Solid Waste Landfill Site Assessment Framework Based on Single-Valued Neutrosophic Hybrid Aggregation and Multi-Criteria Analysis

Abstract

Landfills are an effective way to dispose of waste and appropriate landfill sites can lessen environmental damage during waste treatment. Solid waste landfill site (SWLS) selection has received much attention in the area of multi-criteria decision-making in recent years. However, the uncertainty and complexity of the SWLS selection make it a significant challenge for decision makers (DMs). Since single-valued neutrosophic (SVN) sets have the great advantage of handling complex problems with uncertain and inconsistent information, this paper aims at offering a site planning strategy under the SVN environment. For the SWLS selection problem with interrelated factors, the Schweizer–Sklar power Bonferroni mean operator is first created, which not only considers the possible correlations among attributes but also reduces the adverse effects of anomalous assessment information on decision results. Then, a multi-criteria analysis framework based on the aggregation operator is proposed and then applied to a real-world SWLS selection. DMs can flexibly adjust the parameters in this model to achieve a preferred SWLS that integrates economic, environmental, and social perspectives. The consistent results obtained from the comparative analysis highlight its benefits for selecting proper SWLSs.

1. Introduction

Due to the accelerating urbanization process in China in recent years, a series of continuous and steady urban renewal activities have brought about a high growth in construction waste. Medical and hazardous waste has risen dramatically as a consequence of the COVID-19 outbreak. Rapid national economic prosperity has increased municipal solid waste volumes while gradually raising the quality of people’s life. The increase in productivity has not only brought us economic prosperity and improved living standards, but at the same time, solid waste emissions are increasing year by year [1].

Municipal solid waste refers to various materials that are thrown out owing to the lack of original use value in a range of activities such as manufacturing and living, specifically including general industrial solid, hazardous, and domestic wastes [2]. The majority of solid wastes may pose risks to both the ecological balance and physical health [3]. It is impossible to disregard the limitations that solid waste pollution has on both economic and ecological progress. Disposing of waste effectively, reducing waste accumulation, and building waste-free cities have now become a consensus at home and abroad [4].

The landfill is a mature solid waste treatment technology that can be non-hazardous. It is widely used because of its simple process, high treatment efficiency, and low cost [5]. The landfill is the destination of all waste. The waste is concentrated in landfill sites after treatment by various technologies. The most significant stage related to the procedure of disposing of solid waste using landfill technology is choosing a landfill location. Many secondary pollution phenomena may occur when the waste is landfilled, such as groundwater and soil contamination by leachate generated by waste, and degradation of surrounding air quality by odor from waste dumping. Therefore, solid waste landfill site selection (SWLSS) is an effort to prevent health and environmental problems caused by the generated waste leachate and other contaminants [6]. To a certain extent, a scientifically sound solid waste landfill site (SWLS) can prevent waste of national land resources, extensive pollution of water resources, and threats to the health of the surrounding public.

On the one hand, SWLSS involves factors about economic, geological conditions, and environmental protection [7]; on the other hand, comparative analysis between multiple options needs to be considered. Therefore, it can be described as a multi-criteria decision-making (MCDM) problem. MCDM evaluation methods such as the order of preference by similarity to ideal solution (TOPSIS) [8], Vlse Kriterijumska Optimizacija I Kompromissno Resenje (VIKOR) [9], and evaluation based on distance from average solution (EDAS) [10] are widely used when it comes to waste management. Therefore, suitable MCDM methods can also provide technical support for conducting SWLS preferences.

As for the quantitative and unquantifiable evaluation criteria, it is pretty difficult to give precise evaluation values in the case of SWLSS. The actual decision process frequently exhibits this phenomenon. People find it challenging to convey their opinions with specific values because of the complicated decision environment. Fuzzy numbers exactly provide a powerful tool for solving such problems. Zadeh [11] originally introduced fuzzy sets (FSs), which established a precedent for using FSs to solve uncertain issues. Since only membership degrees are contained in FSs, more complex decision problems cannot be solved. Atanassov [12] ungraded them and defined intuitionistic fuzzy sets (IFSs) which contain both non-membership and membership degrees. IFSs, despite broadening the applicability of FSs, have little accuracy when addressing issues of imperfect and inconstant information. Inspired by IFSs, the definition of neutrosophic sets (NSs) was put forth by Smarandache in 1999 [13]. However, the degrees of truth, falsity, and indeterminacy memberships are all defined in non-standard unit subintervals. For application convenience, the three functions were settled to a standard unit interval subinterval by Wang et al. [14], who also formulated the theory of single-valued neutrosophic sets (SVNSs). They are more effective at handling erroneous, inconsistent, and inadequate evaluation data. Therefore, selecting SWLSs with SVNSs can improve decision rationality since it delivers information more delicately.

