Introduction

The growing urban population, industrialization, and increasing living standard have immensely increased the quality of waste generation and are perceived as the crucial culprits for ever-increasing environmental pollution. The countries are struggling with the glitches of solid waste management (Soni et al. 2022). The generated municipal waste is a cause of concern for the nation, whereas, on the one hand, it also possesses quite the potential as a raw material for new products (Minelgaitė and Liobikienė 2019). To alleviate the environmental condition and to reduce the energy crises, it is imperative to implement the available waste treatment facilities properly (Nkuna et al. 2022). As a result, the effective management of solid wastes has become a crucial element and has motivated the researchers toward the advancement of several practices of waste management (Moh 2017; Shahnazari et al. 2020). Although there are several techniques that are available for the treatment of solid waste, the management of solid wastes is still complex due to the availability of wastes in a wide range of compositions and different features of solid wastes (Seay and Ternes 2022). The identification of an effective waste management technique and efficient utilization of the available resources have become cumbersome tasks in solid waste management (Goulart Coelho et al. 2017). Moreover, an effective waste management remains as one of the core components for the development of a country which influences the socio-eco and environmental dimensions ( Soni et al. 2022c). The growing concerns of the public toward the environment conditions have motivated for an efficient technique for the management of wastes; as a result, the adoption of a systematic and scientific framework is crucial for solid waste management (Tomić et al. 2022). Therefore, the identification of a suitable waste management practice is essential for the promotion of environment health.

There are several techniques that are available for the treatment of solid waste including reduction, recycling, waste-to-energy (WtE) conversions, composting, and disposal (Mayer et al. 2019; Tsui and Wong 2019). An effective waste management can be achieved by adopting a suitable waste treatment facility (Alao et al. 2022). The various thermal energy systems provide an opportunity to convert biomass into renewable energy (Garlapati 2016; Zhou et al. 2018). Moreover, the development of suitable WtE conversion facilities for energy recovery is beneficial from a techno-eco point of view (Mazzei and Specchia 2023). Thermochemical techniques are economical and effectively reduce environmental pollution (Seo et al. 2022). Moreover, the methods can convert sludge into a different product in less time (Jiang et al. 2021). The conversion of solid wastes to biomass is prominent in producing heat. Among these, the disposal of wastes plays a substantial role in the consequences on the health of environment and public. In spite of this, the perusal for the most suitable method of disposal is still unexplored.

The classifications of municipal solid waste are given in Fig. 1, which are generally classified into two categories, namely, organic and inorganic. The inorganic waste may be hazardous or non-hazardous in nature. The sources of organic waste include waste generated due to agricultural activities, domestic wastes, and horticultural wastes. The components of minerals are considered as inorganic wastes. The secondary process of non-ferrous metals mainly generates hazardous inorganic waste, whereas the wastes generated, due to the mining activities, are non-hazardous wastes. The hazardous wastes being toxic negatively affect the quality of the soil at disposal sites and ground wastes. The global generation of municipal solid waste was 2.01 billion tons in 2016, and the generation of solid is anticipated to be 27 billion tons by the year 2050. Asia alone generates one-third of the total solid waste, whereas India and China generate 0.50–0.9 kg/capita/day and 0–0.49 kg/capita/day, respectively (Kumar and Agrawal 2020). According to the report of the World Bank in 2016, the waste generation from the region of lower income was 0.09–0.60 kg/per capita/day, while the lower-middle regions generate 0.16–0.79 kg/per capita/day, the amount increases to 0.1–1.2 kg/per capita/day for upper-middle groups (Afrane et al. 2021), and only a small portion of the produced solid wastes is recycled, as given in Fig. 2. The available WtE methods could decrease the problem of solid waste, energy demand, and emission of greenhouse gases (GHG) and contribute for the circular economy. For the concern of the environment, with technological advancement and organization support, the WtE technology was successful in reducing the energy demand by 5% in developed countries. The energy generated by solid waste depends on the composition and moisture (Foster et al. 2021). It is reported by the European WtE Plant in 2018 that in Germany, 31% of the generated waste is converted into energy by WtE, while countries including Finland, Norway, Denmark, and Sweden have converted 50% of their wastes into energy (Sagnak et al. 2021). Moreover, the USA has successfully generated around 50 MW of electricity by utilizing 2.6 million tons of methane produced from the landfilling. The anaerobic digesters are widely used in the United Kingdom to provide bio methane and electricity to the nation as well as global level. On another side, the sub-Saharan Africans lack the WtE technology (Perna et al. 2018). It was found that in spite of the immense of waste treatment facilities, there is still an insignificant progress in solid waste management. The proper implementation of WtE facilities can provide an avenue for the transformation of open economy to circular economy.

Fig. 1
figure 1

Compositions of municipal solid waste (Ahluwalia and Patel 2018)

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Fig. 2
figure 2

Status of solid waste management methods

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The effective management of solid wastes highly depends upon waste treatment methods. Municipal solid waste (MSW) is heterogeneous and differs in locations (Saravanan et al. 2022). Therefore, any single practice of waste management is inefficient for the treatment of all kinds of solid wastes (Izquierdo-Horna et al. 2022). The optimal selection of waste treatment methods is a critical issue due to the climatic change, deteriorating health of the environment, depleting natural resources, etc. (Y. C. Chen 2018). The techno-economic and environmental consideration for the solid waste management involves multiple conflicting factors; as a result, the identification of a suitable waste treatment method among the various available methods of waste treatment facilities is recognized as a multi-criteria decision-making problem (Garcia-Garcia 2022). The mathematical techniques allow the identification of a suitable waste management practice for an effective treatment of solid wastes. The mathematical model can successfully rank the available methods of waste management under a set of complex criteria. The implementation of mathematical model for the identification of waste treatment methods can save a considerable amount of time and resources. The approach is different from the conventional practices of selecting waste treatment techniques by trial and error methods which consumes a lot of the available valuable resources. The developed mathematical models are reliable and efficient and give robust results which are helpful in effective management of solid wastes.

