and let $T$ be the lattice of parameters $T$ = ${t_{1} = US, t_{2} = G 7, t_{3} = NorthAmerica, t_{4} = globally}$ for which $t_{1} \leq t_{2} \leq t_{4}$ and $t_{1} \leq t_{3} \leq t_{4}$ , but no such relationship established between $t_{3}$ and $t_{4}$ . Here the set of indexes to denote the two assessment categories. $T$ = ${t^{1} = solar panels, t^{2} = wind turbines}$ . The expression of L _{q
^*} q - ROMFS set is presented, and its tabular representation is in Table 1.

Table 1
Tabular Representation of L_{q
^*} $q - {RO}_{MFS} S$

$US (t_{1})$ $G 7 (t_{2})$ $North - America (t_{3})$ $globally (t_{4})$

$z_{1}$ [(0.3,0.7), (0.4,0.8)] [(0.4,0.6), (0.5,0.7)] [(0.5,0.3), (0.8,0.5)] [(0.8,0.2), (0.9,0.2)]

$z_{2}$ [(0.4,0.7), (0.5,0.7)] [(0.7,0.6), (0.5,0.6)] [(0.8,0.3), (0.7,0.6)] [(0.9,0.1), (0.8,0.3)]

$z_{3}$ [(0.3,0.9), (0.2,0.8)] [(0.5,0.7), (0.4,0.6)] [(0.7,0.5), (0.8,0.5)] [(0.7,0.5), (0.8, 0.4)]

	$US (t_{1})$	$G 7 (t_{2})$	$North - America (t_{3})$	$globally (t_{4})$
$z_{1}$	[(0.3,0.7), (0.4,0.8)]	[(0.4,0.6), (0.5,0.7)]	[(0.5,0.3), (0.8,0.5)]	[(0.8,0.2), (0.9,0.2)]
$z_{2}$	[(0.4,0.7), (0.5,0.7)]	[(0.7,0.6), (0.5,0.6)]	[(0.8,0.3), (0.7,0.6)]	[(0.9,0.1), (0.8,0.3)]
$z_{3}$	[(0.3,0.9), (0.2,0.8)]	[(0.5,0.7), (0.4,0.6)]	[(0.7,0.5), (0.8,0.5)]	[(0.7,0.5), (0.8, 0.4)]

$(H, T)$ =

${\begin{matrix} H (t_{1}) = {\frac{z_{1}}{[(0.3, 0.7), (0.4, 0.8)]}, \frac{z_{2}}{[(0.4, 0.7), (0.5, 0.7)]}, \\ \frac{z_{3}}{[(0.3, 0.9), (0.2, 0.8)]}} \\ H (t_{2}) = {\frac{z_{1}}{[(0.4, 0.6), (0.5, 0.7)]}, \frac{z_{2}}{[(0.7, 0.6), (0.5, 0.6)]}, \\ \frac{z_{3}}{[(0.5, 0.7), (0.4, 0.6)]}} \\ H (t_{3}) = {\frac{z_{1}}{[(0.5, 0.3), (0.8, 0.5)]}, \frac{z_{2}}{[(0.8, 0.3), (0.7, 0.6)]}, \\ \frac{\ddot{\hat{z_{3}}}}{[(0.7, 0.5), (0.8, 0.5)]}} \\ H (t_{4}) = {\frac{z_{1}}{[(0.8, 0.2), (0.9, 0.2)]}, \frac{z_{2}}{[(0.9, 0.1), (0.8, 0.3)]}, \\ \frac{z_{3}}{[(0.7, 0.5), (0.8, 0.4)]}} \end{matrix}}$

⊆ L_{q
^*} q - ROM ⁿ FS $(Z)$

Definition 3.3. Let $Z$ = ${z_{1}, z_{2}, . . ., z_{r}}$ be the initial universe and $T$ = ${t_{1}, t_{2}, . . ., t_{s}}$ be the set of attributes. Then the matrix representation of L _{q
^*} q - ROM ⁿ FS $(H, T)$ can be written as $H_{δ, λ}$ = [h _ij] _r×s or simply by [h _ij] _r×s, i=1,2,...,r, j=1,2,...,s and h _ij = $(δ_{H (t_{j})}^{k} (z_{i}), λ_{H (t_{j})}^{k} (z_{i}))$ , where $δ_{H (t_{j})}^{k} (z_{i})$ and $λ_{H (t_{j})}^{k} (z_{i})$ indicates the M - MVs and M - NMVs of $z_{i}^{'} s$ and $t_{j}^{'} s$ . That is

$H_{δ, λ} = [h_{ij}]_{r \times s}$ = $[(δ_{H (t)}^{k}, λ_{H (t)}^{k})]_{r \times s}$ =

${(\begin{matrix} \begin{matrix} [(δ_{11}^{1}, λ_{11}^{1}), \dots \\ , (δ_{11}^{k}, λ_{11}^{k})] \end{matrix} & \dots \dots & \begin{matrix} [(δ_{1 s}^{k}, λ_{1 s}^{k}), \dots \\ , (δ_{1 s}^{k}, λ_{1 s}^{k})] \end{matrix} \\ \begin{matrix} [(δ_{21}^{1}, λ_{21}^{1}), \dots \\ , (δ_{21}^{k}, λ_{21}^{k})] \end{matrix} & \dots \dots & \begin{matrix} [(δ_{2 s}^{k}, λ_{2 s}^{k}), \dots \\ , (δ_{2 s}^{k}, λ_{2 s}^{k})] \end{matrix} \\ \begin{matrix} [(δ_{r 1}^{1}, λ_{r 1}^{1}), \dots \\ , (δ_{r 1}^{k}, λ_{r 1}^{k})] \end{matrix} & \dots \dots & \begin{matrix} [(δ_{rs}^{k}, λ_{rs}^{k}), \dots \\ , (δ_{rs}^{k}, λ_{rs}^{k})] \end{matrix} \end{matrix})}_{r \times s}$

Definition 3.4. Let $(H, T_{1})$ , $(L, T_{2})$ be two L_{q
^*} q-ROMⁿFSSs of dimension n over $Z$ . -

[(i)] $(H, T_{1})$ is said to a L_q
^* q-rung orthopair multi-fuzzy soft subset of $(L, T_{2})$ If $T_{1}$ ⊆ $T_{2}$ and for any $t$ ∈ $T_{1}$ , $H_{T_{1} (t)} (z)$ ⊆ $L_{T_{2} (t)} (z)$ i.e., $δ_{H_{T_{1} (t)}} (z)$ ≤ $δ_{L_{T_{1} (t)}} (z)$ and $λ_{H_{T_{1} (t)}} (z)$ ≥ $λ_{L_{T_{1} (t)}} (z)$ , for all $t$ ∈ $T_{1}$ , $z$ ∈ $Z$ and it is represented as $(H, T_{1})$ ⊆ $(L, T_{2})$ .

