MULTI-ATTRIBUTE GROUP DECISION-MAKING FOR SOLID WASTE MANAGEMENT USING INTERVAL-VALUED 𝑞 -RUNG ORTHOPAIR FUZZY COPRAS

. In this paper, the COPRAS (Complex Proportional Assessment) method is extended for interval-valued 𝑞 -rung orthopair fuzzy numbers (IV 𝑞 -ROFNs) to solve multi-attribute group decision-making (MAGDM) problems. A novel distance measure for IV 𝑞 -ROFNs is proposed, and its properties are also probed. This distance measure is used in an improved weights determination method for decision-makers. A weighted projection optimization model is developed to evaluate the completely unknown attributes’ weights. The projection of assessment values is defined by the positive and negative ideal solutions, which determine the resemblance between two objects by considering their directional angle. An Indian cities’ ranking problem for a better solid waste management infrastructure is solved using the proposed approach based on composite indicators, like recycling waste, greenhouse gas emissions, waste generation, landfilling waste, recycling rate, waste-to-energy rate, and composting waste. Numerical comparisons, sensitivity analysis, and other relevant analyses are performed for validation.


Introduction
In multi-attribute decision-making (MADM) situation, a decision-maker evaluates each alternative for each attribute, weights the characteristics, and chooses the most desirable from all the alternatives.MADM has received wide attention from researchers [37,38,43,51].MADM, a branch of multiple criteria decision-making (MCDM), is now becoming more complex and uncertain due to various vague and uncertain factors.Zadeh's [59] fuzzy set theory was used to deal with uncertain decisionmaking problems.The researchers have extended it into intuitionistic fuzzy [3], Pythagorean fuzzy [54], -rung fuzzy [55], and many other forms [9,22,30].-rung orthopair fuzzy numbers (-ROFNs) are the generalized form of intuitionistic fuzzy numbers (IFNs) and Pythagorean fuzzy numbers (PFNs).In -ROFNs, the membership and non-membership degrees are such that the sum of the th power of membership and non-membership degrees, respectively, should be less than or equal to one.With the advancement in uncertain decision-making methods, DMs can now express more vague and uncertain knowledge about alternatives and attributes.
Through the literature review and looking at the complexities of the decision-making problems, it has been observed that there is neither a single way of expressing uncertainty nor a method that can adequately address all issues of uncertain problems.So, to make the problems more accessible and understandable, various uncertain methods are given in the literature [38,56].Gupta et al. [19] proposed a VIKOR approach in the intuitionistic fuzzy environment for solving the plant location selection problem.Wan et al. [44] proposed multi-objective programming using knowledge measure for Pythagorean fuzzy numbers and determined unknown attributes' weights.Rahman et al. [35] extended Einstein hybrid operators to develop aggregation operators in an intervalvalued Pythagorean fuzzy environment.Anusha et al. [1] proposed Archimedean Copula-based operations for -rung probabilistic dual hesitant fuzzy numbers based on proposed operations; some aggregation operators are also discussed.Wang and Zhou [46] proposed an approach to evaluate the performance of online education platforms using IV-ROFNs.Many researchers contributed to the literature [5,48,50] towards solving uncertain MADM and multi-attribute group decision-making (MAGDM) problems.
In MAGDM, several DMs evaluate the available alternatives concerning the attributes.Thus, in such a way, MAGDM provides more accurate and precise results than problems with a single DM.Gupta et al. [20] proposed a MAGDM approach in a -rung orthopair trapezoidal fuzzy environment; the unknown weights of attributes and DMs were determined using value and ambiguity indexes.Kumar and Chen [28] proposed a MAGDM method to overcome the drawbacks of some existing methods by extending the weighted average aggregation operators for -ROFNs.Joshi et al. [24] introduced the concept of IV-ROFNs and proposed basic operations laws.Recently, IV-ROFNs are getting more attention from researchers [13-15, 23, 25] and are widely used to express uncertain information.An interval-valued -rung orthopair fuzzy (IV-ROF) environment was explored by Gao et al. [13] by extending weighted average operators and the VIKOR method to solve supplier selection problem.Wang et al. [47] proposed some Maclaurin symmetric mean operators and their application in MAGDM with IV-ROF information.Wei et al. [49] extended Heronian mean operators for IV-ROFNs, which considers interdependency among arguments.Wan et al. [45] proposed an integrated group decision-making model for the selection of hypertension follow-up systems in a community hospital using IV-ROFNs; moreover, similarity measure and aggregation operators for the same numbers are also explored.
COPRAS is well known and widely used by researchers [7,39] in uncertain decision-making.It differs from SAW (Simple Additive Weighting) method because it treats benefit and cost attributes separately and avoids the influence of the different attributes' dimensions in decision-making [34].Dorfeshan and Mousavi [10] proposed an integrated TOPSIS and COPRAS approach in the Pythagorean fuzzy environment for solving a critical path selection problem.Krishankumar et al. [27] extended the COPRAS method for the renewable energy technology selection with -rung orthopair fuzzy information.Ramana et al. [36] proposed an integrated COPRAS approach for nonlinear fuzzy data and used the variance method to identify the unknown weights.Narang et al. [31] proposed a hybrid COPRAS and BCM approach using fuzzy set theory for portfolio construction.Jana and Roy [21] proposed a TOPSIS approach for solving a two person game with hesitant fuzzy payoff matrix.Limboo and Dutta [29] constructed a framework for the -rung evidence set and based on that, probability assignment, belief function, and other measures are also discussed.
The distance measure plays an essential role in uncertain decision-making to differentiate two sets of objects; sometimes also used to find similarities between them.Dügenci [11] proposed a generalized distance measure along with proofs of axioms for IFNs and employed it to calculate separation measures.Khan et al. [26] proposed a novel  distance measure for IV-ROFNs and used it in a problem of strategic supplier selection.Zeng et al. [60] proposed two distance measures for -ROFNs and a new method for handling MADM problems.Fu et al. [12] proposed discriminating power in belief distributions to determine the attributes' weights, and the alternatives are compared using the Hurwicz rule.Gupta et al. [18] proposed an optimization model to determine the attributes' weights using the similarity measure; finally, the problem of mobile phone selection is solved in an intuitionistic fuzzy environment.
Projection is used to define the similarity between two objects by taking into account the angle between objects.Xu and Hu [53] proposed a projection model for an intuitionistic fuzzy environment to solve a MADM problem.Xu and Da [52] used a projection model to evaluate the weights of attributes and illustrated the properties of the projection formula.Zhang et al. [62] proposed a MADM approach in a trapezoidal intuitionistic fuzzy environment; alternatives are ranked using relative closeness coefficients defined by the grey relational projection method.In a software quality evaluation problem, Yue [57] proposed a normalized projection measure to identify the distance and angle between two objects.
Waste management is the collection, transportation, processing, and disposal of waste materials.It includes various activities, such as waste reduction, recycling, composting, incineration, and landfilling.The goal of waste management is to reduce the negative impact of waste on the environment and human health.Coelho et al. [17] presented a literature review for waste management using MCDM techniques.In their study, it has been found that the MCDM approaches have become important and convenient supporting tools as they can handle problems with multiple dimensions.Tirkolaee et al. [40] proposed robust optimization techniques to minimize total costs arising in various municipal solid waste management activities.Tirkolaee et al. [41] proposed a mixed integer linear programming to optimize several objectives related to managing municipal solid waste.Ghosh et al. [16] proposed a waste management transportation model for agriculture waste in a Pythagorean hesitant fuzzy environment.Torkayesh et al. [42] proposed a multi-objective optimization model to attain multiple goals regarding cost, environment, and sustainability of healthcare waste management.As the report [6] outlined, some Indian cities have undertaken various initiatives to manage waste.Since these cities' infrastructures are complicated, it can be difficult to evaluate them with crisp information.However, the alternatives (or cities) can be efficiently evaluated using uncertain environments.