The different multi-criteria decision-making (MCDM) techniques can simultaneously synthesize these contradictory criteria and accomplish a trade-off among them. MCDM is an efficient tool for complex decision-making problems which systematically considers the different criteria involved in ranking the alternatives (Mir et al. 2016). The fuzzy based MCDM techniques are capable of handling the vagueness and uncertainty involved in the decision-making and instability in the parameters of real-world problems (Wang et al. 2018). The assignation of criteria weight is the most critical process in any MCDM method which can be successfully performed by the implementation of available methods for weighting criteria such as analytical hierarchy process (AHP) and criteria importance through intercriteria correlation (CRITIC) (Asefi et al. 2020). In the analytical hierarchy process (AHP), the criteria weights are performed by pairwise comparison. It is simple, flexible, and capable of considering both the quantitative and qualitative attributes of the decision framework. The method is limited to application and the influence of results with the incorporation of new alternative. The shortcomings of the AHP are successfully overcome by the best worst method (BWM) where the different comparisons are performed in pairwise. The method is based on the identification of the best and worst criteria as a reference for pairwise comparison from a set of criteria. The methods are incapable of handling the situations of more than one best and worst criteria. CRITIC method requires an accurate calculation of the standard deviation for the estimation of criteria weights. Among these, stepwise weight assessment ratio analysis (SWARA) assigns the weights by prioritizing the criteria and, therefore, gives an actuate and reliable result (Zolfani, Yazdani, & Zavadskas, 2018), whereas the complex proportional assessment (COPRAS) method is based on utility degree and is a relatively new method which is advantageous in simplicity and stability in solution. Moreover, the method eliminates the requirement of additional parameters and, therefore, is an efficient and robust technique for ranking alternatives ( Ji et al. 2023;  Korucuk et al. 2022).

The research aims to develop a mathematical model for ranking of waste management techniques to solve the issues of solid waste management. By keeping the aim of the research in view, an integrated SWARA-COPRAS mathematical model based on q-rung orthopair fuzzy numbers (q-ROPFNs) is proposed for ranking waste management techniques. The proposed SWARA-COPRAS integrates the subjectivity of decision-makers and objectives of the performance of alternatives to give a balanced and comprised solution. In the present study, ten (10) different practices of solid waste management are considered as alternatives which are ranked against the seven (07) different techno-eco and environmental criteria. The assessments for the alternatives and criteria are collected in linguistic terms from the five (05) different decision-makers (DMs). The decision-makers are the experts, researchers, and professionals involved in the practice of waste management and have expertise in mathematical techniques. The criteria weights are assigned by SWARA methods, while COPRAS is applied for the ranking of alternatives. The ranking result of the proposed model is compared with other well-known MCDM techniques, and sensitivity analyses are performed for different criteria weights.

Research gap

The extensive literature survey has shown several methods for the management of generated municipal solid wastes. However, a little work has been reported on the identification of a suitable method for solid waste management. There is still a need for an effective and systematic approach for the ranking of waste management methods. The development of a SWARA-COPRAS mathematical model for rankings of waste management techniques is rarely reported in the literature. The implementation of the multi-criteria decision-making techniques for ranking the waste treatment methods under techno-eco and environmental conditions for the given alternatives is not reported in any of the literature. Moreover, there is an absolute dearth of investigation for the r-rung orthopair fuzzy number-based model, which can effectively tackle the vagueness of the decision.

Statement of novelty

There are several methods for the management of solid waste including recycling, bio-mass conversions, waste-to-energy methods, and incineration. But there is not any particular technique for waste management which is equally applicable for the treatment of different kinds of wastes under a given set of conditions. Therefore, the development of a mathematical model for ranking the waste management techniques under techno-eco and environment consideration is one of the novel approaches for solid waste management. Moreover, the ability of the proposed q-ROPFN-based SWARA-COPRAS approach to deal with the vagueness and uncertainty in the decision is another novelty of the present work. The proposed SWARA-COPRAS model is a unique combination of two well-known efficient techniques which can effectively deal with the multi-criteria decision-making problems in solid waste management. The applicability of the proposed model for the identification of a suitable waste treatment strategy is an innovative way to deal with the issues of waste management.

Research contribution

The availability of a wide variety and quantity of solid waste with different attributes and behavior, geographical locations, awareness, etc. for the generated wastes makes an optimal selection of waste treatment facilities a cumbersome process. Moreover, due to the involvement of several conflicting criteria, no particular waste management practice is equally effective for solid waste management in every condition. The research provides an integrated SWARA-COPRAS mathematical model which can effectively rank the available waste treatment facilities under a given set of criteria. The proposed model is capable of handling the vagueness and uncertainty in decision-makers. The research could aid the decision-makers and authorities in the identification of a suitable practice for solid waste management and is potent to improve the environmental condition by ranking the waste treatment facilities. The research reduces the consumption of time and resources required for solid waste management. Moreover, the versatility of the proposed model by varying the criteria weights favors the applicability of the proposed model in different situations. The widespread application of the proposed model could improve the socio-eco and environmental conditions from local to global levels.

The research is structured as follows: (i) the extensive literature survey is given in the “Literatures” section; (ii) the “Waste-to-energy recovery techniques” section gives an overview of the waste management practices; (iii) the social-eco and environmental complications for the implementation of waste management techniques are discussed in the “Implications in solid waste management” section; (iv) the mathematical preliminaries are given in the “Mathematical preliminaries” section; (v) the “Proposed integrated mathematical model” section gives an overview of the proposed mathematical model; (vi) application of the SWARA-COPRAS model is presented in the “Application of q-ROPFWBMO-based SWARA-COPRAS model” section; (vii) the “Results and discussions” section gives the results and discussions; (viii) finally, in the “Conclusions and future scope” section, the conclusions and future scope recommendations are provided.