For tackling the SWLSS problem with interrelated attributes in a single-valued neutrosophic (SVN) environment, based on the proposed SVN Schweizer–Sklar weighted power Bonferroni mean (SVNSSWPBM) aggregation operator (AO) and a novel evaluation index system combining economic, environmental, and social issues, this study aims to propose an MCDM system to select the SWLS with the least environmental pollution. A numerical sensitivity and contrast analysis is applied to verify the stability and applicability of the proposed model. It not only identifies an SWLS preference for decision makers (DMs) but also offers technical support for solving MCDM problems in other areas. The following are the research’s goals and objectives.

(1)

Integrating economic, environmental, and social perspectives, a new set of index systems for evaluating SWLSs will be constructed.

(2)

Combining the advantages of Schweizer–Sklar (SS) operations and power Bonferroni mean (PBM), SVN SS PBM (SVNSSPBM) and SVNSSWPBM AOs are proposed and all their properties and theorems are proved.

(3)

An MCDM framework based on the SVNSSWPBM AO is constructed which builds a solid foundation for the SWLSS.

(4)

The constructed index system and framework are jointly applied to the actual case study of SWLSS, and a comparative analysis with existing decision systems as well as a sensitivity analysis are implemented.

The rest of this essay is organized as follows. The related literature is reviewed in Section 2. The fundamentals of SVNSs and SS operations are covered in Section 3, including basic concepts, operations, and comparison laws. Section 4 presents the SVNSSPBM and SVNSSWPBM AOs and proves all their valuable properties. With reference to existing studies, Section 5 constructs a new index system for SWLSS from three aspects: economic, environmental, and social. Section 6 constructs a framework for solving MCDM problems based on the SVNSSWPBM AO. In Section 7, the constructed index system and the decision framework are jointly used for an SWLSS problem. Sensitivity and comparison analysis offer a compelling justification for this framework’s efficacy. The conclusion section is covered in Section 8.

2. Literature Review

We will review the research on SWLSS and SVNSs and summarize the limitations of the existing research in this section.

2.1. Solid Waste Landfill Site Selection

Many academics are engaged in the application of MCDM approaches to SWLSS. Mousavi et al. [15] used the analytic hierarchy process (AHP) [16] technology to determine the importance of nine important evaluation criteria for SWLSS. Combining the order weighted average and weighted linear combination (WLC) techniques, it was possible to choose a waste disposal site that is close to three cities out of fifteen potential landfill sites. Rehman et al. [17] suggested using a neutrosophic cubic hesitant Dombi exponential AO to integrate the evaluation data to choose an appropriate SWLS. The benefits of hesitant fuzzy and neutrosophic cubic sets were combined by the AO. Bilgilioglu et al. [18] combined a geographic information system (GIS) and the AHP to establish the attributes’ weights and then developed an SWLS suitability analysis model for Mersin in Turkey. The decision-making trial and evaluation laboratory (DEMATEL) and analytic network process (ANP) were utilized by Eghtesadifard et al. [19] to calculate the interdependence and relative importance between indicators, respectively. Finally, multi-objective optimization on the basis of simple ratio analysis (MOORA) and weighted aggregated sum product assessment (WASPAS) were used to rank six alternative municipal solid waste (MSW) sites obtained by the K-means clustering algorithm. Rahimi et al. [20] identified the weights of 14 indicators from multiple perspectives with the help of the best–worst method (BWM) [21]. After layer overlay analysis for preliminary screening of MSW plant locations by GIS, the fuzzy multi-attribute MOORA determined the most suitable address. Alkan et al. [22] created a more powerful distance measure by combining Euclidean and cosine distance. The multi-distance-based evaluation for aggregation dynamic decision analysis and criteria importance through inter-criteria correlation (CRITIC) were extended to IFSs, which offered a decision model for waste disposal site selection.

2.2. SVNSs

Although SVNSs have only just been suggested, MCDM methods in an SVN environment have been relatively well studied. Mishra et al. [23] proposed a generalized Dombi AO for aggregating SVN information. An integrated MCDM framework infused by the removal effects of criteria (MEREC) and multiple objective optimization on the basis of ratio analysis plus full multiplicative form (MULTIMOOR) was designed and a practical case of a low-carbon tourism strategy illustrated its usefulness. Not coincidentally, Hezam et al. [24] extended MEREC and stepwise weight assessment ratio analysis (SWARA) to derive the integrated weights of assessment criteria, and the SVN complex proportional assessment (COPRAS) helped to select the greatest bioenergy production technology. Hezam et al. [25] developed a traffic project investment evaluation method combining a new discriminant measure and COPRAS. To address the problem of end-of-life electronics recycling partner selection in an SVN environment, Rani and Mishra [26] created a novel similarity measure and introduced it into the combined compromise solution (COCOSO) to jointly identify the best partner. To assist India in selecting the finest renewable energy sources, Azzam et al. [27] presented a new decision evaluation framework based on SVN information, and the study showed that the most suitable was wind energy. Rani et al. [28] integrated SVN CRITIC and MULTIMOOR for food waste treatment technology selection.