Motivation for the study
-ROFNs remove the barriers over the sum of membership and non-membership degrees in IFNs and PFNs.Thus, -ROFNs are more generalized and represent more uncertain data than the IFNs and PFNs.IV-ROFNs provide DMs more freedom to choose a membership and non-membership degree in interval form with the advantage of -ROFNs.The above discussion implies that the IV-ROFNs are more suitable for expressing uncertain data in decision-making problems.To our knowledge, no research considers the projection or weighted projection for IV-ROFNs.The weighted projection between elements provides similarity between the objects; thus, reliable and consistent results are obtained for the decision-making problem.The COPRAS method, used to rank the alternatives when the attributes are of different dimensions, needs to be extended for the IV-ROF environment to explore its advantages further.
In MAGDM, the main problem arises in determining the DMs' weights and the weights of the attributes.Sometimes, these weights are predefined, but well-defined approaches are needed when these are partially known or completely unknown.The weight deriving method discussed in [58] utilizes the average of assessment values from the DMs and the complement for assessing the similarity between DMs.However, the exact fluctuation in assessments of the DMs could not be considered with the complement values of the assessments.So, to overcome this issue, we utilize the positive ideal assessments (PIA) and negative ideal assessments (NIA).Next, the projection method provides similarity between two objects considering their angles; thus, it becomes an adaptive and informative method in decision-making.Finally, we use the projection method to define the weights of the attributes, and alternatives are ranked based on COPRAS, the simplest and most widely used method.The issues discussed above and some other valuable facts motivated us for the current work, and the proposed approach demonstrates the following novelties.