Literatures

The evaluation of smart waste collection facilities based on the Internet of Things (IoT) has been carried out by applying an MCDM technique based on q-rung orthopair fuzzy sets (Seker 2022). AHP was employed for the selection of waste disposal sites by considering the economic, technical, environmental, and legal factors (Zewdie and Yeshanew 2023). A supply chain model was introduced to optimize the incineration and biogas in municipal solid waste management and found that the economic factors are insensitive to the cost of power stations (Abbasi et al. 2022). The implementation of a non-linear mathematical fuzzy-AHP model for the identification of a suitable waste-to-energy strategy has shown the effectiveness of the waste model in recovering 4.2% of the energy demand with the reduction of 97.6% of the carbon footprints from landfills (Abdallah et al. 2021). The researchers have developed a novel multi-objective technique with a mixed integer for quantitative assessment of waste management facilities which supports the development of an efficient MSW management system (Ooi et al. 2021). The issue of heat integration in chemical waste treatment in industrial sites has been solved by multi-objective techniques (Capón-García et al. 2014). The assessment of waste-to-energy technologies, namely, photovoltaic cells and power to gas through genetic algorithms for multi-objective optimization of the parameters was carried out (Ravajiri et al. 2023). A fuzzy interval number-based mathematical model by considering the economic and environmental considerations has been introduced to assess the decision-making in waste-to-energy techniques and has suggested recycling of wastes as a viable approach for the successful management of generated waste (Liang et al. 2022). A mixed integer programming model for a waste clean-up system in the condition of post-disaster proposed the considerations for economics, environment, and time for trading the optimal solution (Cheng et al. 2022). The researcher has proposed a model for the treatment of food waste in trans-esterification and anaerobic digestion by using trapezoidal fuzzy numbers (Deng et al. 2022). The characterizations of the degraded oils and metal wastes were successfully performed by using mathematical models (Soni et al. 2022h; Soni et al. 2022i). The optimization model for an optimal design of a supply chain network in crisp and fuzzy nature minimizes the cost and greenhouse gas emissions by 21.6% and 28.4%, respectively (Li et al. 2022). An integrated model was developed by the combination of dynamic quantitative simulation and multi-objective optimization algorithm for urban bio-waste treatment has successfully optimized the efficiency and reduced the emissions by 14.5% with cost reduction (Fei et al. 2022). A mathematical model for the identification of an optimal technique for solid waste management has shown that the Pareto solutions could successfully reduce the environmental impact and energy consumption by 24.2% and 7.4%, respectively (Chen et al. 2022). The work for the optimization of the economic, environmental, and thermal objective of waste-to-energy systems has highlighted the significance of criteria weights for achieving the best solution. Moreover, the efficiency of the techniques was maximized through a non-dominated algorithm (Mayanti et al. 2021). A bi-level multi-objective model was developed for bioenergy optimization problems and has demonstrated the feasibility of improvement in co-digestion and recovery of resources (Huang and Xu 2022). The application of a multi-objective genetic algorithm and the bee colony algorithm (BCO) to deal with the problem of logistics for the management of solid waste has explored that the bee colony algorithm could successfully achieve the optimal solution (Hashemi 2021). The application of a mathematical model for the management of solid waste by recycling and waste-to-energy techniques had suggested the establishment of recovery and composting to reduce the impact of solid waste (Rabbani et al. 2021). The simulation of integrated biomass gasification was performed by considering the economic emission of particulate and gases (He et al. 2021). The implementation of the analytic hierarchy process (AHP), multi-attribute utility theory (MAUT), outranking procedures, and the technique for order of preference by similarity to ideal solution (TOPSIS) for solid waste management has suggested the integration of different methods for an effective management of solid wastes (Garcia-Garcia 2022). The study has proposed a multi-maximal covering location model to deal with the issues of site location and the number of waste collection sites (Zhuo and Yan 2022). The management of e-waste in IoTs has been performed by using a hybrid model (Ramya and Ramya 2023). A dynamic system model was developed for the maximum recovery of energy from waste (Hosseinalizadeh et al. 2022). The optimization of revenue generated due to the generation of electricity by gasification of solid waste has been performed through life cycle assessment of solid waste (Xu et al. 2022). The allocation of routes for electrically operated waste collection vehicle through a mixed-integer programming (MIP) model has been demonstrated (Erdem 2022). The modified version of TOPSIS has identified the integration of composting, anaerobic digestion, recycling, and landfill as an optimized model for the successful management of solid wastes (Mir et al. 2016).

Waste-to-energy recovery techniques

There are several methods available for the treatment of municipal solid wastes, as classified in Fig. 3. The different methods available for solid waste management are robust, reliable, and efficient. The proper selection of waste treatment methods plays an important role in the successful management of municipal solid waste. The design of the solid waste management method mainly involves three important components including (i) source of origin, (ii) collection and logistics, and (iii) processing and treatment of wastes (Sondh et al., 2022). The techniques for MSW are mainly classified as conventional and non-conventional. Incineration, composting, landfilling, anaerobic digestion, etc. are considered as the conventional methods of waste treatment, whereas pyrolysis, gasification, hydrothermal conversion, etc. are the non-conventional techniques of waste treatment (Khan et al. 2021).

Fig. 3
figure 3

Waste-to-energy techniques

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Upcycling

The upcycling of waste is a value-addition process where waste material is converted into a new product of higher value in every stage of its life cycle. It is an innovative way of reusing waste products (Moh 2017). The upcycling of solid waste is successful in removing the pollutants from the environment. The method is economical but inefficient for handling a wide range of waste products. In recent years, the upcycling of waste plastics has gained a considerable importance due to the low involvement of cost and potential for conversion to higher-value products ( Soni et al. 2022e; Soni et al., 2023a). The method suffers from the limited applications and type of materials available for upcycling (Dogu et al. 2021).

Recycling

Recycling is the reprocessing of waste materials into a product for an original or another purpose in such a way that the identity of the original product is completely lost ( Soni et al. 2022d; Soni et al. 2022g). It is one of the widely accepted methods for the reduction of waste generation. Recycling alters the physical nature of a material and produces a new product from the altered materials (Soni et al. 2022a; Soni et al., 2023b). The practice of recycling is gaining attention as a sustainable method for mitigating environmental pollution. Recycling is favorable in a condition where the considerations for cost and environment are desirable (Soni et al. 2022f). Recycling can divert a significant amount of discarded materials from waste to wealth. Moreover, it could save a considerable landfill space besides providing a source for raw materials (Nguyen et al. 2023; Soni et al. 2022f). Recycling is a waste reduction technique irrespective of the type of recyclable material and is the potential to convert a variety of wastes into value-added products as shown in Table 1, but the practice is rarely performed due to a lack of technological advancement (Dogu et al. 2021).

