Sustainable assessment of plastic and mixed waste disposal problem during COVID-19 pandemic: an integrated multi-criteria decision-making approach

Abstract

This article focuses on India’s inorganic solid waste disposal problem, with a particular emphasis on plastic and mixed waste. It aims to identify the current COVID-19 pandemic situation as well as provide a suitable disposal technique for wastes that are specifically related to municipal solid waste management. We propose an integrated approach to disposing of paper and plastic and mixed wastes in an interval-valued q-rung orthopair fuzzy (IVq-ROF) environment for this problem. In this case, we use the FUCOM method to calculate the weight values of the criteria and the MABAC method to rank the alternatives based on the chosen criteria. To confirm the effectiveness of the proposed method, a numerical illustration is provided, and validation of the suggested method is also shown.

Introduction

In both developing and developed countries, the disposal of inorganic solid waste in urban and suburban regions seems to be a crucial and intractable problem. Municipal solid waste (MSW) collection and disposal is a serious concern in most countries today. MSW management capabilities must be financially sustainable, logistically feasible, socially and legally justifiable, and ecologically beneficial (Patwa et al., 2020). Generally, microbes have a tough time decomposing such waste. As a consequence, trash such as tyres, metals, plastics, and glass waste takes a long time to decay and may take more than 100 years to degrade (Government of India Swachh Bharat Mission, 2016b). The disposable amount of inorganic waste is increasing by the day during the COVID-19 pandemic. Solid waste management is the most detrimental in the vulnerable situation. Particularly, municipal authorities confront the most challenging task when aggregation and degradation of plastics and mixed garbage in both large and small cities (Klemes et al., 2020).

Further, COVID-19 exacerbates waste management issues in developing countries. Inadequate and improper waste disposal have major effects on human health as well as a substantial influence on the environment (Mehran et al., 2021). Some of the plastic and mixed trash is generated by households, workplaces, enterprises, and hospitals (ISWA, 2020). During the pandemic, the amount of garbage grew rapidly across several nations. If corporations upgraded their waste management techniques and services, such as infrastructure needs and disposal methods, they might effectively reduce and eradicate waste. It also assists in determining the best techniques or economic organisations that can be swiftly and conveniently developed in developing nations to reduce the hazard of paper, plastic, and mixed waste (PMW) during the COVID-19 period (Yousefi et al., 2021).

In this paper, we proposed an integrated group MCDM model for this disposal problem. The MABAC method is used to rank the alternatives, and FUCOM is used to determine the criteria weights. When compared to some MCDM methods, these processes are especially robust. The distance between each alternative and the bored approximation area is computed by the MABAC model. The MABAC method has lots of advantages like other MCDM methods (less computational time, more simplistic and more stable, and minimal mathematical calculations). As a result, experts will be able to choose the best alternative in terms of computation time, consistency, and cost by using fuzzy MABAC (Wang et al., 2020). And, the FUCOM model is capable of permitting pairwise comparison of the criteria by using integers, decimals, or values from a constant present. The weight coefficients obtained with FUCOM are more reliable and help with making logical decisions. FUCOM provides more realistic weight coefficient values than other weight finding methods because comparisons are made with a greater level of consistency (Fazlollahtabar et al., 2019).

Considering the benefits of the MABAC and FUCOM methods from literature study, we apply them in an interval-valued q-rung orthopair fuzzy environment. In the context of fuzzy set theory, Yager (2017) introduced the concept of a q-rung orthopair fuzzy set (q-ROFS), which is more impactful and broad than IFS and PFS which deals with uncertain and changing data. In this paper, the q-rung orthopair fuzzy set was transformed into an interval-valued q-rung orthopair fuzzy set (IVq-ROFS) which can resolve the ambiguity better than the IFS and PFS. The sum of positive and negative values meets the condition $(0\le {[{\mathrm{\Omega }}_E^L(x),{\mathrm{\Omega }}_E^U(x)]}^q+{[{\mathrm{\Psi }}_E^L(x),{\mathrm{\Psi }}_E^U(x)]}^q\le 1)$ in IVq-ROF environment. As a result, when it comes to describing fuzziness and uncertainty, the IVq-ROFS clearly outperforms the IFS and PFS. This realisation raises awareness of the IVq-ROFS and stimulates us to develop a proper approach under the q-RPFS. The proposed method has the greatest feasible optimum solution and solves the hazards and fuzziness of this disposal problem. The nomenclature with their abbreviations is in Table 1.

Table 1.

Some important abbreviations

DM	Decision matrix	Alao et al. (2020)
MCDM	Multi-criteria decision-making	Shahnazaria et al. (2020)
MADM	Multi-attribute decision-making	Wang et al. (2020)
IFS	Intuitionistic fuzzy set	Narayanamoorthy et al. (2022b)
PFS	Pythagorean fuzzy set	Ramya et al. (2022)
q-ROFN	q-Rung orthopair fuzzy number	Li et al. (2020)
IVq-ROFN	Interval-valued q-rung orthopair fuzzy number	Wang et al. (2019)
MABAC	Multi-attribute border approximation area comparison	Wei et al. (2020a)
FUCOM	FUll consistency method	Durmic (2019)
AHP	Analytical hierarchical process	Badi and Abdulshahed (2019)
BWM	Best–worst method	Pamucar et al. (2018)
SWM	Solid waste management	Singh (2019)
ISW	Inorganic solid waste	de Paiva et al. (2021)
MSW	Municipal solid waste	Narayanamoorthy et al. (2022a)
MSWM	Municipal solid waste management	Wang et al. (2018)
PMW	Paper & plastic and mixed waste	Geetha et al. (2021) and Browning et al. (2021)

Literature review

This section contains some research on MCDM, plastic waste, and MSW as well as some pertinent studies on MCDM-compliant MSW disposal techniques. The Asir area of Saudi Arabia’s municipal solid waste disposal site selection employing fuzzy AHP and geo-information approaches is described in Mallick (2021). The benefit of anaerobic digestion was first highlighted in Li et al. (2019) by enhancing methane generation through unique mechanisms. Singh (2019) has applied the fuzzy approach to deal with the uncertainty issues with the disposal of MSW. During the COVID-19 epidemic, Kulkarni and Anantharama (2020) suggested MSW management practises and presented alternatives for processing and disposing of MSW in a few developed and developing nations. The AHP and TOPSIS models were suggested by Shahnazaria et al. (2020) for categorising thermo-chemical processes and putting together energy recovery systems from MSW. By employing an interval-valued fuzzy grey relational analysis model, the MSW treatment options were rated (Wang et al., 2018). Composting and anaerobic digestion were the main topics of Patwa et al. (2020)’s analysis of the different methods for disposing of organic solid waste. Energy recovery from MSW was recommended by Neehaul et al. (2020). in Mauritius as a means to improve energy security. The generation of energy-efficient Refuse Derived Fuel (RDF) from municipal solid waste rejects was the focus of Kimambo and Subramanian (2014) in Coimbatore City, India. The report’s findings indicate that RDF from municipal solid waste rejects paired with modifications produced products that were very energy efficient. The MABAC approach and integration of combined weights were used to address green supplier selection in probabilistic uncertain language sets (Wei et al., 2020a). An original MABAC that enables users to examine alternative MADM models using single-valued neutrosophic sets was proposed by Peng and Dai (2018). The fuzzy aggregation methods for q-ROFS in MADM are the subject of the assessment of Liu and Wang (2018). The Best-Worst and MABAC algorithms were suggested to be swapped out for interval-valued fuzzy-rough numbers by Pamucar et al. (2018). To encourage and enhance waste utilisation in cementitious materials, Guedes et al. (2021) suggested doing research on the use of inorganic solid wastes in cement. For the IVq-ROFS, Wang et al. (2019) developed fully new MSM operators to address the issue of choosing eco-friendly vendors.