The novelty of the proposed approach
The novelties of the proposed approach can be listed as follows: (1) In the present study, the COPRAS method is extended for IV-ROFNs with unknown weights of DMs and attributes.
(2) Distance measure developed in [33] is extended for IV-ROFNs, and the related properties are also discussed.It considers the uncertainty of assessment values through parameter .
(3) DMs' weights determination method [58] is improved by utilizing the PIA and NIA in distance measures to evaluate the similarity indexes for the DMs.The improved weight determination method incorporates the lower and upper fluctuations of DMs' assessments in the form of PIA and NIA.(4) Projection and weighted projection formulae are extended for IV-ROFNs.The weighted projection formula is used to formulate an optimization model for evaluating the unknown attributes' weights.(5) The proposed optimization model yields reliable attributes' weights in both scenarios, whether the attributes' weights are partially known or completely unknown.However, the optimization models [18,58], with completely unknown attributes' weights, are unreliable in certain scenarios as without the lower limits of the attributes' weights, some weights tend to be zero.(6) A real-world case study of Indian cities ranking with better solid waste management infrastructure is solved using the proposed approach.

Organization of the rest of the paper
The rest of the paper is presented in the following manner.Section 2 offers the basic properties and definition for IV-ROFNs and other tools like weight determination for DMs and attributes.Section 3 presents the extended COPRAS method for IV-ROFNs and the stepwise approach.In Section 4, a numerical illustration is given to demonstrate the application of the proposed approach.Sensitivity analysis, comparison results, and managerial insights are provided in the same section.Finally, Section 5 concludes the paper.

Preliminaries
In this section, some basic concepts related to interval-valued intuitionistic fuzzy set (IVIFS), interval-valued Pythagorean fuzzy set (IVPFS), and interval-valued -rung orthopair fuzzy set (IV-ROFS) on the universal set Θ are discussed.
The three-dimensional surface diagrams for the degree of indeterminacy or hesitancy for various values of  are shown in Figure 2. The surface of the indeterminacy space with  = 1 changes depending on the membership and non-membership degrees.
The indeterminacy or hesitancy space depicts a curved surface with increasing values of .With an increase in , the surface's curvature grows.The pattern demonstrates that -ROFNs can treat higher indeterminacy or hesitancy.The constraint on membership and non-membership degrees of intuitionistic fuzzy numbers is violated only when the DM is unaware of the precise information pertaining to membership and non-membership degrees.In order to handle this impreciseness in membership and non-membership degrees, the -ROFNs are a convenient and adaptive way of expressing uncertain information.The -ROFNs are adaptable enough to deal with ambiguous and uncertain data with changing .Definition 4 ( [25]).Let there be three IV-ROFNs ⟩, and G = ⟨[  ,   ], [  ,   ]⟩, then the operations rules are defined as follows: (1) [  ,   ]⟩ be an IV-ROFN, then the score (G) and accuracy (G) functions of G are defined as follows: ⟩ be two IV-ROFNs; (G 1 ) and (G 2 ) by the scores of G 1 and G 2 , respectively; and let (G 1 ) and (G 2 ) be the accuracy degrees of G 1 and G 2 , respectively.Then, if . ., ) be the set of IV-ROFNs with weighting vector  = ( 1 ,  2 , . . .,   )  , such that   ∈ [0, 1] and ∑︀  =1   = 1, then the interval-valued -rung orthopair fuzzy weighted average (IV-ROFWA) operator is defined as follows:

Novel distance measure for IV𝑞-ROFNs
We extend the distance measure for IV-ROFNs based on the distance defined for -ROFNs in [33].The proposed distance measure includes the parameter of uncertainty , which depends on the value of .Let ⟩ be two IV-ROFNs, then the novel distance measure is defined as follows: where the parameter . Parameter  is the shape parameter for the area formed with the variation in .The area for the equation 0 ≤   +   ≤ 1 changes with variation in values of , and thus the uncertainty space for any measure varies with  for -ROFNs and their extensions.As discussed in [33], the parameter  provides the center of gravity for the shape formed with th order.For  = 1, -ROFNs become IFNs, and the area becomes a triangle PQR.Then the parameter  = 1 3 using the above formula.For  = 2, -ROFNs become PFNs, and the area becomes a quarter circle.Consequently, using the formula  = 0.4242.For  → ∞, the area tends to a rectangle PQRS shape and by using limits in the formula  ⇒ 1 2 .It can be seen in Figure 3 (for the reference see [33]) how the shape parameter  varies over the variation in .
For three numbers G 1 , G 2 , and G 3 , we have Similarly, the second, third, and fourth terms for distance measure (Eq.( 7)) can be written as follows: and From equations ( 8) to (11), we have Hence the distance measure, equation ( 7) follows all the properties stated in Theorem 1.