SVN AOs also play an important role in solving MCDM problems. Various AOs have been proposed to integrate SVN information to lay a solid foundation for the subsequent decision process. To reduce the impact of anomalous information on data integrity, Yang and Li [29] proposed the SVN power average (SVNPA) AO by considering the supporting degree between data. A practical case of investment selection verified its feasibility. Many AOs are proposed when the attributes are independent of each other, the SVN Bonferroni mean (SVNBM) AO proposed by Liu and Wang [30] fully considered the correlation between different attributes in the actual decision issues. However, both of the above AOs were relatively single in function. The SVN Bonferroni power mean (SVNBPM) AO, which took into account both the association among indicators and data integrity, was developed by Wei and Zhang in 2019 [31]. Within the field of research on fuzzy AOs, the algorithms of fuzzy numbers were also highly valued by scholars, and the study [32] investigated the operation principle of SVN numbers (SVNNs) based on SS operations. SS operations contain very effective triangular norms (TNs) and co-norms (TCs), and the parameters involved give it a high level of flexibility. So far it is widely used in operations on fuzzy information. For the case of SVNSs, Zhang et al. [33] studied the SVN SS Muirhead mean (SVNSSMM) AO to provide decision support for company investment, which considered the correlation between multiple input variables and had great computational flexibility. Liu et al. [34] extended the SS prioritized (SSPR) AO considering the prioritization relationship among indicators to SVNSs, and a practical case of talent introduction verified its feasibility. The SVN SS Hamy mean (SVNSSHM) AO, a potent tool for addressing a wider variety of issues, was defined by Yuan et al. [35].

Through a series of extensive study reviews on SWLSS and SVNSs above, we find that there are still some research gaps.

(1)

In the context of SWLSS, although FSs had come to the sight of scholars [22,36], IFSs [22] contained only affiliation and non-affiliation degrees, which cannot reflect the uncertain psychology of DMs. Pythagorean fuzzy sets (PFSs) [36] offered the benefit of being able to represent more information, but they also shared certain drawbacks with IFSs.

(2)

In the SVN environment, the SVNSSMM [33] and the SVNSSHM AOs [35] only considered the interrelationship between multiple input variables but neglected to focus on the data totality, while the SVN SSPR (SVNSSPR) AO [34] was used to deal with MCDM problems including a certain degree of prioritization among attributes. The above AOs have a sole function. To fill the gap of the overall low research on SS-based AOs, more AOs need to be investigated.

(3)

Searching for MCDM methods research on SWLSS, there are few MCDM methods based on AOs.

To remedy these shortcomings, a novel MCDM framework based on SVN AOs will be proposed to be applied for selecting the optimal SWLS. The primary points of this paper are listed below:

(1)

We allow DMs to evaluate alternative SWLSs with SVN information, which presents more advantages in dealing with incomplete, inconsistent, and imprecise fuzzy information compared to other FSs.

(2)

We incorporate SS operations into the PBM AOs. From the view of data integrity, it not only fully takes into account potential inter-correlations among indicators, but also lessens the impact of anomalous information on the accuracy of decision findings. Numerous parameters contained in the SVNSSPBM AO make it extremely flexible. The SVNSSPBM AO is more general since many existing AOs are its special case.

(3)

To resolve the SWLSS problem, the suggested SVNSSWPBM AO is employed. As an important part of MCDM, AOs are relatively simple to apply. DMs can set parameters according to their risk preferences and adjust them according to the different situations in the actual problem.

3. Preliminaries

We first review the basics of SVNSs, PA, BM, and PBM AOs to prepare for the proposed AOs.

Definition 1. 

([37]). Suppose ${\rm E}=\left\{e_1,e_2,\dots ,e_n\right\}$ is a non-empty set, then

$$\Delta =\left\{\langle e,{\varsigma }_{\Delta }(e),{\iota }_{\Delta }(e),{\upsilon }_{\Delta }(e)\rangle |e\in {\rm E}\right\}$$

(1)

is called a single-valued neutrosophic set (SVNS). ${\varsigma }_{\Delta }\left(e\right):{\rm E}\to [0,1]$, ${\iota }_{\Delta }\left(e\right):{\rm E}\to [0,1]$ and ${\upsilon }_{\Delta }\left(e\right):{\rm E}\to [0,1]$ are the truth, indeterminacy, and falsity memberships of the elements $e$ in $E$ belonging to $\Delta$, respectively. For $\forall e\in {\rm E}$, there is $0\le {\varsigma }_{\Delta }\left(e\right)+{\iota }_{\Delta }\left(e\right)+{\upsilon }_{\Delta }\left(e\right)\le 3$. $\Delta =\left\{{\varsigma }_{\Delta }\left(e\right),{\iota }_{\Delta }\left(e\right),{\upsilon }_{\Delta }\left(e\right)\right\}$ is called an SVNN, abbreviated as $\Delta =\langle {\varsigma }_e,{\iota }_e,{\upsilon }_e\rangle$.