Table 1 Classification and recycling potential of municipal solid waste
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Incineration

Incineration is a method of direct burning of organic material present in solid waste at elevated temperatures of 800–1000 °C to produce heat which can be further utilized in vapor turbines and heat exchangers for different processes. The incineration is successful to reduce the volume of solid waste by 80–85%. The incineration releases several harmful flue gases and toxic materials which require further treatment through suitable methods and, therefore, is less effective (Makarichi et al. 2018).

Pyrolysis

Pyrolysis is the thermal decomposition of biomass by heating the substances in the absence of oxygen to yield charcoal, bio-oils, and fuel gas. The calorific value of the gaseous fuel due to pyrolysis possesses a net calorific value of 10–20 MJ/N-m3. Based on the operating conditions, pyrolysis is mainly classified as carbonization as conventional pyrolysis, fast pyrolysis, and ash pyrolysis (Kwon et al. 2019). The method typically involves a feeding section which can successfully handle the feedstock without any pre-treatments, reactor assembly, and separate liner vessels for the collection of products. The conventional pyrolysis method requires a low heating rate of 300-850 °C to permit the production of pyrolysis products (Zhang et al. 2020).

Gasification

Gasification is a thermal conversion process for the treatment of solid waste. It is a process of the production of alternative fuel from solid waste. The gasification of solid waste produces “syngas” which offers a feasibility to connect in burners, gas engines, boiler, and gas turbine to produce heat or electricity. The gasification is performed in the presence of oxygen, stream, air, etc. and requires a temperature in the middle range of pyrolysis and combustion of above 650 °C (Lopez et al. 2018). The method is advantageous over the traditional combustion process as it offers the property to balance under a wide range of operating conditions and features of a specific reactor to produce syngas. The method is economical and feasible. The components of the process form the basis in the production of valuable chemicals and fuels (Munir et al. 2019).

Composting

Composting is the decomposition of organic matter and is one of the suitable methods for the treatment of waste. It involves the conversion of organic waste into nutrient-rich humus through the microbial activities of bacteria and fungi. The obtained product of composting is rich in humus and plant nutrients besides carbon dioxide, heat, and water as a by-product. It is one of the best methods for the management of organic solid waste. The high cost of operation and maintenance and the difficulty in the separation of non-compostable materials are the limitations of the method (Le Pera et al. 2022).

Landfilling

The dumping of solid waste in an openly is known as a landfill and is the worst method of waste management. Improper landfilling increases the impact of waste on environmental health. The landfilling of waste is a common practice of waste treatment in an underdeveloped countries (Vaverková 2019).

Hydrolysis

Hydrolysis is a chemical reaction which proceeds with the interaction of chemicals with water and leads toward the decomposition of the substance and water. The reaction takes in the presence of salts, protein, fats, etc. with the participation of enzymes. It is the reaction of organic chemicals with water to produce a new substance which is the cleavage of chemical bonds by the addition of water (Varjani et al. 2022). It is the inverse of the condensation of reaction where the molecules are joined to a larger unit and ejects a water molecule. In hydrolysis, the water reacts with a compound without causing decomposition. The conditions like low pressure and long retention time favor the process of enzymatic hydrolysis. The high content of carbohydrates favors the production of ethanol and other biofuels. It does not involve the use of chemical to loss the carbohydrates (Yang et al. 2021).

Bio-technology

Bio-technology is the integration of engineering and natural science for the attainment of the applications of organisms and molecules for the products and services. The modern usage of biotechnology encompasses the genetic engineering and tissue culture technologies (Hussaini 2013). The concept of biotechnology involves a range of practices for the modification of the living organism and improvement through breeding program. These technologies are promising for selective recovery of metal ions with low cost and requirement of low toxic chemicals thereby avoiding the generation of toxic sludge (Burgess et al. 2001).

Thermal plasma pyrolysis

Thermal plasma pyrolysis is the reaction of a carbonaceous solid at high temperature with a small amount of oxygen to yield gaseous products such as hydrogen, carbon dioxide, and carbon monoxide, having a small fraction of solid product. The obtained products of thermal plasma pyrolysis are easy to handle (Sikarwar et al. 2020). The process requires a considerable amount of energy for high processing temperatures which increases their operational cost. The plasma pyrolysis is gaining interest for the management of waste (Prado et al. 2020).

Implications in solid waste management

The nature of municipal solid wastes varies with geographical locations, living standards, attitudes and behavior of people, etc. Therefore, not any particular type of waste management practice is equally suitable for all types of waste management. The available waste-to-energy techniques suffer due to low conversion efficiencies and high operational costs. In contrast, recycling and upcycling are rarely performed due to a lack of proper training facilities, inefficient technologies, and awareness. Furthermore, the different technological, environmental, and cost considerations increase the complexity in the selection of a suitable waste treatment method for the management of municipal solid wastes as given in Fig. 4.

Fig. 4
figure 4

Problem hierarchy of the waste treatment problem

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Mathematical preliminaries

This section discusses the mathematical concepts, including q-ROPFNs, their operational laws, and the q-ROPFN-based BM operator used in the proposed model.

q-rung orthopair fuzzy numbers

A q-ROPFN in a finite universe of a discourse given by Yager is defined as follows:

$$A=\left\{\left(x,{\mu}_A(x),{\upsilon}_A(x)\right)|x\ \epsilon\ X\right\}$$

where μA : X → [0, 1] denotes the MD and υA : X → [0, 1] denotes the NMD of the element x ∊ X to the set A, with the condition 0 ≤ (μA(x))q + (υA(x))q ≤ 1, (q ≥ 1). The degree of indeterminacy is given as follows: πA(x):\({\left({\mu}_A(x)\right)}^q+{\left({\upsilon}_A(x)\right)}^q-{\left({\mu}_A(x)\right)}^q{\left({\upsilon}_A{(x)}^q\right)}^{\frac{1}{q}}\). For convenience, we call(μA(x), υA(x)) a q-ROPFN denoted by A = (μA,υA).