Projection for IV𝑞-ROFNs
The projection model considers the distance between two objects and the angle between them; this study will combine the projection model with a case similarity measurement to find the most comparable instance.The projection formula based on IVIFNs [53] is extended for IV-ROFNs and is used to determine the weights of the attributes.Module values and cosine angles for IV-ROFNs are defined to determine the projection values.
Let K  = (G 1 , G 2 , . . ., G  ),  = 1, 2, . . .,  be the set of IV-ROFNs, then the module of K  is defined by For an IV-ROFN G = ⟨[  ,   ], [  ,   ]⟩, if the deviation between [  ,   ] and [  ,   ] is greater, then the value of membership terms gets bigger, and the value of non-membership terms gets smaller, and thus IV-ROFN G gets greater.Definition 9. Let K  = (G 1 , G 2 , . . ., G  ) be the assessments for th alternative, and the included cosine angle between K  and K * .cos(K  , K * ) reflects only the similarity measure of the direction of K  and K * , as we consider that the vector is composed of direction and module.Below, the projection of K  on K * provides the similarity degree between K  and K * .
Obviously, for the greater value of Proj K * K  , the K  is closer to the IV-ROFN K * , and thus, the th alternative is better.
The weighted module of the alternative K  = (G 1 , G 2 , . . ., G  ) is given by where  = ( 1 ,  2 , . . .,   ) is the attribute weight vector such that   ≥ 0 and ∑︀  =1   = 1.Similarly, the weighted module for IV-ROFNs Definition 10.The weighted projection of K  on the positive ideal solution (PIS) K + and the weighted projection of K  on the negative ideal solution (NIS) K − can be defined, respectively, as follows: and In the above definition, the PIS is defined as follows: Similarly the NIS is defined as follows: Example 1.Consider the following decision matrix given in Table 1.The associated weights of   ,  = 1, 2, 3, 4 are (0.35, 0.3, 0.2, 0.15).All the   are of benefit type, which means more the better type.Then, the weighted projection of elements according to Definition 10 on PIS and NIS, respectively, are given as follows.
Using equations ( 20) and ( 21), the PIS and NIS, respectively, are obtained as For  = 3, Thus, the weighted projection of ℵ 1 on PIS and NIS is calculated using equations ( 18) and ( 19), respectively.Similarly, the weighted projection of ℵ 2 and ℵ 3 on the PIS and NIS solution can be calculated.As we know, in mathematics, projection is defined as the mapping of a set into a subset and is equal to its square for mapping composition.Moreover, in the fuzzy set theory, projection measures the similarity between objects with directional angles.Thus, the above-calculated projection values show the similarity measure of ℵ 1 from PIS and NIS, respectively.
Remark 2. The weight vector  = ( 1 ,  2 , . . .,   ) is unknown in the proposed approach.Therefore, the weighted projection is used to formulate an optimization model to evaluate these unknown weights.

Weights of the decision makers
First, a MAGDM problem is defined in terms of IV-ROFNs.There are  numbers of alternatives ∆ = (ℵ 1 , ℵ 2 , . . ., ℵ  ) to be evaluated by  DMs with respect to  attributes Λ = ( 1 ,  2 , . . .,   ).The th DM provide his assessments in terms of IV-ROFNs to th alternative with respect to th attribute and is denoted by The positive distance matrix × between th DM's assessment matrix   and PIA matrix  + is obtained using equation ( 7) as follows: The negative distance matrix × between th DM's assessment matrix   and NIA matrix  − is obtained using equation (7) as follows: The average positive matrix distance d(︀   ,  + )︀ and average negative matrix distance d(︀   ,  − )︀ are calculated using the following relations: The similarity index (SI  ) for the th DM is calculated as follows: Using the similarity index, the weights of the DMs are defined by the following relation: The similarity index is the ratio of the average of distance matrix from the NIA matrix and the sum of the averages of distance matrices from PIA and NIA.The similarity index considers the deviation of an individual DM's assessment from the best (PIA) and the worst (NIA) assessments given by the DMs.Hence, the weights of the DMs are based on their expertise and knowledge variation in individual assessments.

Attributes' weights determination
Whenever the attribute weights are unknown in a decision-making problem, it becomes essential to determine them to reach the final decision.In this subsection, a weight determination model is constructed considering the case similarities and the uncertainty of emergencies.The nearness degree can be defined using attribute similarity based on projection.Definition 11.If  (C  ) denotes the nearness degree, then the computation formula according to TOPSIS may be stated as follows: the bigger  (C  ), the better th alternative.
Thus, the sum of weighted nearness degrees of all alternatives can be defined as follows: We argue that the maximum entropy principle may be utilized to quantify information uncertainty in determining unknown attributes' weights.The highest entropy indicates that the amount of unclear data is modest.It is ideal for demanding as little information as possible in addressing the issue; hence the solutions with the highest entropy should be chosen.As a result, to determine the attributes' weights, we have the following optimization model: where  ∈ [0, 1] is the balancing factor and is known prior.By solving optimization model (30), the optimal attribute weights may be obtained  = ( 1 ,  2 , . . .,   ).
It is important to note that the projection measure is not just a measure of similarity between two objects but also the inclination (directional angle) of one object towards another.Thus, in determining attributes' weights, the projection measure includes the directional angle of the attribute toward PIS and NIS.Moreover, from the literature [12,18], it has been observed that the optimization models used for the purpose require partial information regarding attributes' weights; without such information, some weights may become zero because there is no lower limit constraint.With the proposed optimization model, the case of completely unknown and partially known attributes' weights can be handled efficiently.