[(ii)] L _q
^* q-rung orthopair multi-fuzzy soft equal $(H, T_{1})$ ⊆ $(L, T_{2})$ if $δ_{H_{T_{1} (t)}} (z)$ = $δ_{L_{T_{1} (t)}} (z)$ and $λ_{H_{T_{1} (t)}} (z)$ = $λ_{L_{T_{1} (t)}} (z)$ , for all $t$ ∈ $T_{1}$ , $z$ ∈ $Z$

Definition 3.5. For any two L _{q
^*} q - ROM ⁿ FS sets, $(H, T)$ , $(L, T)$ , a lattice ordered (L _{q
^*}) q-rung orthopair multi-fuzzy soft similarity measure between $H$ and $L$ is defined as

$\begin{matrix} {Sim}_{L_{q^{*}}} ((H, T), (L, T)) \\ = \frac{1}{n} \sum_{k = 1}^{n} {Sim}_{L_{q^{*}}}^{k} ((H, T), (L, T)) \\ where {Sim}_{L_{q^{*}}}^{k} ((H, T), (L, T)) \\ = \frac{\begin{matrix} \sum_{j = 1}^{m} [((H_{δ^{k} (t_{j})} (z))^{q} \land (L_{δ^{k} (t_{j})} (z))^{q}) + \\ ((H_{λ^{k} (t_{j})} (z))^{q} \land (L_{λ^{k} (t_{j})} (z))^{q})]_{k = 1}^{n} \end{matrix}}{\begin{matrix} \sum_{j = 1}^{m} [((H_{δ^{k} (t_{j})} (z))^{q} \lor (L_{δ^{k} (t_{j})} (z))^{q}) \\ + ((H_{λ^{k} (t_{j})} (z))^{q} \lor (L_{λ^{k} (t_{j})} (z))^{q})]_{k = 1}^{n} \end{matrix}} \end{matrix}$

Example 3.6. Consider the initial universal set $Z$ = ${z_{1}, z_{1}, z_{1}}$ and the set of attributes $T$ = ${t_{1}, t_{1}, t_{1}}$ of dimension 2 over $Z$ , for which $t_{1} \leq t_{2} \leq t_{3}$ . Here $(H_{δ^{k}}, L_{δ^{k}})$ = $[(H_{δ^{1}}, L_{δ^{1}}), (H_{δ^{2}}, L_{δ^{2}})]$ .

Let $(H, T)$

- $= {\begin{matrix} H (t_{1}) = {\frac{z_{1}}{[(0.3, 0.9), (0.4, 0.7)]}, \\ \frac{z_{2}}{[(0.4, 0.7), (0.5, 0.3)]}, \\ \frac{z_{3}}{[(0.6, 0.6), (0.7, 0.2)]}} \\ H (t_{2}) = {\frac{z_{1}}{[(0.4, 0.8), (0.5, 0.6)]}, \\ \frac{z_{2}}{[(0.5, 0.6), (0.7, 0.2)]}, \\ \frac{z_{3}}{[(0.7, 0.5), (0.8, 0.2)]}} \\ H (t_{3}) = {\frac{z_{1}}{[(0.6, 0.7), (0.7, 0.4)]}, \\ \frac{z_{2}}{[(0.7, 0.6), (0.8, 0.5)]}, \\ \frac{z_{3}}{[(0.6, 0.3), (0.9, 0.1)]}} \end{matrix}}$

⊆ L _{q
^*} q - ROM ² FSS $(Z)$

and

Let $(L, T)$ - $= {\begin{matrix} L (t_{1}) = {\frac{z_{1}}{[(0.2, 0.9), (0.3, 0.7)]}, \\ \frac{z_{2}}{[(0.3, 0.7), (0.4, 0.4)]}, \\ \frac{z_{3}}{[(0.5, 0.7), (0.6, 0.3)]}} \\ L (t_{2}) = {\frac{z_{1}}{[(0.3, 0.8), (0.4, 0.7)]}, \\ \frac{z_{2}}{[(0.4, 0.7), (0.6, 0.4)]}, \\ \frac{z_{3}}{[(0.6, 0.7), (0.7, 0.3)]}} \\ L (t_{3}) = {\frac{z_{1}}{[(0.5, 0.6), (0.8, 0.5)]}, \\ \frac{z_{2}}{[(0.7, 0.5), (0.8, 0.5)]}, \\ \frac{z_{3}}{[(0.8, 0.4), (0.9, 0.0)]}} \end{matrix}}$

⊆ L _{q
^*} q - ROM ² FSS $(Z)$

Then we have (suppose q = 3), whose matrices are as follows:

$(H, T)$ $[\begin{matrix} \begin{matrix} [(0.3, 0.9), \\ (0.4, 0.7)] \end{matrix} & \begin{matrix} [(0.4, 0.8), \\ (0.5, 0.6)] \end{matrix} & \begin{matrix} [(0.6, 0.7), \\ (0.7, 0.4)] \end{matrix} \\ \begin{matrix} [(0.4, 0.7), \\ (0.5, 0.3)] \end{matrix} & \begin{matrix} [(0.5, 0.6), \\ (0.7, 0.2)] \end{matrix} & \begin{matrix} [(0.7, 0.6), \\ (0.8, 0.5)] \end{matrix} \\ \begin{matrix} [(0.6, 0.6), \\ (0.7, 0.2)] \end{matrix} & \begin{matrix} [(0.7, 0.5), \\ (0.8, 0.2)] \end{matrix} & \begin{matrix} [(0.6, 0.3), \\ (0.9, 0.1)] \end{matrix} \end{matrix}]$

$(L, T)$ = $[\begin{matrix} \begin{matrix} [(0.2, 0.9), \\ (0.3, 0.7)] \end{matrix} & \begin{matrix} [(0.3, 0.8), \\ (0.4, 0.7)] \end{matrix} & \begin{matrix} [(0.5, 0.6), \\ (0.8, 0.5)] \end{matrix} \\ \begin{matrix} [(0.3, 0.7), \\ (0.4, 0.4)] \end{matrix} & \begin{matrix} [(0.4, 0.7), \\ (0.6, 0.4)] \end{matrix} & \begin{matrix} [(0.7, 0.5), \\ (0.8, 0.5)] \end{matrix} \\ \begin{matrix} [(0.5, 0.7), \\ (0.6, 0.2)] \end{matrix} & \begin{matrix} [(0.6, 0.7), \\ (0.7, 0.3)] \end{matrix} & \begin{matrix} [(0.8, 0.4), \\ (0.9, 0.0)] \end{matrix} \end{matrix}]$