Khan et al. (2019) discussed the factors that affect consumer recycling behaviour when it comes to plastic waste. Wong et al. (2015) reviewed the current position and possible applications for plastic trash as a source of energy. Klemes et al. (2020) looked at how the pandemic and epidemic affected the lifespans of numerous plastic products, particularly those used for personal safety and healthcare. Ragaert et al. (2020) analysed the case study regarding the recycling of any mixed plastic waste. Moharir and Kumar (2019) examined the challenges involved in the process of plastic degradation when disposing of plastic waste. Ragaert et al. (2017) assessed the major challenges in recycling strategies for mechanical and chemical methods of solid plastic waste disposal. Gear et al. (2018) devised a novel method for dealing with mixed plastic waste. Idumah and Nwuzor (2019) discussed the novel routes that were successfully used to convert MSW to energy and other valuable chemicals. Browning et al. (2021) addressed the challenges of plastic waste disposal in developing countries. Shahnawaz et al. (2019) proposed disposal and reuse techniques for plastic waste. Klemes et al. (2021) proposed a novel plastic waste footprint to better understand the net potential impacts of plastic and to aid in the decision-making process for plastic replacement. Verma et al. (2016) proposed the toxic pollution from plastic wastes when its burning and also which increases the risk of health issues. de Paiva et al. (2021) study encouraged the wastes are used in cementitious materials. For Lagos in Nigeria, an appropriate waste-to-energy technology based on TOPSIS method has been developed by Alao et al. (2020). Balwada et al. (2021) used AHP to investigate a better waste collection system.

The idea of q-ROFS was initially put up in Yager (2017), where ${\theta }^q+{\varphi }^q=1$, or the sum of the qth powers for satisfaction and dissatisfaction, is constrained to one. At $q=1$ and $q=2$, the q-ROFS will be reduced to IFS and PFS, respectively. q-ROFS is a superior way for communicating assessment results when compared to IFS and PFS (Kang et al., 2022). Additionally, expanding q-ROFN into IVq-ROFNs aids in handling situations in which expertise are unable to choose from a variety of plausible satisfaction and dissatisfaction values (Yanbing et al., 2019). We are unable to consistently offer reliable evaluation values of alternatives due to expert ambivalence and trouble making judgments when confronted with MADM problems.

Wei et al. (2020b) suggested employing entropy weights and MABAC in uncertain probabilistic linguistic sets to evaluate the green supplier selection problem. Moreover, in 2015 the MABAC (Multi-Attributive Border Approximation Area Comparison) method is described by Pamucar and Cirovic (2015), calculates the distance measurement between each alternative and the bored approximation area (BAA) and has a large number of useful features. Hence, the MABAC approach is also the best tool to obtain realistic decisions from experts. Therefore, the MADM is proposed for disposing of ISW. And, also FUCOM has been used in many decision-making topics, such as forklift selection (Fazlollahtabar et al., 2019), and supplier selection (Durmic, 2019). For this, we consider four alternatives, namely sanitary landfilling (SL), incineration, refuse-derived fuel (RDF), and recycling and recovery (RR). To evaluate this MADM problem, we use an IVq-ROF approach with linguistic scale.

As a result, we found the benefits of the q-ROF set, which is an extended version of the intuitionistic and Pythagorean fuzzy sets, in providing a sophisticated means to describe uncertain information with a flexible large range of intervals (Yager, 2017; Yanbing et al., 2019), FUCOM—a better and simpler replacement for AHP and BWM with simple calculation, less pairwise comparison combinations, more consistency, reliability, and contribution to reasonable judgements (Durmic, 2019), and MABAC—can cope with more intricate and fuzzy situations, indicating that this approach is more suited for use in management activities, because it may enhance the growth of management field by expanding theory study (Wei et al., 2020a, b). But, those methods are very young and needs more applications and integration.

Motivation of the research

• Based on the detailed literature study, we identified that numerous authors have focused on the application of MSWM in MCDM, and several authors have worked on plastic waste disposal problems in the MCDM approach using different fuzzy sets.

• Lack of specified treatment methods for mixed paper and plastic waste made curious to work and the advantages of IVq-ROF set tend to address waste disposal systems during pandemic.

• Integration of the benefits of weight finding technique and outranking method allows expert viewpoints to be freely expressed in IVq-ROFS when providing values for selecting the suitable disposal techniques for these wastes.

• This type of MCDM research drives us to develop appropriate waste disposal methods for reducing waste and enhancing the environment in terms of hygiene, which will assist in waste management during pandemics.

Contribution of the research

• A new approach to MCGDM methods, namely MABAC and the weight detection model FUCOM, has been proposed in the environment of IVq-ROFS. The IVq-ROFS-MABAC procedure is used to rank the alternatives based on the selected criteria.

• IVq-ROF linguistic scale and aggregation operators as interval-valued q-rung orthopair fuzzy weighted averaging (IVq-ROFWA) operator and interval-valued q-rung orthopair fuzzy weighted geometric (IVq-ROFWG) operator were developed.

• The uniqueness of the result from our proposed method is studied and proved by comparing with few currently available aggregation operators for ranking.

• To further comprehend the applicability of the established framework, we altered the expert weight vectors and parameter q values in our suggested technique.

The following is the flow of the paper: Preliminaries are discussed in Sect. 3. The MABAC with IVq-ROFNs and the FUCOM techniques are presented in Sect. 4. In Sect. 5, a numerical application to understand the effectiveness of the established framework is provided. Section 6 shows the validity of the suggested technique. Finally, in Sect. 7, a discussion and conclusion are provided.

Preliminaries

A fundamental definition of the q-ROF set (Definition 3.1), which is then expanded with a finite interval value to form the IVq-ROF set (Definition 3.2). To simplify this IVq-ROFS, Definition 3.3 provides a scoring function, S(h) as well as an accuracy function, A(h) to evaluate the proposed set. A weighted IVq-averaging ROF’s and geometric operators are specified in Definitions 3.4 and 3.5. Theorems 3.1 and 3.2 use mathematical induction to show the theoretical aggregation findings. On Definition 3.6, the normalised distance metric between IVq-ROF sets is as follows:

Definition 3.1

(Yager, 2017) Let us assume that X be a universal set. A q-rung orthopair fuzzy set (q-ROFS) E in X is described as follows:

$$\begin{array}{c}\hfill E=\{<x,({\mathrm{\Omega }}_E(x),{\mathrm{\Psi }}_E(x))>|x\in X\}\end{array}$$ 1

where ${\mathrm{\Omega }}_E(x):X\to [0,1]$ and ${\mathrm{\Psi }}_E(x):X\to [0,1]$ represent the grade of satisfaction and dissatisfaction values of $x\in X$ to the set E, respectively, with the condition $0\le {({\mathrm{\Omega }}_E(x))}^q+{({\mathrm{\Psi }}_E(x))}^q\le 1,(q\ge 1)$. The grade of indeterminacy ${\beta }_E(x)={({({\mathrm{\Omega }}_E(x))}^q+{({\mathrm{\Psi }}_E(x))}^q-({({\mathrm{\Omega }}_E(x))}^q)({({\mathrm{\Psi }}_E(x))}^q))}^{1/q}$. Here, $<{\mathrm{\Omega }}_E(x),{\mathrm{\Psi }}_E(x)>$ is called a q-rung orthopair fuzzy number (q-ROFN).

Definition 3.2

(Yanbing et al., 2019) Given X be a non-empty fixed set, an interval-valued q-rung orthopair fuzzy set (IVq-ROFS) E on X can be described as follows:

$$\begin{array}{c}\hfill E=\{<x,([{\mathrm{\Omega }}_E^L(x),{\mathrm{\Omega }}_E^U(x)],[{\mathrm{\Psi }}_E^L(x),{\mathrm{\Psi }}_E^U(x)])>|x\in X\}\end{array}$$ 2

where $[{\mathrm{\Omega }}_E^L(x),{\mathrm{\Omega }}_E^U(x)]$ and $[{\mathrm{\Psi }}_E^L(x),{\mathrm{\Psi }}_E^U(x)]$ are denote the satisfaction and dissatisfaction values of x to E, respectively, which holds the condition that $[{\mathrm{\Omega }}_E^L(x),{\mathrm{\Omega }}_E^U(x)]\subseteq [0,1]$, $[{\mathrm{\Psi }}_E^L(x),{\mathrm{\Psi }}_E^U(x)]\subseteq [0,1]$, and $0\le {({\mathrm{\Omega }}_E^U(x))}^q+{({\mathrm{\Psi }}_E^U(x))}^q\le 1,(q\ge 1)$. The indeterminacy grade is drawn as

$$\begin{array}{c}\hfill [{\eta }_E^L(x),{\eta }_E^U(x)]=[\sqrt[q]{1-{({\mathrm{\Omega }}_E^U(x))}^q-{({\mathrm{\Omega }}_E^U(x))}^q},\sqrt[q]{1-{({\mathrm{\Psi }}_E^L(x))}^q-{({\mathrm{\Psi }}_E^L(x))}^q}]\end{array}$$

$<[{\mathrm{\Omega }}_E^L(x),{\mathrm{\Omega }}_E^U(x)],[{\mathrm{\Psi }}_E^L(x),{\mathrm{\Psi }}_E^U(x)]>$ is denoted as a IVq-ROFN, which is defined by $\theta =([{\mathrm{\Omega }}_E^L,{\mathrm{\Omega }}_E^U],[{\mathrm{\Psi }}_E^L,{\mathrm{\Psi }}_E^U])$.

