Significant risks to human well-being are associated with the economy, society, and environment posed by the fast urban population growth. The buildup of material waste and the challenge of locating appropriate sites for its management and disposal rank among the most urgent problems. Numerous issues, including a lack of space, poor infrastructure, and the environmental risks associated with inappropriate waste management, exacerbate this issue. To maintain sustainable urban development and the well-being of the populace and ecosystem, these issues require immediate attention [1,2]. The problem of solid waste management, which is commonly perceived as a multi-criteria decision-making (MCDM) problem [3,4], entails assessing multiple criteria to determine the best possible solutions. However, it is better to approach it as a multi-attribute decision-making (MADM) problem because of the complexity that multiple parameters and sub-parameters introduce. This enables a more methodical analysis that concentrates on particular characteristics that influence the results of decisions. Prioritizing features, balancing economic, environmental, and social factors, and incorporating stakeholder preferences are all important issues to take into consideration in this situation as they are essential to making wise decisions [5,6]. The main research challenges are (a) how to address expert concerns about approximating different options, (b) how to deal with the inherent ambiguity of attributes and sub-attributes, and (c) how to support experts in freely expressing their opinions, even when those opinions are somewhat vague. Fuzzy parameterization and possibility-graded settings are crucial tools for addressing these problems. These ideas offer a methodical approach to handling expert subjectivity and uncertainty in decision-making. The main objective of the study is to integrate fuzzy set-based numerical frameworks to address these research questions, providing a mathematical approach to expert judgment and model ambiguity. To give readers a better understanding of the suggested method, this section also provides an overview of the theoretical methodology and addresses the motivational factors and background of the study. For researchers, coping with informative ambiguity and uncertainty has been a critical issue. However, various structures, such as the single-valued neutrosophic set (SV-NS) suggested by Wang et al. [7], have been designed to handle this kind of imprecision and incompleteness. The SV-NS is a unique application of the neutrosophic set (NS) [8], allowing it to be used in a variety of academic disciplines. Although it requires more processing than established structures [9–11] its advantages, such as flexibility and dependability, set it apart. The three main parts true, indeterminate, and false belongings grades are operators that take the range [0, 1] as a co-domain and the set of items under inspection as a domain. These parts are free as opposed to the aforementioned structures, and their sum must fall inside the range [0, 3]. There have been numerous studies on SV-NS reported up to this point. However, it is important to note the research done by the researchers [12–15], about the preparations of various measures and their use in different problems. Biswas et al. [16] have proposed a novel way for different techniques by using the various operations of SV-NS. Farid et al. [17,18] have covered usages of decision-related issues utilizing the aggregation operators of SV-NS. With time, it has been noticed that there are some real-world circumstances where specifications play a crucial role. As a result, the idea of soft sets (SS) [19] was developed to offer a parameterization tool for dealing with such circumstances. Then different structures have been developed by the combination of SS with neutrosophic set to form neutrosophic soft set (NSS) [20] and the single-valued neutrosophic soft set (SV-NSS) [21] are introduced since these two structures are unable to handle circumstances needing an approximate soft setting. By providing lower and upper approximations, Marei [22] applies the SV-NSS concept to rough sets to manage data roughness. Debnath [23] introduced a brand-new idea known as inverse SV-NSS to examine its use in decision-related issues by utilizing its operational advantages. When it came to handling situations when there were numerous expert opinions in a single soft set model, the soft set-like structures that had been developed for single expert opinions entirely broke down. This posed a significant obstacle when making decisions. The concept of a soft expert set (SeS) was provided by Alkhazaleh et al. [24] as a solution to the problem of managing different expert opinions in a model. In actuality, they included the viewpoints of several specialists in the domain set. The opinions of the experts have been employed to