$\begin{matrix} {Sim}_{L_{q^{*}}} ((H, T), (L, T)) \\ = \frac{1}{n} \sum_{k = 1}^{n} {Sim}_{L_{q^{*}}}^{k} ((H, T), (L, T)) \\ where & {Sim}_{L_{q^{*}}}^{k} ((H, T), (L, T)) \\ = \frac{\begin{matrix} \sum_{j = 1}^{m} [((H_{δ^{k} (t_{j})} (z))^{q} \land (L_{δ^{k} (t_{j})} (z))^{q}) + \\ ((H_{λ^{k} (t_{j})} (z))^{q} \land (L_{λ^{k} (t_{j})} (z))^{q})]_{k = 1}^{n} \end{matrix}}{\begin{matrix} \sum_{j = 1}^{m} [((H_{δ^{k} (t_{j})} (z))^{q} \lor (L_{δ^{k} (t_{j})} (z))^{q}) \\ + ((H_{λ^{k} (t_{j})} (z))^{q} \lor (L_{λ^{k} (t_{j})} (z))^{q})]_{k = 1}^{n} \end{matrix}} \\ {Sim}_{L_{q^{*}}}^{1} ((H, T), (L, T)) \\ = \frac{\begin{matrix} 0 . 2^{3} + 0 . 9^{3} + 0 . 3^{3} + 0 . 7^{3} + 0 . 5^{3} + 0 . 6^{3} \\ + 0 . 3^{3} + 0 . 8^{3} + 0 . 4^{3} + 0 . 6^{3} + 0 . 6^{3} + 0 . 5^{3} \\ + 0 . 5^{3} + 0 . 6^{3} + 0 . 7^{3} + 0 . 5^{3} + 0 . 6^{3} + 0 . 3^{3} \end{matrix}}{\begin{matrix} 0 . 3^{3} + 0 . 9^{3} + 0 . 4^{3} + 0 . 7^{3} + 0 . 6^{3} + 0 . 7^{3} \\ + 0 . 4^{3} + 0 . 8^{3} + 0 . 5^{3} + 0 . 7^{3} + 0 . 7^{3} + 0 . 7^{3} \\ + 0 . 6^{3} + 0 . 7^{3} + 0 . 7^{3} + 0 . 6^{3} + 0 . 8^{3} + 0 . 4^{3} \end{matrix}} \\ = \frac{3.66}{5.146} = 0.71 \\ Similarly, {Sim}_{L_{q^{*}}}^{2} ((H, T), (L, T)) = 0.76 \\ {Sim}_{L_{q^{*}}} ((H, T), (L, T)) = \\ \frac{1}{2} {{Sim}^{1} ((H, T), (L, T)) + {Sim}^{2} ((H, T), (L, T))} \\ = \frac{1}{2} (0.71 + 0.76) = 0.735 \end{matrix}$

Lemma 3.7. Let $(H, T)$ , $(L, T)$ , $(M, T)$ be three L _{q
^*} q - ROM ⁿ FSS $(Z)$ over the universe $Z$ with the set of attributes of dimension n. Then the similarity measure defined in Definition 3.5 satisfies the following: -

[(i)] 0 ≤ Sim _{L
_{q
^*}} $((H, T), (L, T))$ ≤ 1.

[(ii)] Sim _{L
_{q
^*}} $((H, T), (L, T))$ = Sim _{L
_{q
^*}} $((L, T), (H, T))$

[(iii)] For $(H, T)$ = $(L, T)$ , if and only if Sim _{L
_{q
^*}} $((H, T), (L, T))$ = 1.

[(iv)] Consider $(H, T)$ ⊆ $(L, T)$ ⊆ $(M, T)$ , then Sim _{L
_{q
^*}} $((H, T), (M, T))$ ≤ Sim _{L
_{q
^*}} $((L, T) (M, T))$

Proof. (i) As multi-membership and multi-non-membership values of both L _{q
^*} q - ROM ⁿ FS sets belongs to [0,1]. So it is obvious that Sim $((H, T), (L, T))$ belongs to [0,1]. $\begin{matrix} (ii) & {Sim}_{L_{q^{*}}}^{k} ((H, T), (L, T)) = \\ \frac{\begin{matrix} \sum_{j = 1}^{m} [((H_{δ^{k} (t_{j})} (z))^{q} \land (L_{δ^{k} (t_{j})} (z))^{q}) \\ + ((H_{λ^{k} (t_{j})} (z))^{q} \land (L_{λ^{k} (t_{j})} (z))^{q})]_{k = 1}^{n} \end{matrix}}{\begin{matrix} \sum_{j = 1}^{m} [((H_{δ^{k} (t_{j})} (z))^{q} \lor (L_{δ^{k} (t_{j})} (z))^{q}) \\ + ((H_{λ^{k} (t_{j})} (z))^{q} \lor (L_{λ^{k} (t_{j})} (z))^{q})]_{k = 1}^{n} \end{matrix}} \\ = \frac{\begin{matrix} \sum_{j = 1}^{m} [((L_{δ^{k} (t_{j})} (z))^{q} \land (H_{δ^{k} (t_{j})} (z))^{q}) \\ + ((L_{λ^{k} (t_{j})} (z))^{q} \land (H_{λ^{k} (t_{j})} (z))^{q})]_{k = 1}^{n} \end{matrix}}{\begin{matrix} \sum_{j = 1}^{m} [((L_{δ^{k} (t_{j})} (z))^{q} \lor (H_{δ^{k} (t_{j})} (z))^{q}) \\ + ((L_{λ^{k} (t_{j})} (z))^{q} \lor (H_{λ^{k} (t_{j})} (z))^{q})]_{k = 1}^{n} \end{matrix}} \\ = {Sim}_{L_{q^{*}}}^{k} ((L, T), (H, T)) \end{matrix}$ we have, $\begin{matrix} {Sim}_{L_{q^{*}}} ((H, T), (L, T)) \\ = \frac{1}{n} \sum_{k = 1}^{n} {Sim}_{L_{q^{*}}}^{k} ((H, T), (L, T)) \\ = & \frac{1}{n} \sum_{k = 1}^{n} {Sim}_{L_{q^{*}}}^{k} ((L, T), (H, T)) \\ = & {Sim}_{L_{q^{*}}} ((L, T), (H, T)) \end{matrix}$ $\begin{matrix} (iii) & {Sim}_{L_{q^{*}}} ((H, T), (L, T)) \\ = \frac{1}{n} \sum_{k = 1}^{n} {Sim}_{L_{q^{*}}}^{k} ((H, T), (L, T)) \\ = \frac{1}{n} {\frac{\begin{matrix} \sum_{j = 1}^{m} [((H_{δ^{k} (t_{j})} (z))^{q} \land (L_{δ^{k} (t_{j})} (z))^{q}) \\ + ((H_{λ^{k} (t_{j})} (z))^{q} \land (L_{λ^{k} (t_{j})} (z))^{q})]_{k = 1}^{n} \end{matrix}}{\begin{matrix} \sum_{j = 1}^{m} [((H_{δ^{k} (t_{j})} (z))^{q} \lor (L_{δ^{k} (t_{j})} (z))^{q}) \\ + ((H_{λ^{k} (t_{j})} (z))^{q} \lor (L_{λ^{k} (t_{j})} (z))^{q})]_{k = 1}^{n} \end{matrix}}} \\ = & \frac{1}{n} {\sum_{j = 1}^{m} 1} (Since H_{δ^{k} (t_{j})} (z) = L_{δ^{k} (t_{j})} (z) \\ and H_{λ^{k} (t_{j})} (z) = L_{λ^{k} (t_{j})} (z)) \\ = & \frac{n}{n} = 1 \end{matrix}$ (iv) Proof is straightforward.□