Definition 3.3

(Yanbing et al., 2019) Let $h=([{\mathrm{\Omega }}_h^L,{\mathrm{\Omega }}_h^U],[{\mathrm{\Psi }}_h^U,{\mathrm{\Omega }}_h^U])$ be an IVq-ROFN, then the score function S(h) and the accuracy function A(h) are defined as below:

$$\begin{array}{cc}& S(h)=\frac{1+{({\mathrm{\Omega }}_h^U)}^q-{({\mathrm{\Psi }}_h^U)}^q+1+{({\mathrm{\Omega }}_h^L)}^q-{({\mathrm{\Psi }}_h^L)}^q}{4}\hfill \end{array}$$ 3

$$\begin{array}{cc}& A(h)=\frac{{({\mathrm{\Omega }}_h^U)}^q+{({\mathrm{\Omega }}_h^L)}^q+{({\mathrm{\Psi }}_h^U)}^q+{({\mathrm{\Psi }}_h^L)}^q}{4}\hfill \end{array}$$ 4

Definition 3.4

(Yanbing et al., 2019) Assume $h_t=([{\mathrm{\Omega }}_t^L,{\mathrm{\Omega }}_t^U],[{\mathrm{\Psi }}_t^L,{\mathrm{\Psi }}_t^U])$ be a collection of IVq-ROFNs, then the interval-valued q-rung orthopair fuzzy weighted averaging (IVq-ROFWA) operator, that is IVq-ROFWA: ${\theta }^n\to \theta$ can be expressed as

$$\begin{array}{c}\hfill IVq-ROFWA(h_1,h_2,...,h_n)={\oplus }_{t=1}^n{\lambda }_t{a}_t\end{array}$$ 5

in which $\theta$ is the set of all IVq-ROFNs and the weight vector $\lambda ={({\lambda }_1,{\lambda }_2,...,{\lambda }_n)}^T$ of $h_t(t=1,2,...,n)$, such that ${\lambda }_t\in [0,1]$ and ${\sum }_{t=1}^n{\lambda }_t=1$.

We can find the specific form of the aggregation result, as seen in the theorem below.

Theorem 3.1

(Yanbing et al., 2019) Let $h_t=([{\mathrm{\Omega }}_t^L,{\mathrm{\Omega }}_t^U],[{\mathrm{\Psi }}_t^L,{\mathrm{\Psi }}_t^U])(t=1,2,...,n)$ be a list of IV-ROFNs, $\lambda ={({\lambda }_1,{\lambda }_2,...,{\lambda }_n)}^T$ is the weight vector of $h_t$, such that ${\lambda }_t\in [0,1]$ and ${\sum }_{t=1}^n{\lambda }_t=1$. Then, their aggregated value by Definition 3.4 is still a IVq-ROFN, and has

$$\begin{array}{cc}\hfill IVq-ROFWA(h_1,h_2,...,h_n)& =\left.〈\left[{\left(1,-,\prod _{t=1}^n,{(1-{({\mathrm{\Omega }}_t^L)}^q)}^{{\lambda }_t}\right)}^{\frac{1}{q}},,,{\left(1,-,\prod _{t=1}^n,{(1-{({\mathrm{\Omega }}_t^U)}^q)}^{{\lambda }_t}\right)}^{\frac{1}{q}}\right],,\right)\hfill \\ \hfill & \left(\left[\prod _{t=1}^n,{({\mathrm{\Psi }}_t^L)}^{{\lambda }_t},,,\prod _{t=1}^n,{({\mathrm{\Psi }}_t^U)}^{{\lambda }_t}\right]\right.〉\hfill \end{array}$$ 6

Definition 3.5

(Li et al., 2020) Let $h_t=([{\mathrm{\Omega }}_t^L,{\mathrm{\Omega }}_t^U],[{\mathrm{\Psi }}_t^L,{\mathrm{\Psi }}_t^U])$ be a collection of IVq-ROFNs, then the interval-valued q-rung orthopair fuzzy weighted geometric (IVq-ROFWG) operator, that is IVq-ROFWG: ${\theta }^n\to \theta$ can be expressed as

$$\begin{array}{c}\hfill IVq-ROFWG(h_1,h_2,...,h_n)={\otimes }_{t=1}^n{\lambda }_t{a}_t\end{array}$$ 7

in which $\theta$ is the set of all IVq-ROFNs and the weight vector is $\lambda ={({\lambda }_1,{\lambda }_2,...,{\lambda }_n)}^T$ of $h_t(t=1,2,...,n)$, such that ${\lambda }_t\in [0,1]$ and ${\sum }_{t=1}^n{\lambda }_t=1$.

We should got the certain form of the aggregation result, as shown in the theorem below.

Theorem 3.2

(Yanbing et al., 2019; Li et al., 2020) Let $h_t=([{\mathrm{\Omega }}_t^L,{\mathrm{\Omega }}_t^U],[{\mathrm{\Psi }}_t^L,{\mathrm{\Psi }}_t^U])(t=1,2,...,n)$ be a list of IV-ROFNs, $\lambda ={({\lambda }_1,{\lambda }_2,...,{\lambda }_n)}^T$ is the weight vector of $h_t$, such that $h_t\in [0,1]$ and ${\sum }_{t=1}^n{h}_t=1$. Then, their aggregated value by Definition 3.5 is still a IVq-ROFN, and has

$$\begin{array}{cc}\hfill IVq-ROFWG(h_1,h_2,...,h_n)& =\left.〈\left[\prod _{t=1}^n,{({\mathrm{\Omega }}_t^L)}^{{\lambda }_t},,,\prod _{t=1}^n,{({\mathrm{\Omega }}_t^U)}^{{\lambda }_t}\right],,\right)\hfill \\ \hfill & \left(\left[{\left(1,-,\prod _{t=1}^n,{(1-{({\mathrm{\Psi }}_t^L)}^q)}^{{\lambda }_t}\right)}^{\frac{1}{q}},,,{\left(1,-,\prod _{t=1}^n,{(1-{({\mathrm{\Psi }}_t^U)}^q)}^{{\lambda }_t}\right)}^{\frac{1}{q}}\right]\right.〉\hfill \end{array}$$ 8

Theorems 3.1 and 3.2 can be proved by the mathematical induction, which is not repeated here.

Definition 3.6

(Wang et al., 2020) Let $h_1=([{\mathrm{\Omega }}_1,{\mathrm{\Omega }}_1],[{\mathrm{\Psi }}_1,{\mathrm{\Psi }}_1])$ and $h_2=([{\mathrm{\Omega }}_2,{\mathrm{\Omega }}_2],[{\varphi }_2,{\varphi }_2])$ be two IVq-ROFNs, then we can obtain the interval-valued q-rung orthopair fuzzy normalised hamming distance (IVq-ROFNHD)is given as follows:

$$\begin{array}{cc}\hfill d_q-IVq-ROFNHD(h_1,h_2)& =\frac{1}{4}\left(|,{({\mathrm{\Omega }}_1^L)}^q,-,{({\mathrm{\Omega }}_2^L)}^q,|+|,{({\mathrm{\Omega }}_1^U)}^q,-,{({\mathrm{\Omega }}_2^U)}^q,|+\right)\hfill \\ \hfill & \left(|,{({\mathrm{\Psi }}_1^L)}^q,-,{({\mathrm{\Psi }}_2^L)}^q,|+|,{({\mathrm{\Psi }}_1^U)}^q,-,{({\mathrm{\Psi }}_2^U)}^q,|+\right)\hfill \\ \hfill & \left(|,{(1-{({\mathrm{\Omega }}_1^L)}^q-{({\mathrm{\Psi }}_1^L)}^q)}^{\frac{1}{q}},-,{(1-{({\mathrm{\Omega }}_2^L)}^q-{({\mathrm{\Psi }}_2^L)}^q)}^{\frac{1}{q}},|+\right)\hfill \\ \hfill & \left(|,{(1-{({\mathrm{\Omega }}_1^U)}^q-{({\mathrm{\Psi }}_1^U)}^q)}^{\frac{1}{q}},-,{(1-{({\mathrm{\Omega }}_2^U)}^q-{({\mathrm{\Psi }}_2^U)}^q)}^{\frac{1}{q}},|\right)\hfill \end{array}$$ 9

Here, the linguistic scale with IV-ROFNs to evaluate the alternatives is presented in Table 2 (Narayanamoorthy et al., 2022a).