Definition 2. 

([38]). Let $e_i=\left({\varsigma }_{e_i},{\iota }_{e_i},{\upsilon }_{e_i}\right)\left(i=1,2\right)$, $e=\left({\varsigma }_e,{\iota }_e,{\upsilon }_e\right)$ be three SVNNs. Assume that $l$ is any real number and $l>0$, the basic algorithms between SVNNs are defined as follows:

(1)

$e_1\otimes e_2=\langle {\varsigma }_{e_1}\times {\varsigma }_{e_2},{\iota }_{e_1}+{\iota }_{e_2}-{\iota }_{e_1}\times {\iota }_{e_2},{\upsilon }_{e_1}+{\upsilon }_{e_2}-{\upsilon }_{e_1}\times {\upsilon }_{e_2}\rangle$;

(2)

$e_1\oplus e_2=\langle {\varsigma }_{e_1}+{\varsigma }_{e_2}-{\varsigma }_{e_1}\times {\varsigma }_{e_2},{\iota }_{e_1}\times {\iota }_{e_2},{\upsilon }_{e_1}\times {\upsilon }_{e_2}\rangle$;

(3)

$e^l=\langle {\left({\varsigma }_e\right)}^l,{\left(1-{\iota }_e\right)}^l,{\left(1-{\upsilon }_e\right)}^l\rangle$;

(4)

$le=\langle 1-{\left(1-{\varsigma }_e\right)}^l,{\left({\iota }_e\right)}^l,{\left({\upsilon }_e\right)}^l\rangle$;

(5)

$e^c=\langle {\varsigma }_e,1-{\iota }_e,{\upsilon }_e\rangle$.

Definition 3. 

([38]). Let $e=\left({\varsigma }_e,{\iota }_e,{\upsilon }_e\right)$ be an SVNN, then its score value $D_e$ and accuracy value $A_e$ are defined as follows:

$$D_e=\left({\varsigma }_e+1-{\iota }_e+1-{\upsilon }_e\right)/3,$$

(2)

$$A_e={\varsigma }_e-{\upsilon }_e.$$

(3)

Definition 4. 

([38]). Let $e_i=\left({\varsigma }_{e_i},{\iota }_{e_i},{\upsilon }_{e_i}\right)\left(i=1,2\right)$ be any two SVNNs, then

(1)

If $D_{e_1}>D_{e_2}$, then $e_1>e_2$;

(2)

If $D_{e_1}=D_{e_2}$, then:

(i)

if $A_{e_1}>A_{e_2}$, then $e_1>e_2$;

(ii)

if $A_{a_1}=A_{a_2}$, then $e_1=e_2$.

Definition 5. 

([39]). Let $e_i=\left({\varsigma }_{e_i},{\iota }_{e_i},{\upsilon }_{e_i}\right)\left(i=1,2\right)$ be any two SVNNs. The Hamming distance HD is defined as

$$HD(e_1,e_2)=\frac{1}{3}\left(\left|{\varsigma }_{e_1}-{\varsigma }_{e_2}\right|+\left|{\iota }_{e_1}-{\iota }_{e_2}\right|+\left|{\upsilon }_{e_1}-{\upsilon }_{e_2}\right|\right)$$

(4)

Definition 6. 

([40]). Let $e_i\left(i=1,2,\dots ,\wp \right)$ be a set of real numbers, then

$$PA\left(e_1,e_2,\dots ,e_{\wp }\right)={\displaystyle \sum _{i=1}^{\wp }\frac{1+\Psi \left(e_i\right)}{{\displaystyle \sum _{t=1}^{\wp }1+\Psi \left(e_t\right)}}{e}_i}$$

(5)

is called the PA operator, where $\Psi \left(e_i\right)={\displaystyle \sum _{j=1,j\ne i}^{\wp }\Xi \left(e_i,e_j\right)}\left(i=1,2,\dots ,\wp \right)$ and $\Xi \left(e_i,e_j\right)=1-HD\left(e_i,e_j\right)$. $\Psi \left(e_i\right)$ is the total support degree between $e_i$ and $e_{j=1,j\ne i}(j=1,2,\dots ,\wp )$ while $\Xi \left(e_i,e_j\right)$ is the support degree between $e_i$ and $e_j$ meets the following conditions:

(1)

$\Xi \left(e_i,e_j\right)\in \left[0,1\right]$;

(2)

$\Xi \left(e_i,e_j\right)=\Xi \left(e_j,e_i\right)$;

(3)

If $HD\left(e_i,e_j\right)\le HD\left(e_m,e_{\wp }\right)$, then $\Xi \left(e_i,e_j\right)\ge \Xi \left(e_m,e_{\wp }\right)$.