3.1 L _{q
^*} q - ROM ⁿ FS weighted similarity measure

Definition 3.8. For any two L _{q
^*} q - ROM ⁿ FS sets, $(H, T)$ , $(L, T)$ , a lattice ordered (L _{q
^*}) q-rung orthopair multi-fuzzy soft weighted similarity measure between $H$ and $L$ is defined as $\begin{matrix} ω {Sim}_{L_{q^{*}}} ((H, T), (L, T)) \\ = \frac{1}{n} \sum_{k = 1}^{n} ω {Sim}_{L_{q^{*}}}^{k} ((H, T), (L, T)) \\ where {Sim}_{L_{q^{*}}}^{k} ((H, T), (L, T)) \\ = \frac{\begin{matrix} \sum_{j = 1}^{m} ω_{j}^{k} [((H_{δ^{k} (t_{j})} (z))^{q} \land (L_{δ^{k} (t_{j})} (z))^{q}) \\ + ((H_{λ^{k} (t_{j})} (z))^{q} \land (L_{λ^{k} (t_{j})} (z))^{q})]_{k = 1}^{n} \end{matrix}}{\begin{matrix} \sum_{j = 1}^{m} [((H_{δ^{k} (t_{j})} (z))^{q} \lor (L_{δ^{k} (t_{j})} (z))^{q}) \\ + ((H_{λ^{k} (t_{j})} (z))^{q} \lor (L_{λ^{k} (t_{j})} (z))^{q})]_{k = 1}^{n} \end{matrix}} \end{matrix}$

Example 3.9. In Example 3.6, let’s assume that the weights assigned to e ₁, e ₂, and e ₃, each having a dimension of 2, are [(0.4, 0.6) , (0.3, 0.7) , (0.2, 0.8)] ^T.

$\begin{matrix} {Sim}_{L_{q^{*}}}^{k} ((H, T), (L, T)) \\ = \frac{\begin{matrix} \sum_{j = 1}^{m} ω_{j}^{k} [((H_{δ^{k} (t_{j})} (z))^{q} \land (L_{δ^{k} (t_{j})} (z))^{q}) \\ + ((H_{λ^{k} (t_{j})} (z))^{q} \land (L_{λ^{k} (t_{j})} (z))^{q})]_{k = 1}^{n} \end{matrix}}{\begin{matrix} \sum_{j = 1}^{m} [((H_{δ^{k} (t_{j})} (z))^{q} \lor (L_{δ^{k} (t_{j})} (z))^{q}) \\ + ((H_{λ^{k} (t_{j})} (z))^{q} \lor (L_{λ^{k} (t_{j})} (z))^{q})]_{k = 1}^{n} \end{matrix}} \\ {Sim}_{L_{q^{*}}}^{1} ((H, T), (L, T)) = \frac{1.1376}{5.146} = 0.22 \\ Similarly {Sim}_{L_{q^{*}}}^{2} ((H, T), (L, T)) = 0.55 \\ {Sim}_{L_{q^{*}}} ((H, T), (L, T)) = \frac{1}{2} {{Sim}^{1} ((H, T), (L, T)) \\ + {Sim}^{2} ((H, T), (L, T))} \\ = & \frac{1}{2} (0.22 + 0.55) = 0.39 \end{matrix}$

Fig. 1

Diagrammatic representation of healthcare waste disposal method problem.

Lemma 3.10. Let $(H, T)$ , $(L, T)$ , $(M, T)$ be three L _{q
^*} q - ROM ⁿ FSS $(Z)$ over the universe $Z$ with the set of attributes of dimension n. Then the weighted similarity measure defined in Definition 3.8 satisfies the following: -

[(i)] 0 ≤ ^ω Sim _{L
_{q
^*}} $((H, T), (L, T))$ ≤ 1.

[(ii)] ^ω Sim _{L
_{q
^*}} $((H, T), (L, T))$ =

^ω Sim _{L
_{q
^*}} $((L, T), (H, T))$

[(iii)] For $(H, T)$ = $(L, T)$ , if and only if ^ω Sim _{L
_{q
^*}} $((H, T), (L, T))$ = 1.

[(iv)] Consider $(H, T)$ ⊆ $(L, T)$ ⊆ $(M, T)$ ,

then ^ω Sim _{L
_{q
^*}} $((H, T), (M, T))$ ≤ ^ω Sim _{L
_{q
^*}} $((L, T) (M, T))$

Proof. Proof similar to Lemma 3.7.□

4 A case study on effective healthcare waste disposal strategies

Healthcare waste disposal is a complex and highly regulated aspect of the healthcare sector. Improper handling and disposal can lead to environmental pollution, public health risks, and legal liabilities. It involves dealing with various types of waste materials, such as used needles, contaminated dressings, and expired medications, which can pose significant risks to both the environment and public health if not handled and disposed of properly. A prominent company in the healthcare industry is embarking on a strategic approach to address the responsible disposal of medical waste. With the increasing importance of environmental sustainability and stringent regulations surrounding medical waste management, the company is committed to establishing a state-of-the-art medical waste disposal site. The decision-makers involved in this endeavor are the board of governors, company executives, environmental experts, and financial analysts, who are tasked with determining the most suitable strategy for this venture. Three distinct strategies are being considered:

-
Strategy A (S _A -Incineration): Incineration serves as a prevalent disposal method, particularly for pathological and pharmaceutical waste within the medical sector. In the course of incineration, medical waste incinerators are engineered to attain staggering temperatures, often reaching up to 2,000 degrees Fahrenheit. It’s noteworthy that California law allows for the incineration of trace chemotherapy waste, further highlighting the versatility and regulatory acceptance of this disposal technique.
-
Strategy B (S _B -Autoclave): Autoclaving, a sterilization method employing steam, is highly effective for sterilizing microbiological waste. However, it may not be suitable for handling other categories of waste, such as pathological materials and hazardous chemicals, where alternative disposal methods are warranted.
-
Strategy C (S _C -Microwaving): Irradiative sterilization leverages technology akin to microwaves to decontaminate waste materials. The waste undergoes a process where it is first shredded and combined with water. Subsequently, controlled heating is applied to ensure the complete neutralization of all biological components within the waste.