Table 2.

Fuzzy scale

Linguistic words	Fuzzy values
Terrible (T)	([0.14, 0.15], [0.94, 0.95])
More penurious (MP)	([0.34, 0.35], [0.74, 0.75])
Penurious (P)	([0.44, 0.45], [0.64, 0.65])
Moderate (M)	([0.54, 0.55], [0.55, 0.56])
Worth (W)	([0.64, 0.65], [0.44, 0.45])
More worth (MW)	([0.74, 0.75], [0.34, 0.35])
Best worth (BW)	([0.94, 0.95], [0.14, 0.15])

Proposed method

The MABAC method with IVq-ROFNs

Consider the s alternatives $\{D_1,D_2,...,D_s\}$, t attributes $\{Y_1,Y_2,...,Y_t\}$ with weight vector be ${\lambda }_t$ and $\alpha$ experts $\{e_1,e_2,...,e_{\alpha }\}$ with weighting vector be $\{{\lambda }_1,{\lambda }_2,...,{\lambda }_{\alpha }\}$, given the interval-valued q-rung orthopair fuzzy evaluation matrix (D),

$$\begin{array}{c}\hfill D=[d_{\mathit{st}}]=([{({\mathrm{\Omega }}_{\mathit{st}}^{\alpha })}^L,{({\mathrm{\Omega }}_{\mathit{st}}^{\alpha })}^U][{({\mathrm{\Psi }}_{\mathit{st}}^{\alpha })}^L,{({\mathrm{\Psi }}_{\mathit{st}}^{\alpha })}^U]),s=1,2,...,m,t=1,2,...,n,\end{array}$$

$[{({\mathrm{\Omega }}_{\mathit{st}}^{\alpha })}^L,{({\mathrm{\Omega }}_{\mathit{st}}^{\alpha })}^U]\in [0,1]$denotes the lower and upper satisfaction degree, $[{({\mathrm{\Psi }}_{\mathit{st}}^{\alpha })}^L,{({\mathrm{\Psi }}_{\mathit{st}}^{\alpha })}^U]\in [0,1]$ denotes the lower and upper dissatisfaction degree, then the steps of IV-ROFS with fuzzy MABAC method can be expressed as follows and the pictorial representation is shown in Fig. 1.

Fig. 1.

Pictorial representation for Interval-valued q-rung orthopair fuzzy MABAC method

Step 1: Create the interval-valued q-rung orthopair fuzzy matrix is

10

where $([{({\mathrm{\Omega }}_{\mathit{st}}^{\alpha })}^L,{({\mathrm{\Omega }}_{\mathit{st}}^{\alpha })}^U][{({\mathrm{\Psi }}_{\mathit{st}}^{\alpha })}^L,{({\mathrm{\Psi }}_{\mathit{st}}^{\alpha })}^U])(s=1,2,...m,t=1,2,...,n)$ denotes the evaluate information of alternative $D_s(s=1,2,...,m)$ based on attribute $M_t(t=1,2,...,n)$ by experts $e_{\alpha }$.

Step 2: According to the IVq-ROFWA aggregation operator (Yanbing et al., 2019), we can get overall $D_{\mathit{st}}^{\alpha }$ to $D_{\mathit{st}}$; the fused IVq-ROFNs matrix D is given as follows:

11

where $({\mathrm{\Omega }}_{\mathit{st}},{\mathrm{\Psi }}_{\mathit{st}})(s=1,2,...,m,t=1,2,...,n)$ denotes the fused IVq-ROP fuzzy information of alternative $D_s(s=1,2,...,m)$ on attribute $M_t(t=1,2,...,n)$.

Step 3: Normalised decision matrix (NDM) is based on the each attribute by following formula:

For positive attributes,

$$\begin{array}{c}\hfill N_{\mathit{st}}=([{({\mathrm{\Omega }}_{\mathit{st}})}^L,{({\mathrm{\Omega }}_{\mathit{st}})}^U][{({\mathrm{\Psi }}_{\mathit{st}})}^L,{({\mathrm{\Psi }}_{\mathit{st}})}^U]),s=1,2,...,m,t=1,2,...,n\end{array}$$ 12

For negative attributes,

$$\begin{array}{c}\hfill N_{\mathit{st}}=([{({\mathrm{\Psi }}_{\mathit{st}})}^L,{({\mathrm{\Psi }}_{\mathit{st}})}^U],[{({\mathrm{\Omega }}_{\mathit{st}})}^L,{({\mathrm{\Omega }}_{\mathit{st}})}^U]),s=1,2,...,m,t=1,2,...,n\end{array}$$ 13

Step 4: According to the NDM $N_{\mathit{st}}=([{({\mathrm{\Omega }}_{\mathit{st}})}^L,{({\mathrm{\Omega }}_{\mathit{st}})}^U][{({\mathrm{\Psi }}_{\mathit{st}})}^L,{({\mathrm{\Psi }}_{\mathit{st}})}^U])(s=1,2,...,m,t=1,2,...,n)$ and attribute’s weight vector ${\lambda }_t(t=1,2,...,m)$, the fuzzy WNDM $W{N}_{\mathit{st}}=([{({\mathrm{\Omega }}_{\mathit{st}}^{\prime })}^L,{({\mathrm{\Omega }}_{\mathit{st}}^{\prime })}^U][{({\mathrm{\Psi }}_{\mathit{st}}^{\prime })}^L,{({\mathrm{\Psi }}_{\mathit{st}}^{\prime })}^U])(s=1,2,...,m,t=1,2,...,n)$ can be calculated as:

$$\begin{array}{c}\hfill W{N}_{\mathit{st}}={\lambda }_t\otimes N_{\mathit{st}}=(\sqrt[q]{1-{(1-[{({\mathrm{\Omega }}_{\mathit{st}}^L)}^q,{({\mathrm{\Omega }}_{\mathit{st}}^U)}^q])}^{{\lambda }_t}},{([({\mathrm{\Psi }}_{\mathit{st}}^L),({\mathrm{\Psi }}_{\mathit{st}}^U)])}^{{\lambda }_t})(s=1,2,...,m,t=1,2,...,n)\end{array}$$ 14

Step 5: Calculate the values of BAA and the BAA matrix $B=[b_t]$ can be defined as follows:

$$\begin{array}{c}\hfill b_t={\left(\prod _{s=1}^m,W,N_{\mathit{st}}\right)}^{\frac{1}{m}}=\left\{{\left(\prod _{s=1}^m,[({\mathrm{\Omega }}_{\mathit{st}}^{\prime L}),({\mathrm{\Omega }}_{\mathit{st}}^{\prime U})]\right)}^{\frac{1}{m}},,,\sqrt[q]{1-{\left({\prod }_{s=1}^m,(1-[({\mathrm{\Psi }}_{\mathit{st}}^{\prime L}),({\mathrm{\Psi }}_{\mathit{st}}^{\prime U})])\right)}^{\frac{1}{m}}}\right\}\end{array}$$ 15

Step 6: Obtain the distance $V=[v_{\mathit{st}}]$ between each alternative and the BAA is using the following expression:

$$\begin{array}{c}\hfill v_{\mathit{st}}=\left\{\begin{array}{cc}v(W{N}_{\mathit{st}},b_t)\hfill & \text{if}W{N}_{\mathit{st}}>b_t\hfill \\ 0\hfill & \text{if}W{N}_{\mathit{st}}=b_t\hfill \\ -v(W{N}_{\mathit{st}},b_t)\hfill & \text{if}W{N}_{\mathit{st}}<b_t\hfill \end{array}\right)\end{array}$$ 16

where $v(W{N}_{\mathit{st}},b_t)$ is the distance from $W{N}_{\mathit{st}}$ to $b_t$ which can be calculated by using Eq. (9).

Step 7: Sum the values of each alternative’s $v_{\mathit{st}}$ as:

$$\begin{array}{c}\hfill Z_s=\sum _{t=1}^n{v}_{\mathit{st}}\end{array}$$ 17

The order of all alternative can be derived, simply, regardless of the comprehensive evaluation outcome $Z_s$, the greater the comprehensive evaluation result is the better choice.