Extended COPRAS method for IV𝑞-ROFNs
In this section, based on the problem described in Section 2.3 for IV-ROFNs, the extended COPRAS is discussed to solve the decision-making problem.
In order to address MAGDM problems, we modify the standard COPRAS method to handle IV-ROFNs.According to Zhang [61], group decision-making problems are solved in two stages (i) aggregation stage, where the individual assessment values are aggregated to obtain collective assessment values, and (ii) exploitation stage, where aggregated assessment values are prioritized to select best (optimal) alternatives.In the same context, the stepwise COPRAS method is described as follows: Step 1.The aggregated assessment matrix (︀ Ḡ )︀ × is obtained by using IV-ROFWA operator (6), where Ḡ is given by Step 2. In this step, cumulative assessments for benefit and cost type attributes are obtained using equation ( 6) as follows: where  ′ = 1, 2, . . .,  ′ ∈ Benefit type attributes.Similarly, for cost type attributes: where  ′′ = 1, 2, . . .,  ′′ ∈ Cost type attributes.
Step 3. By applying the score function defined in Definition 5, the scores for G & Ĝ are calculated for comparison and are denoted by , respectively.Step 4. The relative significance   of th alternative is defined as follows: where . The relative significance shows the level of satisfaction for each alternative.Obviously, the greater the significance of an alternative, the stronger its importance.
Step 5.The utility degree U  can be calculated using relative significance values   of alternatives.The utility degree U  for the th alternative can be calculated as follows: where M = max is all alternatives' maximum relative significance value.For higher utility value U  , the th alternative will rank higher than the available alternatives.Step 6.The alternatives are ranked based on alternatives' utility values U  .
The hierarchical structure of the proposed approach is depicted in Figure 4.The blocks in the diagram show four stages of the proposed approach.