Numerical Demonstration: In the healthcare waste disposal case study, the universal set $Z$ = ${z_{1}, z_{2}, z_{3}, z_{4}}$ represents various waste management scenarios, as detailed in Table 2.

Table 2
Variety of contexts in healthcare waste handling and disposal

Symbol Set of Universal Elements Description

$z_{1}$ Hospital Setting In a hospital setting, there’s diverse medical waste like sharps, infectious materials, and pharmaceuticals, possibly in both inpatient and outpatient facilities.

$z_{2}$ Pharmaceu-tical Waste Disposal This scenario deals with pharmaceutical waste disposal, including expired medications, in healthcare facilities, addressing unique challenges and regulations.

$z_{3}$ Laboratory Waste This scenario addresses safe disposal of laboratory waste, including hazardous and non-hazardous materials like chemicals, biological substances, and glassware in laboratory settings.

$z_{4}$ Veterinary Clinic This scenario handles the disposal of veterinary medical waste generated during animal treatment at veterinary clinics.

To facilitate a comprehensive decision, experts will compare Strategy A (S _A), Strategy B (S _B), Strategy C (S _C), and an alternate technique, Strategy M (S _M). To evaluate these strategies, five attributes are considered, as detailed in Table 3: -
A ₁:Site Location
-
A ₂:Cost Effectiveness
-
A ₃:Community Engagement
-
A ₄:Environmental Impact
-
A ₅:Regulatory Compliance
The assessment of the strategies based on their attributes is represented using the L _{q
^} q - ROM* ⁿ FS information, as shown in Tables 4-7.

Table 3
Evaluation of criteria

Symbol Criteria Description

A ₁ Site Location This attribute evaluates the appropriateness of the proposed site location for each strategy.

A ₂ Operational Efficiency It assesses the operational efficiency and effectiveness of each strategy in handling medical waste.

A ₃ Community Engagement This measures the extent to which each strategy engages with and benefits the local community.

A ₄ Environmental Impact It gauges the environmental impact and sustainability practices associated with each strategy.

A ₅ Regulatory Compliance This attribute assesses each strategy’s compliance with local and national regulations regarding medical waste disposal.

Fig. 2
Procedural framework of L_{q
^} q-ROMⁿFSS

Table 4
L _{q
^} q - ROM ⁴ FS decision matrix for strategy decision-making S _A (assuming q = 3)

$z_{1}$ $z_{2}$ $z_{3}$ $z_{4}$

A ₁ [(0.21,0.94), [(0.14,0.81), [(0.13,0.91), [(0.10,0.97),

(0.33,0.99), (0.46,0.93), (0.32,0.89), (0.22,0.90),

(0.44,0.82), (0.56,0.87), (0.65,0.63), (0.63,0.77),

(0.54,0.80)] (0.35,0.85)] (0.82,0.54)] (0.78,0.84)]

A ₂ [(0.69,0.60), [(0.33,0.84), [(0.25,0.80), [(0.28,0.71),

(0.73,0.52), (0.42,0.85), (0.40,0.79), (0.68,0.70),

(0.57,0.77), (0.57,0.82), (0.71,0.62), (0.74,0.68),

(0.63,0.63)] (0.42,0.73)] (0.84,0.52)] (0.80,0.59)]

A ₃ [(0.49,0.65), [(0.58,0.68), [(0.42,0.76), [(0.35,0.72),

(0.54,0.61), (0.64,0.63), (0.58,0.65), (0.65,0.71),

(0.76,0.64), (0.64,0.73), (0.81,0.48), (0.79,0.54),

(0.84,0.58)] (0.55,0.71)] (0.86,0.41)] (0.84,0.47)]

A ₄ [(0.65,0.43), [(0.67,0.72) [(0.21,0.87), [(0.71,0.62)

(0.32,0.68), (0.56,0.65), (0.52,0.82), (0.78,0.58),

(0.78,0.52), (0.78,0.64), (0.88,0.42), (0.86,0.51),

(0.86,0.45)] (0.62,0.60)] (0.91,0.34)] (0.87,0.36)]

A ₅ [(0.81,0.42), [(0.89,0.51), [(0.73,0.60), [(0.89,0.41),

(0.67,0.62), (0.93,0.42), (0.78,0.58), (0.93,0.33),

(0.93,0.39), (0.99,0.15), (0.73,0.60), (0.97,0.24),

(1,0.33)] (1,0.12)] (0.86,0.51)] (0.98,0.12)]

Table 5
L _{q
^} q - ROM* ⁴ FS decision matrix for strategy decision-making S _B (assuming q = 3)

$z_{1}$ $z_{2}$ $z_{3}$ $z_{4}$

A ₁ [(0.25,0.98), [(0.33,0.84) [(0.21,0.97) [(0.34,0.83)

(0.28,0.91), (0.43,0.84), (0.28,0.88), (0.32,0.78),

(0.31,0.88), (0.58,0.81), (0.34,0.83), (0.48,0.78),

(0.37,0.84)] (0.49,0.75)] (0.32,0.78)] (0.64,0.73)]

A ₂ [(0.34,0.92), [(0.40,0.72), [(0.25,0.93), [(0.64,0.63),

(0.43,0.87), (0.68,0.55), (0.34,0.87), (0.75,0.54),

(0.48,0.78), (0.74,0.43), (0.39,0.78), (0.73,0.60),

(0.64,0.73)] (0.34,0.62)] (0.45,0.73)] (0.78,0.58)]

A ₃ [(0.35,0.72), [(0.42,0.76), [(0.58,0.68), [(0.49,0.65),

(0.65,0.71), (0.58,0.65), (0.64,0.63), (0.54,0.61),

(0.76,0.64), (0.64,0.73), (0.81,0.48), (0.79,0.54),

(0.84,0.58)] (0.55,0.71)] (0.86,0.41)] (0.84,0.47)]