manage the uncertainty in the parameters. These viewpoints have been interpreted with precision. Then Alkhazaleh et al. [25] also used this arrangement in a fuzzy environment by introducing a fuzzy soft expert set. This was the combination of a fuzzy and soft expert environment. In 2015, Broumi et al. [26] made use of the combination by developing the intuitionistic fuzzy soft expert set. After this Şahin et al. [27] developed the structure of the neutrosophic expert set. The soft set cannot deal with the domain having multiple arguments because it is only for dealing with a domain having a single argument. As a result, a new technique of function with the domain of multiple arguments, which is used to create the hypersoft set (HSS) [28] is suggested. These functions regard the order pairing of disjoint subclasses with attribute values as its domain and approximate alternatives. Kamaci [29] investigated the creation of a new combination of HSS and rough set associated procedures. The different researchers explored the different methods and aggregation operations in a neutrosophic HSS context [30,31]. Saqlain et al. [32] suggested some new structures by using the combination of SV-NS and HSS. Ihsan et al. [33,34] presented and applied concepts of neutrosophic soft sets with the combination of hypersoft expert sets (HSES) [35] in decision-related issues in the direction of authorizing the multiple crucial judgments of specialists. The contributions of researchers [36,37] concerning the use of aim-situation systems, combination methods of uncertain situations, and fuzzy forecasting methods are important and worthy of mention. To deal with the informational hesitant temperament, the idea of possibility degree [38] has been proposed as an alternative to the theory of probability. According to Zadeh [39], the fuzzy set (FS)’s various properties serve as the basis for the idea of possibility degree. Fedrizzi [40] managed numerous optimization modules in a mixed setting of FSs and the idea of possibility degree. The emphasis on the amplifying of distinct characteristics of these ideas was made by Dubois et al. [41]. Many academics are quite interested in the idea of possibility degree for measuring approximation-based uncertainty. To evaluate the study limitations and inevitable nature of the provided work, a few pertinent types of literature are evaluated in the following paragraphs. Fuzzy SS and the idea of possibility degree were merged by Alkhazaleh et al. [42] to define possibility fuzzy SS, where each approximation is given a specific possibility grade. The purpose of this grade is to evaluate how precise approximations are. They also talked about how to use it based on how similarity measures are created. The possibility intuitionistic fuzzy soft set was established by Bashir et al. [43] by expanding the idea of a possibility fuzzy soft set. By utilizing the suggested similarity measures between them, Karaaslan [44] contributed to the construction of the possible NSS and explained in different situations. In the literature, Zhao et al. [45] made a new combination of the idea of possibility degree with different structures of neutrosophic sets. They put up an algorithm by using the different operations of this new structure and used it for different purposes. Rahman et al. [46,47] developed some new structures of the idea of possibility degree. Their aggregation processes and similarity metrics were described. They also used the suggested ideas for decision-related issues like recognition and medical diagnosis. It has been noted that selectors may not always be certain of the attributes to select on a preference basis while making decisions. Put differently, they don’t think the settings were chosen with certainty. Fuzzy parameterization, which is intended to evaluate the unsure attitude of decision-makers towards the choice of parameters, is used to manage this scenario. This evaluation is carried out by computing their approximations once a certain fuzzy membership grade has been attached to the parameters. Fuzzy parameterization was employed in a fuzzy soft set context by researchers [48–50] leading to the development of the fuzzy parameterized fuzzy soft set concepts. They talked about the applications for making decisions based on these put-up ideas. This concept was developed by Sulukan et al. [51] to form a new hybrid of fuzzy parameterization and the decision-related issues were covered based on performance values. In Deli’s [52] discussion of several models employing these unsure attributes, he linked the concepts of parameterization and HSS. Qu et al. [53] addressed the resolution of several problems characterized by uncertainty, focusing on approaches that manage ambiguous and imprecise information. The contributions of scholars [54–56] are worth noting regarding the quantification of uncertainties, and indeterminacies.