We can obtain from above the procedure that the IVq-ROF-MABAC model can handle more dynamic and unclear issues, suggesting that this methodology is more suited for use in waste management operations and can facilitate the growth of the MSW sector and also enrich management science.

The FUCOM method

FUCOM is a new age of weight finding method for solving MADM. The principles are based on pairwise comparison and results authentication through deviation from maximum consistency (DMC). It decreases the number of pairwise criteria comparisons relative to AHP (Badi & Abdulshahed, 2019) and also has the potential to verify the outcomes with DMC.

Consider “s” alternatives $(D_1,D_2,...,D_s)$ and “t” decision attribute $(Y_1,Y_2,...,Y_t)$. The decision matrix $D=[d_{\mathit{st}}]$ shows the alternatives for each criterion. $\lambda =({\lambda }_1,{\lambda }_2,...,{\lambda }_t)$ denoted as weight vector for all criteria. The diagrammatic representation of FUCOM methods is shown in Fig. 2.

Fig. 2.

The pictorial representation of FUCOM

Step 1: Create the ranking set from the $(Y_1,Y_2,...,Y_t)$ evaluation criteria provided. As follows, the parameters ordered according to their intended importance are obtained.

$$\begin{array}{c}\hfill Y_{t(1)}>Y_{t(2)}>...>Y_{t(z)},\end{array}$$ 18

where n denotes the order of criterion.

Step 2: Each pair of adjacent attribute is compared, and the comparative priorities ${\chi }_{\frac{(z-1)}{z}},z=1,2,...,n$ are calculated. Here, ${\chi }_{\frac{(z-1)}{z}}$ denotes the value of criterion $Y_{t(z-1)}$ relative to criterion $Y_{t(z-1)}$, where the order of parameters is expressed by $(z-1),z$. Then, extract the assessment criterion vector of comparative preferences as follows:

$$\begin{array}{c}\hfill \chi =\{{\chi }_{\frac{1}{2}},{\chi }_{\frac{2}{3}},...,{\chi }_{\frac{(z-1)}{z}}\}\end{array}$$ 19

Step 3: Obtain the final weight coefficient values $({\lambda }_1,{\lambda }_2,...,{\lambda }_t)$. They should satisfy the two conditions as follows:

• The ratio of weight coefficient is equal to the comparative priority ${\chi }_{\frac{(z-1)}{z}}$ (from step 2). That is,

  $$\begin{array}{c}\hfill \frac{{\lambda }_{z-1}}{{\lambda }_z}={\chi }_{\frac{(z-1)}{z}}\end{array}$$ 20
• The weight coefficients should satisfy the transitivity condition, i.e. ${\chi }_{\frac{(z-2)}{z-1}}\otimes {\chi }_{\frac{(z-1)}{z}}={\chi }_{\frac{(z-2)}{z}}$. Therefore, the second condition is that the coefficient is

  $$\begin{array}{c}\hfill \frac{{\lambda }_{(z-2)}}{{\lambda }_z}={\chi }_{\frac{(z-2)}{(z-1)}}\otimes {\chi }_{\frac{(z-1)}{z}}\end{array}$$ 21

  According to the explanations outlined, a nonlinear constrained programming model is built as follows:

  $$\begin{array}{cc}\hfill \text{min}& \eta ,\hfill \\ \hfill \text{s.t.}& \left|\frac{{\lambda }_{(z-1)}}{{\lambda }_z},-,{\chi }_{\frac{(z-1)}{z}}\right|\le \eta ,\left|\frac{{\lambda }_{(z-2)}}{{\lambda }_z},-,{\chi }_{\frac{(z-2)}{z-1}},\otimes ,{\chi }_{\frac{(z-1)}{z}}\right|\le \eta ,\hfill \\ \hfill & \sum _{t=1}^n{\lambda }_t=1,\forall t\hfill \\ \hfill & {\lambda }_t\ge 0,\forall t\hfill \end{array}$$ 22

  Solving the model provides the optimal values of the weights of the assessment is $({\lambda }_1,{\lambda }_2,...,{\lambda }_t)$.

Numerical example

Inorganic solid waste disposal is a big concern all over the world. Population expansion, industrialisation, and hospitalisation are all contributing to an increase in plastic and mixed trash. As a result, facilities for disposing of the increasing volumes of inorganic waste must be enhanced (Kimambo & Subramanian, 2014). Since then, numerous nations have used traditional garbage disposal methods such as landfills and incineration. These procedures are unsanitary and endanger both human and animal health. MSW collection, storage, and removal should be rigorously adhered to for safe and strong waste management (Government of India Swachh Bharat Mission, 2016a). Appropriate waste management enhances ecological integrity and safeguards critical natural resources such as groundwater, soil quality, and air quality. Because the great majority of wastes are recyclable and reused in some way, ISW categories include paper and plastic trash, mixed rubbish, hazardous waste, and e-waste, for example. As a result, we examine four alternative options for disposing of paper and plastic rubbish, as well as mixed debris. Moreover, this research focuses on and explores four alternative waste disposal treatment methods (Government of India Swachh Bharat Mission, 2016b). As a result, the issue is created as shown in Fig. 3 which are explored more below.

Fig. 3.

PMW disposal method

Sanitary landfills Humans and machinery are needed to dispose solid waste from inception to its final disposal. A technique of discarding solid waste on land without causing inconveniences or threats to public health or safety by using complex ideas to restrict the trash to the minimum possible area, compress it to the lowest practicable density, and cover it with a layer of earth at the completion of every day’s operations or at more periodic times as would be required (Mallick, 2021).

Incineration A moving blade incinerator, also known as a municipal solid waste incinerator, is the most prevalent form of trash incinerator. Rubbish is dumped onto a movable blade that traverses through the various chambers of an incinerator. The continually rotating grate allows for instant, efficient, and comprehensive trash transfer and processing (Narayanamoorthy et al., 2022a).

Refused-derived fuel Refuse-Derive Fuel (RDF) is derived from municipal garbage and is one of the alternative fuels. It is the process of transforming trash into usable energy. As a result, the problem of solid waste management is alleviated, and RDF may be used instead of coal in boilers (Kimambo & Subramanian, 2014).

Recycling and recovery Recycling is the process of converting garbage into a new substance or product, thereby reducing the need for new resources. Anaerobic digestion, incineration with energy recovery, gasification, and pyrolysis are all methods of recovering energy (fuels, heat, and electricity) and materials from garbage (Neehaul et al., 2020).

The proposed MADM technique is utilised in this part to evaluate the method of mixed and paper and plastic trash disposal. For that reason, we chose eight criteria to examine the four alternatives in order to determine the best disposal approach that has the least environmental effect while meeting all social and economic needs. Those attributes are cost $(Y_1)$, health and safety $(Y_2)$, air pollution $(Y_3)$, noise pollution $(Y_4)$, soil and water pollution $(Y_5)$, workers $(Y_6)$, technical efficiency and feasibility $(Y_7)$, and land requirement and equipment facilities $(Y_8)$. We assume three deciders in this case to assess the options for each attribute. Then, the alternatives are sanitary landfilling $(D_1)$, incineration $(D_2)$, refused-derived fuel $(D_3)$, and recycling and recovery $(D_4)$. The selected alternatives and attributes are shown in Fig. 3.

FUCOM weight finding method

Step 1: For $D{M}_1$, experts rank the attribute in descending order of importance:

$$\begin{array}{c}\hfill Y_2>Y_3>Y_5>Y_4>Y_1>Y_7>Y_6>Y_8\end{array}$$

; similarly, for $D{M}_2$ and $D{M}_3$, experts rank are arranged as $Y_2>Y_5>Y_3>Y_1>Y_4>Y_7>Y_6>Y_8$ and $Y_2>Y_5>Y_3>Y_1>Y_4>Y_7>Y_6>Y_8$.

Step 2: From step 1, for $D{M}_1$, $D{M}_2$ and $D{M}_3$, experts perform a pairwise comparison of rating criterion. It is based on the scale [1, 9] (Table 3). The priorities of each attribute are given below.

Table 3.