Definition 7. 

([41]). Let c and d be non-negative real numbers that are not simultaneously equal to 0 and $e_i\left(i=1,2,\dots ,\wp \right)$ be a set of non-negative real numbers, then

$$BM\left(e_1,e_2,\dots ,e_{\wp }\right)={\left(\frac{1}{\wp (\wp -1)}{\displaystyle \sum _{i,j=1;i\ne j}^{\wp }{\left(e_i\right)}^c{\left(e_j\right)}^d}\right)}^{\frac{1}{c+d}}$$

(6)

is called the BM operator.

Definition 8. 

([42]). Let $e_i\left(i=1,2,\dots ,\wp \right)$ be a set of non-negative real numbers and ${\omega }_i=\frac{\left(1+\Psi \left(e_i\right)\right)}{{\displaystyle \sum _{t=1}^{\wp }\left(1+\Psi \left(e_t\right)\right)}}$, then

$$PBM\left(e_1,e_2,\dots ,e_{\wp }\right)={\left(\frac{1}{\wp (\wp -1)}\underset{i,j=1;i\ne j}{\overset{\wp }{\oplus }}\left({\left(\wp {\omega }_i{e}_i\right)}^c\otimes {\left(\wp {\omega }_j{e}_j\right)}^d\right)\right)}^{\frac{1}{c+d}}$$

(7)

is called the PBM operator, where c and d be non-negative real numbers that are not simultaneously equal to 0.

Definition 9. 

([32]). The SS TN $T_{SS,\kappa }$ and TC $T_{SS,\kappa }^{*}$contained in SS operations are defined as

$$T_{SS,\kappa }={\left(p^{\kappa }+q^{\kappa }-1\right)}^{\frac{1}{\kappa }},$$

(8)

$$T_{SS,\kappa }^{*}=1-{\left({\left(1-p\right)}^{\kappa }+{\left(1-q\right)}^{\kappa }-1\right)}^{\frac{1}{\kappa }}.$$

(9)

$\kappa$ controls the intersection and union of real numbers p and q to become different types of TN and TC, where $\kappa <0$ and $p,q\in \left[0,1\right]$. If $\kappa =0$, there is $T_{SS,\kappa }\left(p,q\right)=pq$ and $T_{SS,\kappa }^{*}\left(p,q\right)=p+q-pq$. At this point, the SS operations degenerate into simple algebraic operations.

Based on the above definitions, we can obtain the SS operations for SVNSs.

Definition 10. 

([32]). Let $e_i=\left({\varsigma }_{e_i},{\iota }_{e_i},{\upsilon }_{e_i}\right)\left(i=1,2\right)$ and $e=\left({\varsigma }_e,{\iota }_e,{\upsilon }_e\right)$ be three SVNNs, where $\lambda$ is an arbitrary positive real number. Then

$$(1)e_1{\oplus }_{SS}{e}_2=\left(\begin{array}{l}1-{\left({\left(1-{\varsigma }_{e_1}\right)}^{\kappa }+{\left(1-{\varsigma }_{e_2}\right)}^{\kappa }-1\right)}^{\frac{1}{\kappa }},\\ {\left({\left({\iota }_{e_1}\right)}^{\kappa }+{\left({\iota }_{e_2}\right)}^{\kappa }-1\right)}^{\frac{1}{\kappa }},\\ {\left({\left({\upsilon }_{e_1}\right)}^{\kappa }+{\left({\upsilon }_{e_2}\right)}^{\kappa }-1\right)}^{\frac{1}{\kappa }}\end{array}\right);$$

$$(2)e_1{\oplus }_{SS}{e}_2=\left(\begin{array}{l}{\left({\left({\varsigma }_{e_1}\right)}^{\kappa }+{\left({\varsigma }_{e_2}\right)}^{\kappa }-1\right)}^{\frac{1}{\kappa }},\\ 1-{\left({\left(1-{\iota }_{e_1}\right)}^{\kappa }+{\left(1-{\iota }_{e_2}\right)}^{\kappa }-1\right)}^{\frac{1}{\kappa }},\\ 1-{\left({\left(1-{\upsilon }_{e_1}\right)}^{\kappa }+{\left(1-{\upsilon }_{e_2}\right)}^{\kappa }-1\right)}^{\frac{1}{\kappa }}\end{array}\right);$$