A ₄ [(0.54,0.63), [(0.67,0.72), [(0.69,0.60), [(0.71,0.62),

(0.62,0.68), (0.56,0.65), (0.73,0.52), (0.78,0.58),

(0.78,0.52), (0.78,0.64), (0.88,0.42), (0.86,0.51),

(0.86,0.45)] (0.62,0.60)] (0.91,0.34)] (0.87,0.36)]

A ₅ [(0.72,0.51), [(0.89,0.51), [(0.73,0.60), [(0.89,0.41),

(0.81,0.43), (0.93,0.42), (0.78,0.58), (0.93,0.32),

(0.93,0.39), (0.99,0.15), (0.86,0.51), (0.96,0.26),

(1,0.33)] (1,0.12)] (0.87,0.36)] (0.98,0.20)]

Table 6
L _{q
^} q - ROM* ⁴ FS decision matrix for strategy decision-making S _C (assuming q = 3)

$z_{1}$ $z_{2}$ $z_{3}$ $z_{4}$

A ₁ [(0.13,0.91), [(0.21,0.94), [(0.14,0.81), [(0.10,0.97),

(0.32,0.89), (0.33,0.99), (0.46,0.93), (0.22,0.90),

(0.44,0.82), (0.56,0.87), (0.65,0.63), (0.63,0.77),

(0.54,0.80)] (0.35,0.85)] (0.82,0.54)] (0.78,0.84)]

A ₂ [(0.21,0.87), [(0.33,0.84), [(0.25,0.80), [(0.28,0.71),

(0.52,0.82), (0.42,0.85), (0.40,0.79), (0.68,0.70),

(0.57,0.77), (0.57,0.82), (0.71,0.62), (0.74,0.68),

(0.63,0.63)] (0.42,0.73)] (0.84,0.52)] (0.80,0.59)]

A ₃ [(0.35,0.72), [(0.42,0.76), [(0.58,0.68), [(0.49,0.65),

(0.65,0.71), (0.58,0.65), (0.64,0.63), (0.54,0.61),

(0.76,0.64), (0.64,0.73), (0.81,0.48), (0.79,0.54),

(0.84,0.58)] (0.55,0.71)] (0.86,0.41)] (0.84,0.47)]

A ₄ [(0.54,0.63), [(0.67,0.72), [(0.69,0.60), [(0.71,0.62),

(0.62,0.68), (0.56,0.65), (0.73,0.52) (0.78,0.58),

(0.78,0.52), (0.78,0.64), (0.88,0.42), (0.86,0.51),

(0.86,0.45)] (0.62,0.60)] (0.91,0.34)] (0.87,0.36)]

A ₅ [(0.73,0.62), [(0.74,0.67), [(0.81,0.82), [(0.78,0.54),

(0.58,0.62), (0.77,0.42), (0.84,0.49), (0.83,0.38)

(0.82,0.48), (0.86,0.42), (0.91,0.36), (0.87,0.34),

(0.89,0.41)] (0.89,0.41)] (0.99,0.10)] (0.89,0.0)]

Table 7
L _{q
^} q - ROM* ⁴ FS decision matrix for strategy decision-making S _M (assuming q = 3)

$z_{1}$ $z_{2}$ $z_{3}$ $z_{4}$

A ₁ [(0.25,0.93), [(0.21,0.97), [0.12,0.92), (0.13,0.91),

(0.28,0.91), (0.32,0.84), (0.23,0.87), (0.32,0.89),

(0.39,0.78), (0.34,0.83), (0.25,0.88), (0.44,0.82),

(0.45,0.73)] (0.32,0.78)] (0.38,0.84)] (0.54,0.88)]

A ₂ [(0.33,0.84), [(0.35,0.81), [(0.25,0.91), [(0.35,0.72),

(0.43,0.84), (0.46,0.78), (0.34,0.87), (0.65,0.71),

(0.58,0.81), (0.48,0.76), (0.39,0.78), (0.76,0.64),

(0.49,0.75)] (0.64,0.73)] (0.45,0.73)] (0.84,0.58)]

A ₃ [(0.45,0.71), [(0.40,0.72), [(0.35,0.81), [(0.42,0.76),

(0.56,0.68), (0.68,0.55), (0.46,0.78), (0.58,0.65),

(0.68,0.76), (0.74,0.43), (0.48,0.76), (0.59,0.77),

(0.74,0.73)] (0.34,0.62)] (0.64,0.73)] (0.63,0.64)]

A ₄ [(0.50,0.72), [(0.69,0.59), [(0.64,0.63), [(0.54,0.63),

(0.68,0.55), (0.74,0.51), (0.75,0.54), (0.62,0.61),

(0.74,0.43), (0.89,0.45), (0.76,0.51), (0.63,0.54),

(0.84,0.62)] (0.94,0.41)] (0.82,0.47)] (0.79,0.52)]

A ₅ [(0.89,0.51), [(0.92,0.31), [(0.84,0.53), [(0.86,0.51),

(0.93,0.42), (0.97,0.23), (0.87,0.49), (0.87,0.34),

(0.98,0.21), (0.98,0.21), (0.92,0.33), (0.89,0.20),

(0.99,0.15]) (0.99,0.11)] (0.94,0.24)] (0.99,0.15)]

The similarity measure of three alternatives S _A, S _B and S _C with S _M are calculated by using L_{q
^} q-ROMⁿFS similarity measure, which are shown in Table 8. This innovative approach considers the degrees of multi-membership values (M-MVs) and non-membership values (M-NMVs) within the domain of L_{q
^} q-ROMⁿFSSs. From Table 8 we can get S _A is the best strategy for medical waste disposal.

Table 8
L _{q
^} q - ROM* ⁴ FS similarity measure for strategy decision-making(assuming q = 3)

Similarity Measure (S _A, S _M) (S _B, S _M) (S _C, S _M)

L _{q
^} q - ROM* ⁴ FS 0.94 0.82 0.71

similarity measure

L _{q
^} q - ROM* ⁴ FS 0.91 0.80 0.68

weighted similarity measure

In practical decision-making problems, it’s essential to consider the element weights. Suppose we assign criteria weights as ω = [(0.2, 0.3, 0.1, 0.4) , (0.07, 0.2, 0.13, 0.07) , (0.07, 0.2, 0.13, 0.6) , (0.4, 0.3, 0.2, 0.1) , (0.05, 0.15, 0.7, 0.1)] ^T respectively. In this context, we have applied the weighted similarity measures proposed in this study to calculate similarity. The results are presented in Table 3 (assuming q = 3). Based on the findings in Table 3, it’s evident that the degree of similarity between S _A and S _M is the highest when compared to other similarity measures. This suggests that among the two similarity measures, alternative S _A is the closest to being the optimal choice compared to S _M according to the principle of maximum similarity within L_{q
^*} q-ROMⁿFSSs (see Fig. 3). Consequently, S _M emerges as the most favorable strategy.