The greatest environmental change caused by the massive increase in human population is urbanization, which has a detrimental effect on the environment’s capacity to retain its natural stability. As per reports, a substantial quantity of solid waste, which is garbage collected from farms, businesses, municipalities, and other hazardous sources, is generated in every urban area due to various activities conducted by humans and other related species. For municipal administrations, figuring out how to handle and dispose of this much solid waste has proven to be a challenging task. The 3R (Reduce, Reuse, Recycle) method cannot, even when applied for this purpose, get rid of solid waste, including different kinds of materials that fall under specific categories. This specific kind of solid waste is typically handled by land-filling, incineration, and composting, among other well-known methods. But installing all these systems also means considering a lot of other factors, some of which might be social, political, economic, ecological, or environmental. Solid waste can contain a significant amount of organic compounds that can be used to create premium fertilizer for exceptional agricultural yields because it includes both industrial and agricultural waste. Similarly, setting up incinerators is a great way to generate energy, and you can dispose of the waste they generate by putting it in the appropriate type of landfill. Solid waste management and disposal have become recognized as intricate, multidisciplinary issues that need to be taken into account from social, technical, economic, and environmental perspectives.

Since several criteria and sub-criteria are involved, selecting suitable locations is also thought to be an essential step in putting any solid waste settlement techniques (SWST) into place (i.e., which are described in Fig. 1). Price, consistency, practicality, air problem mechanism, and its related factors were taken into consideration by Ekmekçioǧlu et al. [57] and Yildirim et al. [58] while determining the best location for solid waste management.

Since the participation of numerous attributes and sub-attributes, they have viewed the solid waste management challenge as an MCDM. Different techniques have been employed in many research studies to date to solve various environmental issues. In a Pythagorean hesitant fuzzy environment, researchers like using various techniques, Ghosh et al. [59] solved a multi-purpose material mobility issue for solid waste management. Luo et al. [60] used different techniques to evaluate the optimal selection of sites for the installation of an incineration plant. The choice of dump sites was covered by Liu et al. [61] utilizing an integrated strategy involving information skills. By using MADM, a contribution to recycling for environmentally friendly growth was made by Wang et al. [62]. It has been noted that the choice of solid waste site selection attributes and sub-attributes is illogical and vague. As a result, this issue has been covered extensively by authors who used decision-making techniques in contexts resembling fuzzy sets. But more notable and pertinent to this investigation are the studies of Rahimi et al. [63], Liu et al. [64], Mallick [65], Torabi-Kaveh et al. [66], Karasan et al. [67], Ali et al. [68], Hanine et al. [69,70], and Kahraman et al. [71]. To select a suitable solid waste site in uncertain environments, various techniques for decision-making have been applied in these structures. Max-min operators of neutrosophic hypersoft sets have recently been employed by Jafar et al. [72] to choose sites for SWST. A careful examination of the aforementioned research contributions reveals that the subsequent issues cannot be solved together by any analytical framework to solve the solid waste management issue:

(a). Since a typical theoretical approach offers an organized framework for the analysis of complex systems, it is necessary to address parameters and their sub-parametric values simultaneously. Using this method allows for the systematic consideration of interdependencies between parameters and their sub-levels, which facilitates more accurate problem modeling and assessment. This approach ensures a more comprehensive and effective decision-making process by allowing the identification of important factors, the measurement of their effects, and the creation of solutions that account for the intricate relationships between parameters.

(b). Since fuzzy parameterization enables the representation of imprecise information, it is essential for assessing the uncertain and ambiguous nature of multi-argument tuples. Each tuple’s fuzzy membership grade allows for the quantification of the data’s inherent uncertainty and vagueness. With values that are not strictly binary but rather fall within a range that reflects differing degrees of truth or relevance, this approach allows for a more flexible analysis. In situations of uncertainty, fuzzy parameterization enables more accurate modeling and decision-making by offering a nuanced understanding of complex systems.

(c). Since it measures how plausible or feasible these approximations are, the idea of possibility grades is crucial for assessing the degree to which decision-makers approximations are accepted. Rather than giving a definitive answer, possibility grades provide a means to handle uncertainty by assigning values that reflect how likely or possible a given approximation is. This enables more informed and adaptable decision-making in complex and uncertain environments by enabling decision-makers to evaluate and compare various options or solutions based on their potential effectiveness.

(d). A single-valued neutrosophic setting, or a three-dimensional membership function, allows decision-makers to independently express their opinions along various dimensions of analysis, capturing various aspects of the decision-making process. A composite “true grade” reflecting the decisions made by the decision-makers is produced by the function that results from the representation of individual criteria or attributes by each dimension. Because this method takes into account different viewpoints and levels of uncertainty, it can approximate the objects or scenarios being evaluated in a more nuanced manner.

The issue of choosing the solid waste site is a MADM problem, as was earlier mentioned. Thus, if the aforementioned crucial elements are taken into account as a whole, a reliable decision support framework can be created, which ultimately results in the development of a new conceptual framework to satisfy this requirement. The purpose of this study is to present the concepts of ρ^-set, which are fully able to address the previously described issues as one unit. These research’s significant contributions include but are not limited to:

(1). A unique mathematical structure, called ρ^-set, is constructed and characterized to address the potential decision-making concerns related to the degree of possibility, employing a specific degree to parameters, and different settings like SV-NS and hypersoft approximate mapping. By addressing these challenges collectively, it is qualified to aid in the creation of a reliable selecting framework.

(2). By calculating their pertinent fuzzy parameterized grades using the necessary criteria, the inspected attributes and sub-attributes uncertain behaviors are assessed.