$$(3)\lambda e=\left(\begin{array}{l}1-{\left(\lambda {\left(1-{\varsigma }_e\right)}^{\kappa }-\left(\lambda -1\right)\right)}^{\frac{1}{\kappa }}\\ {\left(\lambda {\left({\iota }_e\right)}^r-\left(\lambda -1\right)\right)}^{\frac{1}{\kappa }}\\ {\left(\lambda {\left({\upsilon }_e\right)}^{\kappa }-\left(\lambda -1\right)\right)}^{\frac{1}{\kappa }}\end{array}\right);$$

$$(4)e^{\lambda }=\left(\begin{array}{l}{\left(\lambda {\left({\varsigma }_e\right)}^{\kappa }-\left(\lambda -1\right)\right)}^{\frac{1}{\kappa }}\\ 1-{\left(\lambda {\left(1-{\iota }_e\right)}^{\kappa }-\left(\lambda -1\right)\right)}^{\frac{1}{\kappa }}\\ 1-{\left(\lambda {\left(1-{\upsilon }_e\right)}^{\kappa }-\left(\lambda -1\right)\right)}^{\frac{1}{\kappa }}\end{array}\right).$$

Definition 11. 

([32]). Let $e_i=\left({\varsigma }_{e_i},{\iota }_{e_i},{\upsilon }_{e_i}\right)\left(i=1,2\right)$, $e=\left({\varsigma }_e,{\iota }_e,{\upsilon }_e\right)$ be three SVNNs, where $\lambda$ and ${\lambda }_i(i=1,2)$ are three arbitrary positive real numbers. Then

(1)

$e_1{\oplus }_{SS}{e}_2=e_2{\oplus }_{SS}{e}_1$;

(2)

$e_1{\otimes }_{SS}{e}_2=e_2{\otimes }_{SS}{e}_1$;

(3)

$\lambda \left(e_1{\oplus }_{SS}{e}_2\right)=\lambda e_2{\oplus }_{SS}\lambda e_1$;

(4)

${\lambda }_1{e}_1{\oplus }_{SS}{\lambda }_2{e}_1=\left({\lambda }_1+{\lambda }_2\right)e_1$;

(5)

$e_1^{{\lambda }_1}{\otimes }_{SS}{e}_1^{{\lambda }_2}=e_1^{\left({\lambda }_1+{\lambda }_2\right)}$;

(6)

$e_1^{\lambda }{\otimes }_{SS}{e}_2^{\lambda }={\left(e_1{\otimes }_{SS}{e}_2\right)}^{\lambda }$.

4. SVN Schweizer–Sklar Power Bonferroni Mean Operator

In this part, we combine the advantages of SS operations, PA, and BM AOs to propose the SVNSSPBM and SVNSSWPBM AOs. Their properties and theorems will be listed.

Definition 12. 

Suppose p and q are non-negative real numbers that are not simultaneously 0, let $e_i=\left({\varsigma }_{e_i},{\iota }_{e_i},{\upsilon }_{e_i}\right)\left(i=1,2,\dots ,\wp \right)$ is a collection of SVNNs and ${\omega }_i=\frac{\left(1+\Psi \left(e_i\right)\right)}{{\displaystyle \sum _{t=1}^{\wp }\left(1+\Psi \left(e_t\right)\right)}}$, then

$$SVNSSPBM\left(e_1,e_2,\dots ,e_{\wp }\right)={\left(\frac{1}{\wp (\wp -1)}\underset{i,j=1;i\ne j}{\overset{\wp }{{\oplus }_{SS}}}\left({\left(\wp {\omega }_i{e}_i\right)}^c{\otimes }_{SS}{\left(\wp {\omega }_j{e}_j\right)}^d\right)\right)}^{\frac{1}{c+d}}$$

(10)

is called an SVNSSPBM operator, where $\Psi \left(e_i\right)={\displaystyle \sum _{j=1,j\ne i}^{\wp }\Xi \left(e_i,e_j\right)}\left(i=1,2,\dots ,\wp \right)$. $\Xi \left(e_i,e_j\right)$ represents the support degree between $e_i$ and $e_j$ as well as satisfies the conditions in Definition 6.

Theorem 1. 

Suppose c and d are non-negative real numbers that are not simultaneously 0. Let $e_i=\left({\varsigma }_{e_i},{\iota }_{e_i},{\upsilon }_{e_i}\right)\left(i=1,2,\dots ,\wp \right)$ be a collection of SVNNs, then the value $SVNSSPBM(e_1,e_2,\dots ,e_{\wp })$ is also an SVNN.

In addition, the SVNSSPBM AO has several valuable properties such as idempotency, boundedness, and commutativity.