Numerical illustration
Solid waste is one of the most pressing issues facing the world today, and it is being produced at an alarming rate.The rise in industrialization and population density contributed significantly to this initiative in nations like China and India.In general, solid waste can be divided into three types.Municipal solid waste comes up at the top, followed by industrial solid waste and biomedical/hospital solid waste.Solid waste contains various dangerous compounds, including poisonous, highly combustible, and corrosive ones.In addition, these substances can occasionally react with the environment.As a result, these wastes have various effects on human and animal life and the environment by emitting GHGs into the surrounding area.
Solid waste management must be adequately implemented.Every country faces environmental changes and current epidemic effects on human life.In such circumstances, every country is making its best effort to manage solid waste adequately.For example, India has started many initiatives to lessen the consequences of solid waste, including recycling waste, properly disposing of municipal and biomedical waste, and establishing limits on the amount of solid waste that specific industries are allowed to produce.
In this study, five Indian cities are compared to determine which have the best and worst solid waste management infrastructures.In 2021, the Centre for Science and Environment and NITI Aayog, India published a report about the best practices in municipal solid waste management [6].The report discusses various efforts and technologies and their combined effects on solid waste reduction in different cities.Therefore, we selected some cities in the field to get the insides of the waste management infrastructure and to rank the cities in their practices.A brief description of the selected cities in the present study follows.
(1) Bengaluru (ℵ 1 ): Over the past few years, monitoring waste management practices has been a significant challenge.To monitor waste management services and synchronize coordination among different concessionaires in 2020, the Bruhat Bengaluru Mahanagara Palike (BBMP) implemented some information, com-munication, and technology (ICT) solutions, including a radio frequency identification-based attendance system and geotagging of collection routes.A mobile-based application called Ezetap has also been designed to monitor garbage-vulnerable points and impose penalties.As a result of these ICT-based solutions, the BBMP has achieved 100% door-to-door garbage collection and has eliminated all garbage-vulnerable points in the city.( 2) Pune (ℵ 2 ): In a first-of-its-kind initiative, citizens, workers, and administration all took responsibility for the Red Dot campaign, setting an example for other cities.A well-planned collection, channelization, and disposal system were used to accomplish this.In order to produce value-added products from their sanitary waste, the city administration is exploring state-of-the-art technology.(3) Gangtok (ℵ 3 ): In order to minimize the environmental and health hazards associated with plastic waste pollution, Gangtok adopted an alternative strategy.The state of Sikkim was the first in India to ban disposable plastic bags in June 1998.The city government could enforce the ban by following up with awareness and enforcement activities.Stakeholders were equipped with the knowledge and skills to effectively contribute to reducing waste in the city.( 4) Mysuru (ℵ 4 ): In Mysuru, the zero-waste management plants received solid waste from five wards on average and segregated it into biodegradable fractions.In the field of biodegradable waste recycling, the city is a torchbearer.Science-based methods are used to convert collected biodegradable waste into compost.
The local farmers and horticulture departments further utilize the waste compost.( 5) Indore (ℵ 5 ): The city had a robust communications strategy to affect behavioral change at the mass level.
Citizens were motivated to embrace segregation in order to achieve this goal.The city developed a route plan based on population and waste generated in each ward after segregation was achieved.
The five selected cities denoted by ℵ 1 , ℵ 2 , ℵ 3 , ℵ 4 , and ℵ 5 are evaluated by four DMs concerning seven attributes taken from the literature [4] and are briefly defined as follows: (a) Recycling waste ( 1 ): This attribute measures the proportion of recyclable waste produced annually in a particular city.The attribute is categorized as a benefit type since recycled materials are no longer dangerous.(b) GHG emission from waste ( 2 ): This attribute estimates the amount of  2 and other GHG released into the environment per million people.Since these GHG are dangerous to humans and the environment, the attribute is of cost category.(c) Waste generation ( 3 ): The attribute measures the amount of waste a specific city produces; it contains all types of waste, whether in any form.The maximum amount of waste is more hazardous to the surroundings and environment; thus, the attribute is cost type.(d) Landfilling waste ( 4 ): The attribute measures the total amount of waste from the total waste generated and disposed of on landfill sites.The more landfilling waste, the more land requires for disposing of; the attribute is of cost type.(e) Recycling rate ( 5 ): This attribute measures the amount of waste recycled as a proportion of the total waste generated.The attribute is of benefit type.(f) Waste to energy rate ( 6 ): A portion of the waste produced by various sources is utilized to produce energy, either directly or indirectly.This attribute measures the waste-to-energy generating ratio.The attribute is of benefit type.(g) Composting waste ( 7 ): It measures the total amount of waste that can be composed.Composting is the natural process of transforming waste materials into compost, an ingredient for humus-rich soil.The attribute is of benefit type.
Four DMs evaluate the available five alternatives concerning the given seven attributes using IV-ROFNs.The IV-ROF assessment values provided by DMs are given in Table 2.
In the present decision-making problem, the weights of the DMs and attributes are unknown.Therefore, in order to proceed to choose the best available alternative, we need to identify the numerical weights of DMs and attributes.DMs' weights determination: the weights of the DMs are calculated using the process given in Section 2.3.
A detailed stepwise calculation is given as follows.All the given calculations are performed for parameter  = 3; results for other values of  are discussed in the sensitivity analysis section.First, using equation ( 22), the PIA and NIA matrices are calculated for the DMs, given in Table 3.The distance between individual DMs' assessments and respective PIA and NIA assessments are calculated using equations ( 23) and (24), respectively, and the collective distance matrices are given in Table 4.
Next, as the weights of the DMs are evaluated, now the proposed approach is illustrated stepwise to solve the given problem as follows: Step 1.The aggregated assessment matrix is obtained using equation (31); the above-calculated weights of the DMs are used.The obtained aggregated matrix is given in Table 5.
Attributes weights determination: as the weights of the attributes are unknown, we follow Section 2.4 to determine the unknown attribute weights.First, the PIS and NIS for attributes are calculated using equations ( 20) and ( 21), respectively, and are given in Table 6.In the present study, the attributes  2 ,  3 ,  4 and  5 are the cost type attributes; the rest are of benefit type.With the help of weighted projection values of assessments to the PIS and NIS calculated using equations ( 18) and (19), respectively, the optimization model ( 30) is formulated as follows: Then for  = 0.5, the above optimization model ( 36) is solved using LINGO 18.0, and the obtained optimal attributes' weights are ( 1 ,  2 , . . .,  7 ) = (0.1327, 0.1776, 0.1665, 0.1541, 0.1202, 0.1192, 0.1297).We now have the attributes' numerical weights to move to the next step of the COPRAS method.
Step 2. The cumulative assessments for the benefit and cost type attributes are calculated using equations (32) and (33), respectively.The cumulative assessment values for benefit and cost type attributes are given in Table 7.
Step 3. The score values for G and Ĝ are calculated using Definition 5.As a result of the cumulative assessment of benefit and cost type attributes are  (︁ G )︁ = (0.4150, 0.4078, 0.3776, 0.4036, 0.4518) and  (︁ Ĝ )︁ = (0.4286, 0.4240, 0.4141, 0.4060, 0.4257), respectively.Step 4. The relative significance   for all the alternatives is calculated using equation (34).The obtained values of relatives closeness for alternatives are:  1 = 0.8258,  2 = 0.8231,  3 = 0.8027,  4 = 0.8372, and  5 = 0.8654.Step 5. Finally, using equation (35), the utility degrees for the alternatives are calculated and are (in %): U 1 = 95.4188,U 2 = 95.1059,U 3 = 92.7543,U 4 = 96.7412,and U 5 = 100.Step 6.The alternatives are ranked according to their utility degrees.The ranking of alternatives is From the above analysis, Indore (ℵ 5 ) is the best alternative in terms of considered attributes and has better solid waste management.The second best alternative is ℵ 4 , Mysuru, having better infrastructure for solid waste management.The ranking results may differ in different scenarios.These results depend on the used parameters, techniques, and environment, mainly on the assessments from the DMs.