Pairwise comparison ratings based on preference scale

$D{M}_1$	Attribute	$Y_2$	$Y_3$	$Y_5$	$Y_4$	$Y_1$	$Y_7$	$Y_6$	$Y_8$
	${\omega }_{Y_t}$	1	2.5	3.9	4.7	5.6	6.5	7.1	8
$D{M}_2$	Attribute	$Y_2$	$Y_5$	$Y_3$	$Y_1$	$Y_4$	$Y_7$	$Y_6$	$Y_8$
	${\omega }_{Y_t}$	1	2.9	3.4	4.2	4.5	6.8	7.5	7.9
$D{M}_3$	Attribute	$Y_2$	$Y_5$	$Y_3$	$Y_1$	$Y_4$	$Y_7$	$Y_6$	$Y_8$
	${\omega }_{Y_t}$	1	2.9	3.4	4.2	4.5	6.3	7.7	8

Next, we calculate the comparative priorities based on the priorities of attribute as follows:

$$\begin{array}{c}\hfill \begin{array}{ccc}D{M}_1& D{M}_2& D{M}_3\\ {\chi }_{\frac{Y_2}{Y_3}}=\frac{2.5}{1}=2.50;& {\chi }_{\frac{Y_2}{Y_5}}=\frac{2.9}{1}=2.90;& {\chi }_{\frac{Y_2}{Y_5}}=\frac{2.9}{1}=2.90;\\ {\chi }_{\frac{Y_3}{Y_5}}=\frac{3.9}{2.5}=1.56;& {\chi }_{\frac{Y_5}{Y_3}}=\frac{3.4}{2.9}=1.17;& {\chi }_{\frac{Y_5}{Y_3}}=\frac{3.4}{2.9}=1.17;\\ {\chi }_{\frac{Y_5}{Y_4}}=\frac{4.7}{3.9}=1.20;& {\chi }_{\frac{Y_3}{Y_1}}=\frac{4.2}{3.4}=1.23;& {\chi }_{\frac{Y_3}{Y_1}}=\frac{4.2}{3.4}=1.23;\\ {\chi }_{\frac{Y_4}{Y_1}}=\frac{5.6}{4.7}=1.19;& {\chi }_{\frac{Y_1}{Y_4}}=\frac{4.5}{4.2}=1.07;& {\chi }_{\frac{Y_1}{Y_4}}=\frac{4.5}{4.2}=1.07;\\ {\chi }_{\frac{Y_1}{Y_7}}=\frac{6.5}{5.6}=1.16;& {\chi }_{\frac{Y_4}{Y_7}}=\frac{6.8}{4.5}=1.51;& {\chi }_{\frac{Y_4}{Y_7}}=\frac{6.3}{4.5}=1.40;\\ {\chi }_{\frac{Y_7}{Y_6}}=\frac{7.1}{6.5}=1.09;& {\chi }_{\frac{Y_7}{Y_6}}=\frac{7.5}{6.8}=1.10;& {\chi }_{\frac{Y_7}{Y_6}}=\frac{7.7}{6.3}=1.22;\\ {\chi }_{\frac{Y_6}{Y_8}}=\frac{8}{7.1}=1.12;& {\chi }_{\frac{Y_6}{Y_8}}=\frac{7.9}{7.5}=1.05;& {\chi }_{\frac{Y_6}{Y_8}}=\frac{8}{7.7}=1.03;\end{array}\end{array}$$

Step 3: Obtain the final weights for $D{M}_1$, which should satisfy the two conditions, that is

$$\begin{array}{cc}& \frac{{\lambda }_2}{{\lambda }_3}=2.50;\frac{{\lambda }_3}{{\lambda }_5}=1.56;\frac{{\lambda }_5}{{\lambda }_4}=1.20;\frac{{\lambda }_4}{{\lambda }_1}=1.19;\hfill \\ \hfill & \frac{{\lambda }_1}{{\lambda }_7}=1.16;\frac{{\lambda }_7}{{\lambda }_6}=1.09;\frac{{\lambda }_6}{{\lambda }_8}=1.12\hfill \\ \hfill & \frac{{\lambda }_2}{{\lambda }_5}=2.50\times 1.56=3.90;\frac{{\lambda }_3}{{\lambda }_4}=1.56\times 1.20=1.87;\frac{{\lambda }_5}{{\lambda }_1}=1.20\times 1.19=1.42;\hfill \\ \hfill & \frac{{\lambda }_4}{{\lambda }_7}=1.19\times 1.16=1.38;\frac{{\lambda }_1}{{\lambda }_6}=1.16\times 1.09=1.26;\frac{{\lambda }_7}{{\lambda }_8}=1.09\times 1.12=1.22;\hfill \end{array}$$

Using Eq. (22), we can obtain the final weight for coefficients of $D{M}_1$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \text{Min}& \eta ;\hfill \\ \hfill \mathrm{subject}\mathrm{to}& \\ \hfill & \left|\frac{{\lambda }_2}{{\lambda }_3},-,2.50\right|\le \eta ,\left|\frac{{\lambda }_3}{{\lambda }_5},-,1.56\right|\le \eta ,\left|\frac{{\lambda }_5}{{\lambda }_4},-,1.20\right|\le \eta ,\left|\frac{{\lambda }_4}{{\lambda }_1},-,1.19\right|\le \eta ,\left|\frac{{\lambda }_1}{{\lambda }_7},-,1.16\right|\le \eta ,\hfill \\ \hfill & \left|\frac{{\lambda }_7}{{\lambda }_6},-,1.09\right|\le \eta ,\left|\frac{{\lambda }_6}{{\lambda }_8},-,1.12\right|\le \eta ,\hfill \\ \hfill & \left|\frac{{\lambda }_2}{{\lambda }_5},-,3.90\right|\le \eta ,\left|\frac{{\lambda }_3}{{\lambda }_4},-,1.87\right|\le \eta ,\left|\frac{{\lambda }_5}{{\lambda }_1},-,1.42\right|\le \eta ,\left|\frac{{\lambda }_4}{{\lambda }_7},-,1.38\right|\le \eta ,\left|\frac{{\lambda }_1}{{\lambda }_6},-,1.26\right|\le \eta ,\hfill \\ \hfill & \left|\frac{{\lambda }_7}{{\lambda }_8},-,1.22\right|\le \eta ,\hfill \\ \hfill & \sum _{t=1}^n{w}_t=1,{\lambda }_t\ge 1.\hfill \end{array}\end{array}$$

By solving this model, we can get the optimal values of the weight coefficients as (0.072, 0.403, 0.161, 0.086, 0.103, 0.057, 0.062, 0.051) and DFC of the result is $\eta =0.00$. In a similar way, $D{M}_2$ and $D{M}_3$ can be solved and the obtained optimal weight coefficients are (0.095, 0.396, 0.116, 0.089, 0.136, 0.054, 0.059, 0.051) and (0.095, 0.395, 0.116, 0.088, 0.136, 0.052, 0.063, 0.051) with DFC of the results, $\eta =0.00$ and $\eta =0.00$. Finally, the optimal weights are obtained by applying a geometric mean; we can get (0.086, 0.397, 0.129, 0.087, 0.123, 0.054, 0.061, 0.051). These models are solved using the LINGO software.

Ranking of paper and plastic waste disposal method

When plastic is recycled af,ter its value is over, it is referred to as plastic waste. For many years plastic waste never degrades and persists on the land. Plastic waste is largely recyclable, but since it includes plastics and colours, recycled items are more detrimental to the atmosphere (Klemes et al., 2021). The recycling of new plastic material can only be achieved two to three times since the plastic material deteriorates due to thermal pressure after any recycling, and its life cycle is shortened. Therefore, recycling is not permanent but it helps to reduce plastic wastes (Moharir & Kumar, 2019). It has been noted that plastic waste disposal is a major problem due to an inappropriate method of collection and segregation. Its wide variety of uses includes films for insulation, carry bags, binding materials, containers for fluids, toys, household and industrial goods, and construction materials. This sector is the most proven means of avoiding paper and plastic waste from landfills or from being burned in incinerators. Plastic burning results in the production of a class of flame retardants called halogens, which is affecting to humans and the environment badly (Verma et al., 2016).

Planners analyse paper and plastic trash using the attributes provided by IVq-ROFNs. The IVq-ROFN decision matrix is designed to handle paper and plastic trash. The linguistic scale (Table 1) can be used to evaluate techniques to help specialists explain their findings and points of view clearly. The proposed approach is now being used to analyse e-waste.

Step 1: For evaluation, the following alternatives are being declared an account of criteria and are developed by specialists$e_{\alpha }$: $D_1$—sanitary landfills, $D_2$—incineration, $D_3$—refused-derived fuel, and $D_4$—recycling and recovery, as indicated in Table 4. The weight vector for experts is (0.45, 0.25, 0.30) and attribute weight values are (0.086, 0.397, 0.129, 0.087, 0.123, 0.054, 0.061, 0.051) and the initial DM for experts with q=4.