Property 1. 

(Idempotency) Let $e_i=\left({\varsigma }_{e_i},{\iota }_{e_i},{\upsilon }_{e_i}\right)\left(i=1,2,\dots ,\wp \right)$ be a collection of SVNNs, if $e_1=e_2=\dots =e_{\wp }=e$, then

$$SVNSSPBM(e_1,e_2,\dots ,e_{\wp })=e$$

(11)

Property 2. 

(Commutativity) Let $e_i=\left({\varsigma }_{e_i},{\iota }_{e_i},{\upsilon }_{e_i}\right)\left(i=1,2,\dots ,\wp \right)$ be a collection of SVNNs and $(e_1^{’},e_2^{’},\dots ,e_{\wp }^{’})$ is an arbitrary substitution of $(e_1,e_2,\dots ,e_{\wp })$, then

$$SVNSSPBM(e_1,e_2,\dots ,e_{\wp })=SVNSSPBM(e_1^{’},e_2^{’},\dots ,e_{\wp }^{’})$$

(12)

Property 3. 

(Boundedness) Let $e_i=\left({\varsigma }_{e_i},{\iota }_{e_i},{\upsilon }_{e_i}\right)\left(i=1,2,\dots ,\wp \right)$ be a collection of SVNNs, $f^{+}=\underset{i}{\max }{f}_i$ and $f^{-}=\underset{i}{\min }{f}_i$, then

$$f^{-}\le SVNSSPBM(e_1,e_2,\dots ,e_{\wp })\le f^{+}$$

(13)

Next, we propose an SVNSSWPBM AO to aggregate SVN evaluation information with different indicator weights.

Definition 13. 

Suppose c and d are non-negative real numbers that are not simultaneously 0. Let $e_i=\left({\varsigma }_{e_i},{\iota }_{e_i},{\upsilon }_{e_i}\right)\left(i=1,2,\dots ,\wp \right)$ be a collection of SVNNs and the weight vector is $ƛ={\left({ƛ}_1,{ƛ}_2,\dots ,{ƛ}_{\wp }\right)}^T$, then

$$\begin{array}{l}SVNSSWPBM\left(e_1,e_2,\dots ,e_{\wp }\right)\\ ={\left(\frac{1}{\wp (\wp -1)}\underset{i,j=1;i\ne j}{\overset{\wp }{{\oplus }_{SS}}}\left({\left(\frac{\wp {ƛ}_i\left(1+\Psi \left(e_i\right)\right)}{{\displaystyle \sum _{t=1}^{\wp }{ƛ}_t\left(1+\Psi \left(e_t\right)\right)}}{e}_i\right)}^c{\otimes }_{SS}{\left(\frac{\wp {ƛ}_j\left(1+\Psi \left(e_j\right)\right)}{{\displaystyle \sum _{t=1}^{\wp }{ƛ}_t\left(1+\Psi \left(e_t\right)\right)}}{e}_j\right)}^d\right)\right)}^{\frac{1}{c+d}}\end{array}$$

(14)

is called the SVNSSWPBM operator, where $\Psi \left(e_i\right)={\displaystyle \sum _{j=1,j\ne i}^{\wp }{ƛ}_j\Xi \left(e_i,e_j\right)}\left(i=1,2,\dots ,\wp \right)$ and $\Xi \left(e_i,e_j\right)$ represent the support degree between $e_i$ and $e_j$ and meets the conditions in Definition 6. Moreover,${ƛ}_i\in \left[0,1\right]$ and ${\displaystyle \sum _{i=1}^{\wp }{ƛ}_i}=1$.

Theorem 2. 

Suppose c and d are non-negative real numbers that are not simultaneously 0. Let $e_i=\left({\varsigma }_{e_i},{\iota }_{e_i},{\upsilon }_{e_i}\right)\left(i=1,2,\dots ,\wp \right)$ be a collection of SVNNs, then the result $SVNSSWPBM(e_1,e_2,\dots ,e_{\wp })$ is also an SVNN.

The SVNSSWPBM AO shares the same characteristics as the SVNSSPBM AO, such as boundedness, idempotency, and commutativity.

Property 4. 

(Idempotency) Let $e_i=\left({\varsigma }_{e_i},{\iota }_{e_i},{\upsilon }_{e_i}\right)\left(i=1,2,\dots ,\wp \right)$ be a set of SVNNs. If $e_1=e_2=\cdots =e_{\wp }=e$, then

$$SVNSSWPBM(e_1,e_2,\dots ,e_{\wp })=e.$$

(15)

Property 5. 