Operational laws

Let, \({\tilde{a}}_1=\left({\mu}_{1,}{\upsilon}_1\right)\), \({\tilde{a}}_2=\left({\mu}_{2,}{\upsilon}_2\right)\) be two q-ROFNs, then the following operational laws are defined:

$$\left(\mathrm{i}\right){\tilde{a}}_1v{\tilde{a}}_2=\left(\max \left\{{\mu}_{1,}{\mu}_1\right\},\min \left\{{v}_{1,}{v}_1\right\}\right)$$
(1)
$$\left(\mathrm{ii}\right){\tilde{a}}_1v{\tilde{a}}_2=\left(\min \left\{{\mu}_{1,}{\mu}_1\right\},\max \left\{{v}_{1,}{v}_1\right\}\right)$$
(2)
$$\left(\mathrm{iii}\right){\tilde{a}}_1\oplus {\tilde{a}}_2=\left({\left({\mu}_1^q+{\mu}_2^q-{\mu}_1^q{\mu}_2^q\right)}^{\frac{1}{q}},\kern0.5em {\upsilon}_1{\upsilon}_1\right)$$
(3)
$$\left(\mathrm{iv}\right){\tilde{a}}_1\otimes {\tilde{a}}_2=\left({\mu}_1{\mu}_2,{\left({v}_1^q+{v}_2^q-{v}_1^q{v}_2^q\right)}^{\frac{1}{q}}\right)$$
(4)
$$\left(\mathrm{v}\right)\lambda {\tilde{a}}_1=\left({\left(1-{\left(1-{\mu}_1^q\right)}^{\lambda}\right)}^{{~}^{1}\!\left/ \!{~}_{q}\right.},{\upsilon}_1^{\lambda}\right),\lambda >0$$
(5)
$$\left(\mathrm{vi}\right){\tilde{a}}_1^{\lambda }=\left({\mu}_1^{\lambda },{\left(1-{\left(1-{v}_1^q\right)}^{\lambda}\right)}^{{~}^{1}\!\left/ \!{~}_{q\kern0.5em }\right.}\right),\lambda >0$$
(6)

Bonferroni mean operator (BMO)

The BMO was introduced by Bonferroni, which is basically a mean-type aggregation operator. It will give an aggregation lying between the min and max operators and the logical “or” and “and” operators, which can be defined as follows:

Let p, q ≥ 0 be the parameters, and ai(i = 1, 2, ……. m) be a collection of non-negative real numbers. Then, the aggregation function is as follows:

$${BM}^{s,t}\left({a}_{i,\kern0.5em }{a}_2,\dots, {a}_m\right)={\left(\frac{1}{m\left(m-1\right)}\sum_{\begin{array}{c}i,j=1\\ {}i\ne j\end{array}}^m{a}_i^s{a}_j^t\right)}^{\frac{1}{s+t\ }}$$

It is called a BM operator.

q-ROPFBM operator

Let, \({\tilde{a}}_k=\Big({a}_{k,}{b}_k\)), (k = 1, 2, …. . m) is a collection of q-ROFNs and s, t ≥ 0, q ≥ 1, and q − ROFBM : Ωm → Ω if

$$q-{ROFNs}^{s,t}\left({\tilde{a}}_{1,}{\tilde{a}}_{2,}\dots \dots \dots {\tilde{a}}_m\right)={\left(\frac{1}{m\left(m-1\right)}\begin{array}{c}\begin{array}{c}m\\ {}\oplus \end{array}\\ {}\begin{array}{c}i,j=1\\ {}i\ne j\end{array}\end{array}\left({\tilde{a}}_t^s\otimes {\tilde{a}}_j^t\right)\right)}^{\frac{1}{s+t}}$$
(9)

Theorem: Suppose \({\tilde{a}}_k=\Big({a}_k\), ak) (k = 1, 2, ……m) is a collection of q-ROFNs and s, t≥0, p ≥ 1; then, the result aggregated from Eq. 10 is still a q-ROFN, and even,

$$q-{ROPWBM}^{s,t}\left({\tilde{a}}_{1,}{\tilde{a}}_{2,}\dots \dots \dots {\tilde{a}}_m\right)=\left({\left(1-{\left(\prod_{\begin{array}{c}i,j=1\\ {}i\ne j\end{array}}^m\left(1-{\left(1-{\left(1-{a}_i^q\right)}^{w_i}\right)}^s\ {\left(1-{\left(1-{a}_j^q\right)}^{w_j}\right)}^t\right)\ \right)}^{\frac{1}{m\left(m-1\right)}}\right)}^{\frac{1}{q\left(s+t\right)}},{\left(1-{\left(1-{\left(\prod_{\begin{array}{c}i,j=1\\ {}i\ne j\end{array}}^m\ \left(2-{\left(1-{b}_i^{q{w}_i}\right)}^s-{\left(1-{b}_j^{q{w}_j}\right)}^t-\left(1-{\left(1-{b}_i^{q{w}_i}\right)}^s\right)\left(1-{\left(1-{b}_j^{q{w}_j}\right)}^t\right)\right)\kern0.5em \right)}^{\frac{q}{m\left(m-1\right)}}\right)}^{\frac{1}{s+t}}\right)}^{\frac{1}{q}}\right)$$
(10)

Proposed integrated mathematical model

The discussions of the q-rung orthopair fuzzy weighted Bonferroni mean operator- (q-ROPFWBMO)-based SWARA and MABAC are provided in the section. Suppose there be m alternatives {A1, A2………….Am}, n attributes {C1, C2,……….,Cn}, and weighting vector W = (W1, W2……. Wλ) assigned to λ experts {d1, d2,……….,dλ}.

q-ROPFWBMO-based SWARA

The stepwise weight assessment ratio analysis (SWARA) method was proposed by Zavadskas in 2010 (Zavadskas et al. 2010). In the SWARA method, the most significant criteria are ranked at the top, and the least important criteria are ranked at the bottom. Therefore, the method gives the ranking of the alternatives based on the mean value of the rank. The method provides an opportunity to make decisions under different situations and prioritize the criteria on the basis of their requirement (Alinezhad and Khalili 2019). The method has been recognized as a suitable tool for making decisions. The method allows the decision-makers to exclude the insignificant criteria and formulate the ranking according to the requirement (Hashemkhani Zolfani et al. 2018). The advantages of SWARA methods can be given as follows: (i) the methods handle the expert opinion based on the relative importance in the determination of criteria weights; (ii) the methods assist in the collection of information from the experts; (iii) it is a straightforward, simple, and easily applicable method; (iv) the methods prioritize the problem based on the goals and eliminates the requirement of the criteria assessment.