Sensitivity analysis
In IV-ROFNs, the parameter  plays a more significant role in assessing the assessment values.The choice of parameter  depends on the available assessment values.For IVROFNs,  is chosen such that the restriction  over the membership and non-membership degrees is satisfied.Here we show the sensitivity of the proposed approach for different values of parameter .The obtained ranking for various  shows that the best alternative remains the same for all considered values of , while other alternatives' rank positions for higher and lower values of  vary.In the present problem, the assessments are not so vague that a higher value of  will be needed; therefore, the difference in ranking occurs due to the parameter choice.Detailed ranking results for different values of  are given in Table 8.In Figure 5, the variation of alternatives ranking can be seen for the considered values of parameter .
The second parameter used in the proposed approach is  in the attributes' weights determination optimization model.The variation in this parameter can affect the attributes' weights, and simultaneously the ranking of the alternatives can also be affected.We solve the optimization model (36) for different values of .The results show that for different  values, the ranking changes at  = 0.6 and then becomes constant, while a variation in the attributes' weights has been observed for all values of .The varying attributes' weights are given in Table 9, and the alternatives' rankings are also shown.Figure 6 indicates that every attribute weight follows a trend either in ascending or descending order, with an increase in the value of .In Figure 6, on the -axis, the values of  and on  -axis, the attributes' weights (  ,  = 1, 2, . . ., 7) are shown.Regardless of benefit or cost types, attributes' weights vary depending on assessment values.
Random attributes' weights variation: By generating attributes' weights at random, we examine how the attributes' weights affect the ranks of the alternatives.The ranks of relevant alternatives are observed for one  hundred randomly generated attributes' weights; all other analytic procedures and settings are left unaltered.Twenty situations with significantly different rankings of alternatives are taken into consideration.Table 10 shows that when the weight of the seventh attribute increases while the weight of the third and fifth attributes decreases, a significant shift occurs in the rank position of the best-performing alternatives.As a result, it is possible to forecast that the main characteristics of the problem under consideration are waste generation, recycling rate, and composite waste.Changes in the weights of these attributes significantly impact the rankings of the alternatives.Figures 7 and 8, respectively, demonstrate the variation in the attributes' weights and the ranks of the relevant alternatives.

Numerical comparison
This section compares the proposed approach with some of the similar MAGDM approaches in the literature.
Comparison with Biswas and Sarkar [5]: Using the proposed approach, we solved an investor selection problem from [5] with IVPFNs.For comparing both approaches, the MAGDM problem from [5] is solved with  = 2 and equal DMs' weights.The ranking with the proposed approach is ℵ 4 ≻ ℵ 5 ≻ ℵ 1 ≻ ℵ 2 ≻ ℵ 3 and the ranking of alternatives from [5] is Both approaches provide the same best alternative and overall alternatives' ranking with a higher closeness coefficient of 0.7, even though the uncertain environment and used approaches are different.
Comparison with Wang et al. [48]: A supplier selection problem in a green supply chain system is solved to compare the proposed approach with [48].The weights of the DMs are known from [48], and attributes' weights are calculated using the proposed approach.The ranking of the alternatives from the proposed approach is The ranking from both approaches shows a higher correlation coefficient of 0.9; both approaches give similar results with higher correlation.
Comparison with Zhao et al. [63]: To compare the proposed approach with [63], we solve the problem of green supplier selection discussed in [63] with IVPFNs.The proposed approach considers known weights of the DMs from [63], parameter  = 2, and the attributes both as benefit and cost types.The ranking of the alternatives from proposed approach is ℵ 3 ≻ ℵ 5 ≻ ℵ 2 ≻ ℵ 1 ≻ ℵ 4 and from [63] is The best alternative is found the same in both approaches; the closeness coefficient between both sets of alternatives shows a higher degree of correlation, 0.9.
Comparison with Gao et al. [13]: MAGDM problem of supplier selection of medical products considered in [13] is solved using the proposed approach.The weights of the DMs are known from [13] and attributes' weights are evaluated using the projection method.While using the COPRAS approach to the problem, we find that all the attributes are of benefit type.The ranking of the alternatives using proposed approach is The best alternative is the same from both approaches, and the rank correlation coefficient between the two rankings is 0.9, showing a higher correlation between the two ranking results.
Comparison with Garg [15]: A MADM problem is solved using the proposed approach, and the results are compared with [15].Using the proposed approach, the alternatives' rankings are similar to in [15].The obtained alternatives' ranking is ℵ 4 ≻ ℵ 2 ≻ ℵ 1 ≻ ℵ 3 .In the compared study, a single DM's assessments were available to evaluate the alternatives, the attributes' weights were unknown, and the attributes were of both benefit and cost types.
The comparative study shows that the proposed approach performs well in different uncertain environments and provides approximate similar ranking results.The variation in the ranking of alternatives is due to the used techniques and tools like aggregation operators and weight determination methods or sometimes due to the biasedness of the known attributes' and DMs' weights.An optimization model determines the attributes' weights in the proposed approach.Thus the weighting properties remain preserved, and the DMs may use preferences over the specific attribute's weights while solving the optimization model.