Among several concerns regarding the incineration approach, it remains the most practical alternative for securely disposing of healthcare waste.

Fig. 3
Ranking of the alternatives.
5 Conclusion

Symbol	Set of Universal Elements	Description
$z_{1}$	Hospital Setting	In a hospital setting, there’s diverse medical waste like sharps, infectious materials, and pharmaceuticals, possibly in both inpatient and outpatient facilities.
$z_{2}$	Pharmaceu-tical Waste Disposal	This scenario deals with pharmaceutical waste disposal, including expired medications, in healthcare facilities, addressing unique challenges and regulations.
$z_{3}$	Laboratory Waste	This scenario addresses safe disposal of laboratory waste, including hazardous and non-hazardous materials like chemicals, biological substances, and glassware in laboratory settings.
$z_{4}$	Veterinary Clinic	This scenario handles the disposal of veterinary medical waste generated during animal treatment at veterinary clinics.

Symbol	Criteria	Description
A ₁	Site Location	This attribute evaluates the appropriateness of the proposed site location for each strategy.
A ₂	Operational Efficiency	It assesses the operational efficiency and effectiveness of each strategy in handling medical waste.
A ₃	Community Engagement	This measures the extent to which each strategy engages with and benefits the local community.
A ₄	Environmental Impact	It gauges the environmental impact and sustainability practices associated with each strategy.
A ₅	Regulatory Compliance	This attribute assesses each strategy’s compliance with local and national regulations regarding medical waste disposal.

	$z_{1}$	$z_{2}$	$z_{3}$	$z_{4}$
A ₁	[(0.21,0.94),	[(0.14,0.81),	[(0.13,0.91),	[(0.10,0.97),
	(0.33,0.99),	(0.46,0.93),	(0.32,0.89),	(0.22,0.90),
	(0.44,0.82),	(0.56,0.87),	(0.65,0.63),	(0.63,0.77),
	(0.54,0.80)]	(0.35,0.85)]	(0.82,0.54)]	(0.78,0.84)]
A ₂	[(0.69,0.60),	[(0.33,0.84),	[(0.25,0.80),	[(0.28,0.71),
	(0.73,0.52),	(0.42,0.85),	(0.40,0.79),	(0.68,0.70),
	(0.57,0.77),	(0.57,0.82),	(0.71,0.62),	(0.74,0.68),
	(0.63,0.63)]	(0.42,0.73)]	(0.84,0.52)]	(0.80,0.59)]
A ₃	[(0.49,0.65),	[(0.58,0.68),	[(0.42,0.76),	[(0.35,0.72),
	(0.54,0.61),	(0.64,0.63),	(0.58,0.65),	(0.65,0.71),
	(0.76,0.64),	(0.64,0.73),	(0.81,0.48),	(0.79,0.54),
	(0.84,0.58)]	(0.55,0.71)]	(0.86,0.41)]	(0.84,0.47)]
A ₄	[(0.65,0.43),	[(0.67,0.72)	[(0.21,0.87),	[(0.71,0.62)
	(0.32,0.68),	(0.56,0.65),	(0.52,0.82),	(0.78,0.58),
	(0.78,0.52),	(0.78,0.64),	(0.88,0.42),	(0.86,0.51),
	(0.86,0.45)]	(0.62,0.60)]	(0.91,0.34)]	(0.87,0.36)]
A ₅	[(0.81,0.42),	[(0.89,0.51),	[(0.73,0.60),	[(0.89,0.41),
	(0.67,0.62),	(0.93,0.42),	(0.78,0.58),	(0.93,0.33),
	(0.93,0.39),	(0.99,0.15),	(0.73,0.60),	(0.97,0.24),
	(1,0.33)]	(1,0.12)]	(0.86,0.51)]	(0.98,0.12)]

	$z_{1}$	$z_{2}$	$z_{3}$	$z_{4}$
A ₁	[(0.25,0.98),	[(0.33,0.84)	[(0.21,0.97)	[(0.34,0.83)
	(0.28,0.91),	(0.43,0.84),	(0.28,0.88),	(0.32,0.78),
	(0.31,0.88),	(0.58,0.81),	(0.34,0.83),	(0.48,0.78),
	(0.37,0.84)]	(0.49,0.75)]	(0.32,0.78)]	(0.64,0.73)]
A ₂	[(0.34,0.92),	[(0.40,0.72),	[(0.25,0.93),	[(0.64,0.63),
	(0.43,0.87),	(0.68,0.55),	(0.34,0.87),	(0.75,0.54),
	(0.48,0.78),	(0.74,0.43),	(0.39,0.78),	(0.73,0.60),
	(0.64,0.73)]	(0.34,0.62)]	(0.45,0.73)]	(0.78,0.58)]
A ₃	[(0.35,0.72),	[(0.42,0.76),	[(0.58,0.68),	[(0.49,0.65),
	(0.65,0.71),	(0.58,0.65),	(0.64,0.63),	(0.54,0.61),
	(0.76,0.64),	(0.64,0.73),	(0.81,0.48),	(0.79,0.54),
	(0.84,0.58)]	(0.55,0.71)]	(0.86,0.41)]	(0.84,0.47)]
A ₄	[(0.54,0.63),	[(0.67,0.72),	[(0.69,0.60),	[(0.71,0.62),
	(0.62,0.68),	(0.56,0.65),	(0.73,0.52),	(0.78,0.58),
	(0.78,0.52),	(0.78,0.64),	(0.88,0.42),	(0.86,0.51),
	(0.86,0.45)]	(0.62,0.60)]	(0.91,0.34)]	(0.87,0.36)]
A ₅	[(0.72,0.51),	[(0.89,0.51),	[(0.73,0.60),	[(0.89,0.41),
	(0.81,0.43),	(0.93,0.42),	(0.78,0.58),	(0.93,0.32),
	(0.93,0.39),	(0.99,0.15),	(0.86,0.51),	(0.96,0.26),
	(1,0.33)]	(1,0.12)]	(0.87,0.36)]	(0.98,0.20)]