The steps followed for the calculation of the criteria weight are given as follows:

Step 1: the linguistic assessments by the decision-makers (DMs) are aggregated by using Eq. 10.

Step 2: calculate the relative importance of the criteria by considering the aggregation results.

Step 3: obtain the value of relative importance (Sj).

Step 4: calculate the value of (kj) by using Eq. 11.

$${k}_j=\left\{\begin{array}{c}1,\kern0.75em j=1\\ {}{S}_j+1,\kern1em j>1\end{array}\right\}$$
(11)

Step 5: obtain the recalculated weight (qj) by using Eq. 12.

$${q}_j=\left\{\begin{array}{c}1,\kern0.75em j=1\\ {}\frac{q_{j-1}}{k_j},\kern1em j>1\end{array}\right\}$$
(12)

Step 6: calculate the criteria weight (wj) by using Eq. 13.

$${\omega}_j=\frac{q_j}{\sum_{k=1}^n{q}_k}$$
(13)

q-ROPFWBMO-based COPRAS

Complex proportional assessment (COPRAS) was introduced by Zavadskas, Kaklauskas, and Sarka (Alinezhad and Khalili 2019). The method utilizes the concept of stepwise ranking and evaluation procedure of alternatives as utility degrees. In this method, a separate consideration is made for the effect of minimizing and maximizing criteria on the assessment of results (Gholami et al. 2020). The extension of the techniques to an uncertain environment increases the applicability of this method. The method estimates the uniqueness of one alternative against another (Zavadskas et al. 2009). The method is easy to apply and understand; therefore, the technique can be successfully applied to several decision-making problems in diverse engineering fields.

Step1: construct the q-ROPF evaluation matrix \(R={\left[{A}_{ij}^{\lambda}\right]}_{m\times n}=\left({\mu}_{ij}^{\lambda },{\upsilon}_{ij}^{\lambda}\right)\), i = 1, 2, ……. m, j = 1, 2, …. . n, as below:

$${C}_1\kern4.25em {C}_2\kern2em \dots .\kern4.25em {C}_n$$
$$R={\left[{A}_{ij}^{\lambda}\right]}_{m\times n}={\displaystyle \begin{array}{c}{A}_1\\ {}{A}_2\\ {}\vdots \\ {}{A}_3\end{array}}\left[\begin{array}{cccc}\left({\mu}_{11}^{\lambda },{\upsilon}_{11}^{\lambda}\right)& \left({\mu}_{12}^{\lambda },{\upsilon}_{12}^{\lambda}\right)& \dots & \left({\mu}_{1n}^{\lambda },{\upsilon}_{1n}^{\lambda}\right)\\ {}\left({\mu}_{21}^{\lambda },{\upsilon}_{21}^{\lambda}\right)& \left({\mu}_{22}^{\lambda },{\upsilon}_{22}^{\lambda}\right)& \dots & \left({\mu}_{2n}^{\lambda },{\upsilon}_{2n}^{\lambda}\right)\\ {}\vdots & \vdots & \vdots & \vdots \\ {}\left({\mu}_{m1}^{\lambda },{\upsilon}_{m1}^{\lambda}\right)& \left({\mu}_{m1}^{\lambda },{\upsilon}_{m2}^{\lambda}\right)& \dots & \left({\mu}_{mn}^{\lambda },{\upsilon}_{mn}^{\lambda}\right)\end{array}\right]$$
(14)

where \(R={\left[{A}_{ij}^{\lambda}\right]}_{m\times n}=\left({\mu}_{ij}^{\lambda },{\upsilon}_{ij}^{\lambda}\right)\), i = 1, 2, ……. m, j = 1, 2, …. . n,   denotes the q − ROPF information of alternatives Ai(i=1, 2,……m) on attributes Ci(j = 1, 2, ……m) by expert dλ.

Step 2: using q-ROPFWBMO, we can utilize overall \({A}_{ij}^{\lambda }\) to get Aij. The aggregated q-ROFN matrix r = [Aij]m × n is given below:

$$r={\left[A^\lambda\right]}_{m\times n}=\begin{array}{c}\;\;\;\;\;C_{1\;}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;C_2\;\;\;\;\;\;\;\;\;\;\dots\;\;\;\;\;\;\;\;C_3\\ \begin{array}{c}A_1\\A_2\\ \vdots\\A_m\end{array}\begin{bmatrix}\left(\mu_{11},\upsilon_{11}\right)&\left(\mu_{11},\upsilon_{12}\right)&\dots&\left(\mu_{1n},\upsilon_{1n}\right)\\ \left(\mu_{21},\upsilon_{21}\right)&\left(\mu_{22},\upsilon_{22}\right)&\dots&\left(\mu_{2n},\upsilon_{2n}\right)\\ \vdots&\vdots&\vdots&\vdots\\ \left(\mu_{m1},\upsilon_{m1}\right)&\left(\mu_{m2},\upsilon_{m2}\right)&\dots&\left(\mu_{mn},\upsilon_{mn}\right)\end{bmatrix}\end{array}$$
(15)

where Aij = (μij,υij,) denotes the fused q-ROPF information Ai(i = 1, 2, ……m) on attributes Gj(j = 1, 2, ……n).