(3). A mathematical approach is used to establish the important possibility degrees of alternatives to check the calculation of substitutes related to attributes and sub-attributes depending upon the opinions offered by the experts.

(4). Sanchez’s method, a well-known strategy for making decisions, is altered in the ρ^-set setting.

(5). The ρ^-set aggregates and the modified Sanchez’s approach are combined to create a decision-assisted framework. A strong site selection algorithm is then proposed to control solid waste.

Three main sections comprise the remainder of the paper. To make sure readers are familiar with the fundamental terms used throughout the study, Section 2 offers a review of key terminology and concepts from the pertinent literature. In Section 3, the methodology is presented in detail, including the steps and real-world application of the suggested technique, making it easy to understand how the approach functions. Section 4 presents a discussion of the obtained results, comparison, and sensitivity analysis. Lastly, Section 5 highlights the study’s contributions and possible ramifications while providing a succinct synopsis of the research findings and key conclusions.

This part attempts to go over a few keywords for a correct comprehension of the study that is being presented. The subsequent parts of the paper represent Θ, ω^, Υ, and Ξ as the initial universe of objects, collection of parameters, collection of experts and their opinions, respectively.

Definition 2.1. [7] Consider three different functions βs,βn, and βi defined from Θ to unit closed interval. These are the representation for membership, non-membership and indeterminate functions respectively. If the condition 0≤βs(ϖ)+βi(ϖ)+βn(ϖ)≤3 is satisfied for ϖ∈Θ. Then a set NSV={ϖ,≺βs(ϖ),βi(ϖ),βn(ϖ)≻ϖ∈Θ} is named as a SV-NS set.

Definition 2.2. [35] If ω^1, ω^2, ω^3,…,ω^n, are sub-attribute-valued non-overlapping collections corresponding n different attributes and let Ω be the cartesian product of ω^1, ω^2, ω^3,…,ω^n, and Π=Ω×Υ×Ξ, then HSES represented by Γ can be defined as Γ={(δ,πΓ)∣δ∈Π} where πΓ:Π→Θ.

This section provides a quick description of the stages included in the approved methodology.

This section is meant to discuss the sensitivity analysis of the results obtained and their comparison with the existing models.

This study has provided a novel approach to the selection of suitable SWSTs and locations for the management of solid waste collected from multiple sources within a city. By defining the concepts of ρ^-set, the anticipated ambiguities regarding the attributes, sub-characteristics valued pairs, expert advice based on approximations of substitutes regarding attributes, and sub-attribute grade pairs are addressed. To connect the SWSTs, sites, and sub-attribute valued pairs in the context of the ρ^-set, the traditional method, also known as Sanchez’s method, is modified. To make this adjustment, a ρ^-set matrix representation is used. An explanation and simple example are provided for a method that combines the ideas of ρ^-set and Sanchez’s approach to find the best SWST and location for solid waste control. This approach takes into account the fact that the solid waste management problem is a MADM problem. Two different opinions about the outcomes that were obtained have been expressed: (a) The interdependencies between sites and SWSTs are taken into account when answering the question, “Which of the sites is the most appropriate for any kind of SWST?” (b) The rankings of the site and the SWST are analyzed as independent variables. By comparing the proposed strategy with the most relevant, previously developed structure, it is examined. It is concluded that because of the new approach’s flexibility and reliability in effectively managing uncertainty, it has more positive aspects than previous research. Although this study uses pseudo data on expert opinions and other relevant variables to give a general impression, it can be successfully applied to some case study-based efforts using real data sets.

This work focuses on the fuzzy parameterized degree of sub-characteristic pairs to approximate alternatives; however, numerous other scenarios may require the use of neutrosophic, intuitionistic fuzzy, or picture fuzzy parameterizations of sub-characteristic pairs to evaluate their ambiguous and imprecise characteristics. Since the study employs Sanchez’s technique and expert-graded ideas to measure and estimate SWST with sites, other decision-making methodologies that provide true, value-based feedback from experts through review are also appropriate for these kinds of analyses. Additionally, to give some basic context, researchers only considered the characteristics that the peer literature evaluation deemed most important for MADM-based solid waste development. This avoids mathematical difficulties with the Cartesian product in a hypersoft expert setting. The writers plan to build upon this basic framework in the future by taking into account sets of traits or criteria, making it more global in scope, and utilizing machine learning and other pertinent computer-assisted software. This is because adding more criteria creates numerical challenges that necessitate the use of artificial intelligence and machine learning. Applied mathematics and artificial intelligence are both included in this research.