Comparison of simulated results by different aggregation operators:
We generated random decision matrices one hundred times, and each MAGDM problem was solved using the proposed approach and other available aggregation operators in the literature.The comparison is made between results obtained from the used aggregation operators and some other aggregation operators from the literature, IV-ROFA [24], IV-ROFMSM [48], and IV-ROFHMA [8].To differentiate, the standard deviations between utility values of four compared aggregation operators are collected every time.The standard deviations of utility values for each scenario (i.e., 100 random MAGDM problems) are shown on the line graph (Fig. 9). Figure 9 indicates a higher standard deviation for the used IV-ROFWA operators in maximum scenarios.Thus, it can be seen from Figure 9 the proposed decision framework generates a comprehensive and acceptable set of rank values, which helps DMs prepare backup management for rational decision-making in emergencies.

Managerial implications
The current study assesses the infrastructure for solid waste management in five major Indian cities.We all know that it is impossible to reduce solid waste substantially.However, this amount increases daily due to enhanced industrial production and population density.The study aims to identify the city with the best solid waste management infrastructure among those considered so that other cities can also help create a cleaner and greener environment.Per the attributes mentioned above in the proposed study, city ℵ 5 has the best solid waste management infrastructure.The management or a government organizer might follow the same infrastructure in other cities.As discussed in [6], a clean and green city is not just through the efforts of management infrastructure but also the prominent role of its cooperative citizens.The report shows that better-performing cities follow some modern waste disposal techniques that increase waste recycling; thus, other factors are also adjusted accordingly.
Better-performing cities have been found to use various contemporary waste disposal methods that improve waste recycling; as a result, other criteria are also altered accordingly.Similarly, GHG emissions, waste generation, landfilling, and composite waste will decrease while recycling and waste-to-energy rates will increase.Systematic waste processing includes transportation management and material processing; the main component is sorting material according to various types of plastic waste, biodegradable, landfilling, etc.Several recycling firms, such as Cashify, Attero Recycling, TrashCon Labs, and GSM Plastic Industries, are active in India, intending to reduce waste.The cities which are strict with the system and are constantly making efforts to reduce or utilize the waste are ranked higher using the proposed approach.To better understand the infrastructures and their effects, readers should refer to the report [6].The proposed approach is not limited to the subject at hand; it may also be used in other real-world MADM or MAGDM scenarios in compatible ambiguous settings.

Conclusion
This paper extended the COPRAS method, a well-known MADM/MCDM in literature for IV-ROFNs.In the proposed approach, we use a novel distance measure to evaluate the weights of the DMs.The novel distance measure considers the uncertainty of assessments with the help of parameter .In decision-making problems, the DM's other concern is evaluating the weights of the attributes.We extended the projection method for IV-ROFNs.The projection method evaluates the similarity between two objects and the directional angle between the objects.The extended projection formulae are used to formulate an optimization model using the TOPSIS method to evaluate the attributes' unknown or partially known weights.As a result, the obtained attributes' weights follow all weighting properties and are evaluated based on the similarity and directional angle between the attributes' assessment values.
Finally, a problem of India's cities' ranking in terms of better solid waste management concerning considered composite indicators is solved using the extended COPRAS method.This study aims to identify cities with better infrastructure so that those with worse infrastructure can take action.The best alternative in all the scenarios is Indore (ℵ 5 ); the city has received the cleanest city award six times in a row.As with MADM/MAGDM approaches, it is difficult to say that one approach is better.Comparing the proposed approach with similar approaches in the literature allows us to check its performance and validation.An analysis of the proposed approach's sensitivity and comparative effectiveness is presented in the paper for validation purposes.
Likewise, with all other approaches in the literature, the proposed approach also has some limitations.The proposed approach is not flexible to use uncertain linguistic environments and does not incorporate hesitancy or indeterminacy degree in the process.As a result, we are sometimes restricted to environmental flexibility and lose some information during decision-making due to not incorporating hesitancy.For future work in this direction, the bidirectional projection approaches for IV-ROFNs utilizing the hesitancy degree may contribute to decision-making and provide more accurate and precise results.The projection method has properties that allow it to be used for various ranking approaches.Some statistical tools, such as correlation and covariance, will also be addressed in the future for IV-ROFNs.

Figure 3 .
Figure 3. Shape parameter  in used distance measure.

Figure 4 .
Figure 4.The hierarchical structure of the proposed approach.

Figure 5 .
Figure 5. Ranking for different values of .

Figure 6 .
Figure 6.Weights of the attributes for different values of .

Figure 7 .
Figure 7. Random weights of the attributes.

Figure 8 .
Figure 8. Ranking of alternatives for random attributes' weights.

Figure 9 .
Figure 9.Standard deviations of utility values for different scenarios.

Table 1 .
Assessment values in the form of IV-ROFNs.

Table 3 .
Positive Ideal and Negative Ideal Assessment Values of the DMs.

Table 4 .
Distance Matrices of the DMs' Assessments from Positive Ideal and Negative Ideal Assessments.

Table 6 .
PIS and NIS corresponding to attributes.

Table 7 .
Cumulative assessment values for benefit and cost type attributes.

Table 8 .
Ranking of alternatives for different .

Table 9 .
Sensitivity results for parameter .

Table 10 .
Randomly generated attributes' weights and ranking of alternatives.