	$z_{1}$	$z_{2}$	$z_{3}$	$z_{4}$
A ₁	[(0.13,0.91),	[(0.21,0.94),	[(0.14,0.81),	[(0.10,0.97),
	(0.32,0.89),	(0.33,0.99),	(0.46,0.93),	(0.22,0.90),
	(0.44,0.82),	(0.56,0.87),	(0.65,0.63),	(0.63,0.77),
	(0.54,0.80)]	(0.35,0.85)]	(0.82,0.54)]	(0.78,0.84)]
A ₂	[(0.21,0.87),	[(0.33,0.84),	[(0.25,0.80),	[(0.28,0.71),
	(0.52,0.82),	(0.42,0.85),	(0.40,0.79),	(0.68,0.70),
	(0.57,0.77),	(0.57,0.82),	(0.71,0.62),	(0.74,0.68),
	(0.63,0.63)]	(0.42,0.73)]	(0.84,0.52)]	(0.80,0.59)]
A ₃	[(0.35,0.72),	[(0.42,0.76),	[(0.58,0.68),	[(0.49,0.65),
	(0.65,0.71),	(0.58,0.65),	(0.64,0.63),	(0.54,0.61),
	(0.76,0.64),	(0.64,0.73),	(0.81,0.48),	(0.79,0.54),
	(0.84,0.58)]	(0.55,0.71)]	(0.86,0.41)]	(0.84,0.47)]
A ₄	[(0.54,0.63),	[(0.67,0.72),	[(0.69,0.60),	[(0.71,0.62),
	(0.62,0.68),	(0.56,0.65),	(0.73,0.52)	(0.78,0.58),
	(0.78,0.52),	(0.78,0.64),	(0.88,0.42),	(0.86,0.51),
	(0.86,0.45)]	(0.62,0.60)]	(0.91,0.34)]	(0.87,0.36)]
A ₅	[(0.73,0.62),	[(0.74,0.67),	[(0.81,0.82),	[(0.78,0.54),
	(0.58,0.62),	(0.77,0.42),	(0.84,0.49),	(0.83,0.38)
	(0.82,0.48),	(0.86,0.42),	(0.91,0.36),	(0.87,0.34),
	(0.89,0.41)]	(0.89,0.41)]	(0.99,0.10)]	(0.89,0.0)]

	$z_{1}$	$z_{2}$	$z_{3}$	$z_{4}$
A ₁	[(0.25,0.93),	[(0.21,0.97),	[0.12,0.92),	(0.13,0.91),
	(0.28,0.91),	(0.32,0.84),	(0.23,0.87),	(0.32,0.89),
	(0.39,0.78),	(0.34,0.83),	(0.25,0.88),	(0.44,0.82),
	(0.45,0.73)]	(0.32,0.78)]	(0.38,0.84)]	(0.54,0.88)]
A ₂	[(0.33,0.84),	[(0.35,0.81),	[(0.25,0.91),	[(0.35,0.72),
	(0.43,0.84),	(0.46,0.78),	(0.34,0.87),	(0.65,0.71),
	(0.58,0.81),	(0.48,0.76),	(0.39,0.78),	(0.76,0.64),
	(0.49,0.75)]	(0.64,0.73)]	(0.45,0.73)]	(0.84,0.58)]
A ₃	[(0.45,0.71),	[(0.40,0.72),	[(0.35,0.81),	[(0.42,0.76),
	(0.56,0.68),	(0.68,0.55),	(0.46,0.78),	(0.58,0.65),
	(0.68,0.76),	(0.74,0.43),	(0.48,0.76),	(0.59,0.77),
	(0.74,0.73)]	(0.34,0.62)]	(0.64,0.73)]	(0.63,0.64)]
A ₄	[(0.50,0.72),	[(0.69,0.59),	[(0.64,0.63),	[(0.54,0.63),
	(0.68,0.55),	(0.74,0.51),	(0.75,0.54),	(0.62,0.61),
	(0.74,0.43),	(0.89,0.45),	(0.76,0.51),	(0.63,0.54),
	(0.84,0.62)]	(0.94,0.41)]	(0.82,0.47)]	(0.79,0.52)]
A ₅	[(0.89,0.51),	[(0.92,0.31),	[(0.84,0.53),	[(0.86,0.51),
	(0.93,0.42),	(0.97,0.23),	(0.87,0.49),	(0.87,0.34),
	(0.98,0.21),	(0.98,0.21),	(0.92,0.33),	(0.89,0.20),
	(0.99,0.15])	(0.99,0.11)]	(0.94,0.24)]	(0.99,0.15)]

Similarity Measure	(S _A, S _M)	(S _B, S _M)	(S _C, S _M)
L _{q ^} q - ROM* ⁴ FS	0.94	0.82	0.71
similarity measure
L _{q ^} q - ROM* ⁴ FS	0.91	0.80	0.68
weighted similarity measure

Utilizing the framework of lattice ordered q-rung orthopair multi-fuzzy soft sets (L_{q
^*} q-ROMⁿFSSs), we have introduced a novel approach to measuring similarity. This innovative method takes into consideration the degrees of multi-membership values (M-MVs) and non-membership values (M-NMVs) within the realm of L_{q
^*} q-ROMⁿFSSs. The resultant L_{q
^*} q-ROMⁿFSSs similarity model stands out as a more effective and versatile approach compared to other techniques, owing to its adept utilization of lattice order in attribute management. Further, within the context of L_{q
^*} q-ROMⁿFSSs, we have not only provided a comprehensive set of similarity measures but also introduced weighted similarity measures that account for both M-MVs and M-NMVs. By incorporating the concept of raising M-MVs and M-NMVs to the power of q, we have established a more adaptable and efficient foundation for modeling fuzzy systems and facilitating decision-making under uncertain conditions. This approach effectively bridges the gap between existing structural models and the broader domain of M-MVs and M-NMVs, enhancing its overall applicability and adaptability. It is important to highlight that an increase in the value of q results in the expansion of the decision-making space, providing greater latitude to represent a wider array of fuzzy data. We have successfully implemented the proposed Multiple Criteria Decision-Making (MCDM) methods, as demonstrated through a case study involving the selection of a healthcare waste disposal strategy. Our findings affirm the suitability and stability of the current decision-making approach, showcasing its efficacy in addressing complex multi-criteria group decision-making challenges.

In future research Research related to L_{q
^*} q-ROMⁿFSSs could explore diverse categories of decision-making complexities including product development strategy,supply chain optimization, environmental impact analysis,resource allocation in healthcare, inventory management and restocking strategy.

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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Selection of healthcare waste disposal strategies with L q * q-rung orthopair multi-fuzzy soft set similarity measures

Abstract

Keywords

1 Introduction

3 L q * q-Rung orthopair multi-fuzyy soft similarity measure

Footnotes

Acknowledgments

References

3 L _{q
^*} q-Rung orthopair multi-fuzyy soft similarity measure