Table 4.

Decision matrix using linguistic scale for paper and plastic waste $D{M}_1,D{M}_2,D{M}_3$

	$Y_1$	$Y_2$	$Y_3$	$Y_4$	$Y_5$	$Y_6$	$Y_7$	$Y_8$
$D{M}_1$
$D_1$	G	W	P	VP	W	P	F	W
$D_2$	F	W	W	VP	W	F	F	P
$D_3$	P	VP	F	F	F	G	G	P
$D_4$	VG	VG	EG	G	EG	VG	VG	G
$D{M}_2$
$D_1$	F	W	F	VP	W	F	F	W
$D_2$	W	W	W	P	W	F	VP	W
$D_3$	VG	VG	VG	G	VG	G	VG	G
$D_4$	VG	G	G	G	F	F	G	F
$D{M}_3$
$D_1$	F	W	P	VP	W	VP	F	W
$D_2$	W	W	W	P	W	VP	P	W
$D_3$	G	VG	F	G	F	G	F	G
$D_4$	VG	VG	G	VG	G	G	VG	G

Step 2: The fused IVq-ROFNs matrix may be obtained using the IVq-ROFWA aggregation operator and expert weight.

Step 3: Normalise the fused results matrix $D=[d_{\mathit{st}}],s=1,2,...,m,t=1,2,...,n$ based on the attributes by Eqs. (12) and (13) ($Y_1,Y_7,Y_8$ are cost attributes); then the NDM is denoted as $N_{\mathit{st}}$.

Step 4: According to the NDM $N_{\mathit{st}}$ and attribute’s weights ${\lambda }_t$, the WNDM $W{N}_{\mathit{st}}$ can be calculated using Eq. (14).

Step 5: Calculate the values of BAA using Eq. (15) and the BAA matrix $B=[b_t]$ results are listed below:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill b_1=& ([0.3197,0.3255],[0.9557,0.9570])\hfill \\ \hfill b_2=& ([0.4727,0.4804],[0.8309,0.8367])\hfill \\ \hfill b_3=& ([0.4211,0.4310],[0.9196,0.9223])\hfill \\ \hfill b_4=& ([0.3083,0.3138],[0.9512,0.9527])\hfill \\ \hfill b_5=& ([0.4083,0.4181],[0.9368,0.9392])\hfill \\ \hfill b_6=& ([0.2908,0.2959],[0.9648,0.9659])\hfill \\ \hfill b_7=& ([0.2624,0.2676],[0.9684,0.9694])\hfill \\ \hfill b_8=& ([0.4049,0.4145],[0.9502,0.9517])\hfill \\ \hfill \end{array}\end{array}$$

Step 6: Obtain the distance $V=[v_{\mathit{st}}]$ between alternatives and the BAA using Eq. (16); the distance values are given below.

	$Y_1$	$Y_2$	$Y_3$	$Y_4$	$Y_5$	$Y_6$	$Y_7$	$Y_8$
$D_1$	− 0.0052	− 0.3834	− 0.0584	− 0.0871	− 0.2433	− 0.0490	0.0208	0.1361
$D_2$	0.0741	-0.3834	− 0.2892	− 0.0586	-0.2433	-0.0330	0.0455	0.0052
$D_3$	− 0.0150	0.1230	0.0315	0.0380	0.0677	0.0304	-0.0245	− 0.0830
$D_4$	− 0.0733	0.2145	0.1827	0.0799	0.2067	0.0411	0.0530	− 0.0984

Step 7: Calculate the sum of the values of each alternatives using Eq. (17) as given below and Fig. 4.

Alternatives	$Z_i$
$Z_1$	−  0.6695
$Z_2$	−  0.8827
$Z_3$	0.1681
$Z_4$	0.6062

Fig. 4.

Ranking values for paper and plastic waste

Ranking of Mixed solid wastes disposal method

Any mixture of waste types with varying properties is referred to as mixed waste. Industrial and urban wastes are mostly a combination of chemicals, steel, bottles, biodegradable waste such as paper and textiles, and other unidentifiable garbage. It is made up of a variety of wastes from various locations which consists of general household waste, workplace waste, waste from department stores or other enterprises, other miscellaneous waste, and non-hazardous waste (Klemes et al., 2020). Examples of wastes are drywall, timber (wood), metal, plastic (plumbing pipe, PVC siding, Styrofoam insulation), concrete, etc. In dealing with mixed waste, it is often difficult to recycle and reuse due to processing costs (Sardarmehni & Levis, 2021) (Figs. 4, 5).

Table 5.

Decision matrix using linguistic scale for mixed waste $D{M}_1,D{M}_2,D{M}_3$

	$Y_1$	$Y_2$	$Y_3$	$Y_4$	$Y_5$	$Y_6$	$Y_7$	$Y_8$
$D{M}_1$
$D_1$	F	VG	VG	G	P	F	F	P
$D_3$	G	VG	VG	G	VG	G	F	F
$D_4$	G	G	VP	W	VP	F	G	F
$D{M}_2$
$D_1$	G	VP	P	VP	F	F	G	P
$D_3$	VG	VG	VG	G	EG	G	F	G
$D_4$	G	P	W	F	G	P	F	F
$D{M}_3$
$D_1$	VG	W	VP	P	F	VP	F	P
$D_3$	VG	VG	G	F	VG	VG	G	VG
$D_4$	G	G	W	VP	VP	P	F	VP

Experts examine the construction wastages by each criterion that is mentioned like an IVq-ROFNs. The IVq-ROFN decision matrix is constructed for construction waste. The linguistic scale (Table 1) will be used to measure alternatives which could assist experts in explaining their information and perspectives more effectively. The proposed methodology is being used to analyse e-waste.

Step 1: The proposed alternatives are taken for calculation: $D_1$—sanitary landfills, $D_3$—refused-derived fuel, and $D_4$—recycling and recovery with regard of every criterion by experts $e_{\alpha }$ (weight vector for experts is 0.45, 0.25, 0.30) and attribute’s weight values (0.086, 0.397, 0.129, 0.087, 0.123, 0.054, 0.061, 0.051) and the basic DM for experts with q=4, as shown in Table 5.

Step 2: According to the IVq-ROFWA aggregation operator and experts weight, we can obtain the fused IVq-ROFNs matrix.

Step 3: Normalise the fused results matrix $D=[D_{\mathit{st}}],s=1,2,...,m,t=1,2,...,n$ based on the attributes by Eqs. (12) and (13) ($Y_1,Y_7,Y_8$ are cost attribute); then the NDM is denoted as $N_{\mathit{st}}$.

Step 4: According to the NDM $N_{\mathit{st}}$ and attribute’s weights ${\lambda }_t$, the WNDM $W{N}_{\mathit{st}}$ can be calculated using Eq. (14).

Step 5: Calculate the values of BAA using Eq. (15) and the BAA matrix $B=[b_t]$ results are listed below:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill b_1=& ([0.2323,0.2378],[0.9652,0.9664])\hfill \\ \hfill b_2=& ([0.5392,0.5479],[0.7293,0.7360])\hfill \\ \hfill b_3=& ([0.3790,0.3853],[0.9232,0.9256])\hfill \\ \hfill b_4=& ([0.2996,0.3051],[0.9547,0.9561])\hfill \\ \hfill b_5=& ([0.4197,0.4288],[0.9115,0.9142])\hfill \\ \hfill b_6=& ([0.2841,0.2891],[0.9660,0.9670])\hfill \\ \hfill b_7=& ([0.2745,0.2798],[0.9639,0.9650])\hfill \\ \hfill b_8=& ([0.2806,0.2857],[0.9675,0.9684])\hfill \\ \hfill \end{array}\end{array}$$

Step 6: Obtain the distance $V=[v_{\mathit{st}}]$ between alternatives and the BAA using Eq. (16); the distance values are given below.

	$Y_1$	$Y_2$	$Y_3$	$Y_4$	$Y_5$	$Y_6$	$Y_7$	$Y_8$
$D_1$	0.0073	−  0.0874	0.0214	0.0128	−  0.0716	−  0.0285	−  0.0096	0.0323
$D_3$	−  0.0196	0.0856	0.1082	0.0580	0.1377	0.0487	−  0.0119	−  0.045
$D_4$	0.0109	−  0.0310	−  0.1919	−  0.0889	−  0.1070	− 0.0271	0.0207	0.0076

Step 7: Calculate the sum of the values of each alternatives using Eq. (17) as given below and Fig. 5.