Step 3: normalize the matrix [Aλ]m × n based on the type of attributes by the given formula:

For beneficial attributes:

$${N}_{ij}={A}_{ij}=\left({\mu}_{ij},{\upsilon}_{ij}\right)$$
(16)

For cost attributes:

$${N}_{ij}={\left({A}_{ij}\right)}^c=\left({\upsilon}_{ij},{\mu}_{ij}\right),i=1,2,\dots \dots m,j=1,2,\dots \dots n$$
(17)

Step 4: obtain the sum of the weighted normalized values for beneficial and non-beneficial attributes by using Eqs. 18 and 19, respectively.

$${D}^{'}= dij=\times ij\ x. wj$$
$${\mathrm{S}}_{\mathrm{i}}^{+}=\sqrt{\sum_{j=1}^k\ {d}_{ij}}\kern0.5em \mathrm{j}=1,2,\dots \dots, \mathrm{k}\ \mathrm{maximizing}\ \mathrm{index}$$
(18)
$$\mathrm{S}{\mathrm{i}}^{-}=\kern0.75em \sqrt{\sum_{j=k+1}^n\ {d}_{ij}}\kern0.5em {\displaystyle \begin{array}{cc}\mathrm{j}=\mathrm{k}+1,\mathrm{k}+2\dots \dots, & \mathrm{n}\ \mathrm{minimizing}\ \mathrm{index}\end{array}}$$
(19)

Step 5: obtain the value of relative weight Qi for the ith by using Eq. 20

$$\kern3em {Q}_i={S}_{+i}+\frac{\min {S}_{-i}\sum_{i=1}^m{S}_{-i}}{S_{-i}\sum_{i=1}^m\frac{\min {S}_{-i}}{S_{-i}}}$$
(20)
$$\mathrm{A}\ast =\left\{{\mathrm{A}}_{\mathrm{i}}|\ \max\ {\mathrm{Q}}_{\mathrm{i}}\right\}$$

Step 6: calculate the value of the performance index Pi for each alternative by using Eq. 21.

$${P}_i=\frac{Q_i}{Q_{\mathrm{max}}}\times 100$$
(21)

Application of q-ROPFWBMO-based SWARA-COPRAS model

The ranking of solid waste management methods is performed against several techno-eco and environmental factors. The different methods of solid waste management, including upcycling (A1), recycling (A2), pyrolysis (A3), gasification (A4), core plasma pyrolysis (A5), biotechnological (A6), hydrolysis (A7), composting (A8), incineration (A9), and landfill (A10), are considered as alternatives. The alternatives are evaluated under a set of criteria, namely, cost (C1), eco-friendly (C2), suitability (C3), efficiency (C4), feasibility (C5), capacity (C6), and acceptability (C7). The details of the alternatives and criteria are given in Table 2 and Table 3, respectively. The algorithm followed for ranking the waste management techniques is given in Fig. 5. The linguistic assessments are collected from the decision-makers (DMs) by using a nine-point scale of relative importance given in Table 4. The alternatives and criteria are aggregated by the q-rung fuzzy number-based Bonferroni mean operator (BMO). The aggregated alternative and criteria are given in Table 5 and Table 6, respectively. The relative importance (Sj) is obtained from the aggregated criteria matrix. The coefficient (Kj) is calculated by using Eq. 11, and recalculated weight (qj) is calculated by using Eq. 12. The criteria weight (Wj) is obtained by using Eq. 13, and the results are furnished in Table 7. The calculated values for the criteria weight follow the sequences as C2 > C3 > C7 > C5 > C4 > C6 > C1, having values of 0.134, 0.152, 0.147, 0.140, 0.142, 0.135, and 0.146 for the criteria as C1, C2, C3, C4, C5, C6, and C7, respectively. The aggregated decision matrix for the assessments is given in Table 8. The aggregated decision matrix is normalized by using Eqs. 16 and 17 for beneficial and non-beneficial criteria, respectively which gives the normalized decision matrix in Table 9. The criteria weights are multiplied by the corresponding elements to obtain the weighted normalized matrix in Table 10. The values of relative importance (Qi) is obtained by using Eq. 20, and the performance index (Pi) of the alternatives is calculated by using Eq. 21. The alternatives are ranked according to the value of (Pi), and the obtained rankings are furnished in Table 11.

Table 2 Details of alternatives
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Table 3 Details of criteria
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Fig. 5
figure 5

Algorithm of the proposed q-ROPFBMO-based SWARA-COPRAS

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Table 4 The scale of relative importance for q-ROPFNs
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Table 5 Aggregation of alternatives
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Table 6 Aggregation of criteria
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Table 7 Weight of the criteria by using SWARA
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Table 8 Aggregated decision matrix
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Table 9 Normalized decision matrix
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Table 10 Weighted normalized decision matrix
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Table 11 Ranking of alternatives by COPRAS method
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Results and discussions

The rankings of the alternatives through the proposed q-ROPF number-based SWARA-COPRAS model are plotted in Fig. 6, which indicates that the ranking results follow the sequences as A1 > A2 > A3 > A7 > A6 > A5 > A9 > A8 > A4 > A10. The alternatives A1 and A2 obtained approximately equal values of Pi; therefore, the observation establishes upcycling and recycling of wastes as suitable options for managing solid waste. Conversely, the lowest value of Pi for the alternative A10, i.e., landfilling, is the least suitable practice of waste management. Upcycling is an effective approach for the treatment of waste with less requirement for fresh energy and resources and is comparable to recycling; this practice is rarely performed due to the lack of awareness and technical advancement. Moreover, the discarded products are of very low quality, which creates a hindrance for the development of new product, besides increasing the overall cost of the final product.

Fig. 6
figure 6

Ranking of the alternative by q-ROPFN-based SWARA-COPRAS method

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The benefits obtained from the upcycling and recycling of solid waste are as follows:

  • The upcycling and recycling of solid wastes consume less amount of energy and resources for the transformation of waste materials into new products.

  • The economic viability of the waste product can be enhanced by upcycling and recycling waste. Moreover, the developed products through upcycling and recycling are novel and sustainable.

  • The upcycling and recycling of solid waste produce less amount of by-products or residues; therefore, these are eco-friendly. Moreover, the methods are feasible and capable of handling a wide range of waste products.