Alternatives	$Z_i$
$Z_1$	− 0.1233
$Z_3$	0.3608
$Z_4$	− 0.4067

Fig. 5.

Ranking values for mixed waste

Validity of the proposed method

Comparison analysis

   In this study, the proposed interval-valued q-ROF-MABAC approach is compared with IVq-ROFWA, IVq-ROFWG operators (using Theorems 3.1 and 3.2), and the IVq-ROFWG-MABAC model. The combined values of the aggregation operators are described in Table 6 based on the attribute weight and the decision matrix. The results of the proposed approach are compared with the above-mentioned aggregation operators, the results of the sorting are somewhat different in terms of the values and the ordering of alternatives, the ranking of alternatives is listed in Table 6, and the graphical representations are given in Figs. 6 and 7. However, the proposed model has the useful characteristics of measuring the distance between each alternative and the BAA and can be achieved by using the suggested solution relative to other MADM models to achieve more accurate results.

Table 6.

Comparative analysis in an IVq-ROF environment

Wastage types	Operators	$S(D_i)$	$D_i$	$R_i$
Mixed waste	IVq-ROFWA	$S(D_1)$ = 0.5088, $S(D_3)$ = 0.6184, $S(D_4)$ = 0.4781	$D_3$ > $D_1$ > $D_4$	$D_3$
	IVq-ROFWG	$S(D_1)$ = 0.3866, $S(D_3)$ = 0.5711, $S(D_4)$ = 0.3869	$D_3$ > $D_4$ > $D_1$	$D_3$
	IVq-ROFWG MABAC model	$S(D_1)=-0.064$, $S(D_3)$ = 0.6104, $S(D_4)$ = $-0.3107$	$D_3$ > $D_1$ > $D_4$	$D_3$
	Proposed method	$S(D_1)=-0.1233$, $S(D_3)$ = 0.3608, $S(D_4)$ = $-0.4067$	$D_3$ > $D_1$ > $D_4$	$D_3$
Paper and plastic waste	IVq-ROFWA	$S(D_1)$ = 0.4102, $S(D_2)$ = 0.3592, $S(D_3)$ = 0.5379, $S(D_4)$ = 0.6427	$D_4$ > $D_3$ > $D_1$ > $D_2$	$D_4$
	IVq-ROFWG	$S(D_1)$ = 0.2029, $S(D_2)$ = $-0.2726$, $S(D_3)$ = 0.4917, $S(D_4)$ = 0.5450	$D_4$ > $D_3$ > $D_2$ > $D_1$	$D_4$
	IVq-ROFWG MABAC model	$S(D_1)$ = $-0.2633$, $S(D_2)$ = $-0.2988$, $S(D_3)$ = 0.5598, $S(D_4)$ = 0.7855	$D_4$ > $D_3$ > $D_1$ > $D_2$	$D_4$
	Proposed method	$S(D_1)$ = $-0.6695$, $S(D_2)$ = 0.8827, $S(D_3)$ = 0.1681, $S(D_4)$ = 0.6062	$D_4>D_3>D_1>D_2$	$D_4$

Fig. 6.

Mixed wastage ranking results

Fig. 7.

Paper and plastic wastes ranking results in comparative analysis

Sensitivity analysis

   The sensitivity of “q” parameters and experts weight vector (0.30, 0.45, 0.25) is analysed in the IVq-ROFWA operator. Various scores and ranking outcomes could be achieved by assigning numerous entries to the parameter q. The sensitivity results are shown in Table 7. Here, we denote score values as $S(D_i)$, order as $(D_i)$ and the ranking result as $R_i$. The graphical representations are given in Figs. 8 and 9.

Table 7.

Analysis of sensibleness of the proposed method

Wastage types	q values	Xi values	$D_i$	$R_i$
Mixed waste	$q=3$	$S(D_1)=-0.208$, $S(D_3)=0.4264$, $S(D_4)=-0.3768$	$D_3$ > $D_1$ > $D_4$	$D_3$
	$q=7$	$S(D_1)=-0.1436$, $S(D_3)=0.4015$, $S(D_4)=-0.391$	$D_3$ > $D_1$ > $D_4$	$D_3$
	$q=10$	$S(D_1)=-0.1522$, $S(D_3)=0.3097$, $S(D_4)=-0.3881$	$D_3$ > $D_1$ > $D_4$	$D_3$
Paper and plastic waste	$q=3$	$S(D_1)=-0.6144$, $S(D_2)=-0.7467$, $S(D_3)=0.1693$, $S(D_4)=0.3971$	$D_4$ > $D_3$ > $D_1$ > $D_2$	$D_4$
	$q=7$	$S(D_1)=-0.6608$, $S(D_2)=-0.9196$, $S(D_3)=0.0521$, $S(D_4)=0.3168$	$D_4$ > $D_3$ > $D_1$ > $D_2$	$D_4$
	$q=10$	$S(D_1)=-0.5737$, $S(D_2)=-0.9687$, $S(D_3)=-0.1064$, $S(D_4)=0.2504$	$D_4$ > $D_3$ > $D_1$ > $D_2$	$D_4$

Fig. 8.

Sensitivity check of mixed wastes

Fig. 9.

Sensitivity check of paper and plastic wastage

Results evaluation and discussion

The alternatives for disposal of paper and plastic wastes are ranked by the proposed method as shown in Fig. 4. There are many benefits of recycling and the reuse of waste plastics. This leads to a decline in the use of new products and the use of energy, thereby also reducing the emission of carbon dioxide. Therefore, recycling is a good disposal method and it helps to reduce the plastic wastes. The alternative $D_4$—recycling and recover is the best suitable disposal method for paper and plastic waste. Also, the final ranking of the alternatives from mixed solid waste disposal problem proposed with unique method (Fig. 5) explicits that $D_3$—refused-derived fuel is the best suitable disposal method for mixed waste. The removal and treatment of mixed waste are costly and necessitate special handling. Moreover, most vendors must have a permit for the kind of waste they are disposing of; mixed waste cannot be disposed of by regular vendors. The RDF system is used to dispose of mixed wastes; it separates all types of waste and converts it into an environmentally friendly fuel. RDF contributes to the reduction of mixed waste.

Since experts faced struggle to evaluate choices among the various different satisfaction and displeasure degrees to predict uncertainty information, this research study managed information using the IVq-ROFS. An improved MABAC model appears to be more accurate and appropriate for solving a variety of other MADM issues based on the end ranking findings (Fig. 3) (Pamucar & Cirovic, 2015). The findings of the comparison between MABAC and other ranking techniques show that MABAC offers a more feasible and appropriate solution that is more sustainable. The calculated criteria weights demonstrate this strategy was coupled with the FUCOM weight finding method, which is sophisticated, quick, and precise. By altering the value of the parameter “q” in the IVq-ROFS preferences, the stability and viability of the FUCOM method are also examined, and a comparison of the observed findings shows that FUCOM is more precise and stable under any unclear circumstances. As a result, we used the COVID-19 problem as a research study topic and provided information that is very helpful to the ruling authorities.

Conclusion

The current study seeks to improve plastic and mixed trash disposal practices in India. A lack of information about the adverse impacts, poor pollution control, environmental damage, and a lack of human and material wealth are the most prominent concerns in waste management (Singh, 2019). The MABAC model is a better alternative technique in a way to determine the decision and compare the alternatives (Wei et al., 2020a). Clearly, the proposed method has the advantages of less pairwise comparing simplified calculated sophisticated way of handling uncertain information with flexible range of problem spaces definitely producing stable and sustainable results. In this paper, we recommend waste treatment approaches that concentrate on the sort of waste that can lead to better maintenance and effective waste management. The fostering of public awareness, the enforcement of laws and regulations, and the development of appropriate treatment are the main remedial measures to ensure sound environmental protection. If we follow these measures in India, we would be able to reduce PMW while simultaneously ensuring safe waste management during the pandemic. In the future, we plan to use the proposed method to address other issues, such as biomedical waste treatment and plastic recycling. Furthermore, the suggested method’s application in the spherical q-ROFS setting is an intriguing direction to examine.

Data availability

The authors declare that our manuscript has no associated data.

Declarations

Conflicts of interest

The authors declare that there is no conflict of interest regarding this research work.