  • These practices possess an ample opportunity for employment generation through the engagement of more workers in the development of new products and improve the social condition of a nation. Moreover, the methods improve the environmental conditions in an economical manner.

  • The methods are more sustainable as compared to the other methods in the study and pave the way for the socio-eco-environmental development of the globe.

The ranking results of the proposed q-ROPF number-based SWARA-COPRAS model are compared with the ranking of the other well-known techniques, namely, q-ROPF number-based SWARA-TOPSIS, SWARA-MOORA, SWARA-ARAS, SWARA-WSM, SWARA-WPM, and SWARA-WASPAS. The ranking of the alternatives obtained through the other techniques is summarized in Table 12 and plotted in Fig. 7, which has identified the alternatives Al and A2 as suitable methods for solid waste management, whereas alternative A10 is the least preferable method for waste management. Moreover, a majority of the techniques favor upcycling, followed by recycling as an effective technique for waste management. The disposal of waste as dumping or landfilling is the least suitable method for the disposal of solid waste. Incineration is recognized as the second least preferable strategy for waste management by the majority of the well-known techniques. The comparative analyses of the ranking results of the proposed model and other techniques found a similarity in the ranking sequences for the best and worst alternative and, therefore, verifies the reliability of the proposed q-ROPF number-based SWARA-COPRAS model. Furthermore, a slight deviation in the ranking sequences is observed, the difference in algorithms and calculations involved could be the possible cause for such variations in ranking sequences. The robustness of the ranking results is estimated by the calculations of Spearman’s rank correlation coefficient (SRCC) between the proposed SWARA-COPRAS methods and other techniques.

Table 12 Ranking of alternatives through MCDMs
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Fig. 7
figure 7

Ranking of alternatives through MCDM techniques

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The plot for SRCC between the q-ROFN-based SWARA-COPRAS and q-ROFN-based other MCDM techniques in Fig. 8 shows that the SRCC between the q-ROPF number-based SWARA-COPRAS and SWARA-TOPSIS is 0.90303. The SRCC between the q-ROPF number-based SWARA-COPRAS and SWARA-MOORA is 0.9272. Moreover, the SRCC between the q-ROPF number-based SWARA-COPRAS and SWARA-WPM is 0.8787. Due to a similarity in the ranking sequences of q-ROPF number-based SWARA-ARAS, SWARA-WSM, and SWARA-WASPAS, an equal value of SRCC between the proposed model and these methods with SRCC of 0.85454 is observed. The SRCC between the proposed mathematical model and other well-known techniques is more than 0.7; therefore, the results of SRCC verify the robustness of the proposed SWARA-COPRAS model.

Fig. 8
figure 8

SRCC between the proposed q-rung COPRAS and MCDMs

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The influence of the criteria weight in the ranking results is investigated by performing sensitivity analysis (SA) for the different cases of the criteria weights as given in Table 13. The rankings of the alternatives for the considered cases of criteria weights are plotted in Fig. 9, which shows that the ranking results are influenced by the criteria weights. However, for small changes in the criteria weights, i.e., for cases 3, 4, and 5, a similarity in the ranking sequences as A3 > A2 > A1 > A7 > A6 > A9 > A5 > A4 > A8 > A10 is observed. The overall observation for the considered cases established that the criteria weight has a significant role in determining the overall ranking of the alternatives. Thus, the study shows that the proposed q-ROFN-based SWARA-COPRAS can satisfactorily rank the techniques for waste management.

Table 13 Sensitivity analysis with respect to criteria weights
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Fig. 9
figure 9

Ranking of the alternative for the cases of different criteria weights

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Conclusions and future scope

The growing population, coupled with industrialization, urbanization, and industrialization are the responsible factors for an exponentially increasing rate of waste generation. The detrimental effect of solid wastes on habitats and lives necessitates the establishment of waste management plans. The compositions and nature of solid wastes vary with the geographical locations and public behavior toward the wastes, which has encouraged the development of several waste-to-energy techniques and methods for the management of solid wastes. The involvement of several factors in waste management increases the complexity in the identification of suitable techniques for municipal solid waste management and, therefore, can be viewed as a multi-criteria decision-making problem. The research is performed for the identification of a suitable practice for solid waste management under technical, economic, and environmental factors. The proposed q-rung fuzzy number-based integrated SWARA-COPRAS is a reliable approach for ranking waste management techniques. The study has suggested upcycling and recycling as suitable methods for the management of solid waste as these techniques require low energy in their operation and generate fewer by-products for further treatment. The techniques such as pyrolysis and core plasma pyrolysis suffer due to high operational costs and low efficiency. The open burning of wastes as incineration leaves several toxic gases which is harmful to the environment. Hydrolysis and composting are less feasible, while the biotechnical methods are not economic methods of waste treatment. The model recognizes landfilling as the least favorable method for the disposal of waste due to its detrimental effect on the environment. The study has shown that not any single practice of waste management could effectively fulfill all the considered criteria and require trade-offs. The comparative analysis of the results found a good correlation coefficient between the ranking of the proposed model and other MCDM techniques. Moreover, the sensitivity analysis for the criteria weights provides an understanding of the effect of criteria weight on ranking. The model is capable of handling the vagueness and uncertainty in the decision. The study highlights the potential of mathematical models for the identification of suitable practice for the effective management of municipal solid wastes.

The research could assist the decision-makers in the formulation of recommendations for solid waste management with scientific rigor for reliable decision outcomes. The implementation of the proposed model for the identification of a suitable waste management practice is significant for the improvement of the environmental condition. The research could assist policymakers and professionals in making decisions involved in the sector of waste management. The findings of the research could provide a practical guidance in the development of effective waste management practices.

The study is limited to ranking waste management techniques in techno-eco and environmental considerations, whereas there are still ample opportunities for research in different stages of waste management, such as collection, sorting, transportation, and handling. The research can be extended for more reliable mathematical models through the implementation of advanced uncertainty sets for more accurate results. Moreover, other novel methods for aggregations of the assessments can be implemented for more accuracy of the results.