(Commutativity) Let $e_i=\left({\varsigma }_{e_i},{\iota }_{e_i},{\upsilon }_{e_i}\right)\left(i=1,2,\dots ,\wp \right)$ be a collection of SVNNs. If $(e_1^{’},e_2^{’},\dots ,e_{\wp }^{’})$ is an arbitrary substitution of $(e_1,e_2,\dots ,e_{\wp })$, then

$$SVNSSWPBM(e_1,e_2,\dots ,e_{\wp })=SVNSSWPBM(e_1^{’},e_2^{’},\dots ,e_{\wp }^{’}).$$

(16)

Property 6. 

(Boundedness) Let $e_i=\left({\varsigma }_{e_i},{\iota }_{e_i},{\upsilon }_{e_i}\right)\left(i=1,2,\dots ,\wp \right)$ is a collection of SVNNs, ${\tilde{f}}^{+}=\underset{i}{\max }{\tilde{f}}_i$ and ${\tilde{f}}^{-}=\underset{i}{\min }{\tilde{f}}_i$. We have

$${\tilde{f}}^{-}\le SVNSSPBM(e_1,e_2,\dots ,e_{\wp })\le {\tilde{f}}^{+},$$

(17)

where the weight vector is $ƛ={\left({ƛ}_1,{ƛ}_2,\dots ,{ƛ}_{\wp }\right)}^T$, ${ƛ}_i\in [0,1]$ and ${\displaystyle \sum _{i=1}^{\wp }{ƛ}_i}=1$.

Since the proving process for these theorems and properties are straightforward, they are omitted. Several special cases of the SVNSSWPBM AO are listed in Appendix A.

5. SWLSS Evaluation Index System

To effectively reduce the total amount of waste, some solid waste is transported to landfill sites for disposal. The selection of suitable SWLS is crucial as improper landfill sites can seriously affect urban planning, the surrounding ecology, and the living environment. Whether the SWLS evaluation index is comprehensive or not directly affects the decision results. From the perspective of pollution prevention, we consider human safety and ecological environmental protection, and from the economic perspective, we also need to consider the economic benefits. Referring to the existing studies on SWLSS, we extracted six attributes from three dimensions: total cost, elevation and slope, land use, distance from surface and groundwater, safe distance, and public acceptance.

(1)

Economic: SWLSS should not only comply with regulations and ensure environmental safety, but saving transportation, construction, operation, and land acquisition costs as much as possible is equally important. For cost saving, we choose total cost as an attribute under the economic dimension.

Total cost ($Z_1$): The total cost mainly includes land cost [36] and transportation cost [43]. Land cost is a relatively large part of the cost in the process of SWLSS. The cost is typically influenced by the land type and distance from a metropolitan region. Generally speaking, the cost decreases with increasing distance from a metropolitan center. Using wasteland and abandoned land can not only reduce the cost but also increase the abandoned land utilization rate [14]. Transportation costs are the costs incurred in transferring solid waste to the landfill site which are proportional to the transportation distance. It would be best to locate landfills close to busy roads [44]. The total cost can be chosen as a reference for the price of land and the distance of the landfill site from roads, railroads, waste transfer stations, etc.

(2)

Environmental: Since the SWLS directly affects the surrounding air, water, and natural environment, environmental protection should be guaranteed when selecting a waste disposal site. The environmental factors affecting the SWLSS are elevation and slope, distance from surface and groundwater, and land use.

Elevation and slope ($Z_2$): Too large a slope and elevation will not only increase the preliminary site construction difficulty but also increase the later waste transportation difficulty [45] and more costs will arise from climbing the slope [46]. In addition, the increase in slope and elevation will accelerate the flow of surface water and groundwater. Moreover, pollutants from the waste piled will contaminate surface water and groundwater at a faster rate through subsurface infiltration and surface runoff [47].

Distance from surface and ground water ($Z_3$): There is a large number of pollutants in wastewater and exhaust gases. The wastewater can contaminate groundwater in the form of permeate through subsurface infiltration [48]. In case of improper disposal, rainwater may bring pollutants in the landfill area to the surface and pollute surface water by surface runoff [49]. Landfills far from water sources may be more successful in preventing water contamination from the point of view of protecting surface water and groundwater resources [50].

Land use ($Z_4$): When engineered barriers fail partially or completely, leachate from landfills can have a smaller environmental impact on locations with sound bedrock. Materials with strong adsorption in clayey soils or mud-like rocks have a certain barrier effect [51]. Geologies such as loose rock formations with weak permeability or hard rock formations are suitable for landfill construction [52]. Arid areas are more suitable as landfill sites than wetlands [53].

(3)

Social: The choice of SWLS should not only be subject to the local city’s environmental, general, and sanitation planning but also the consent of the government and the public.

Safe distance ($Z_5$): SWLS construction is often accompanied by a series of problems such as sewage, mosquitoes, flies, and bad odors. The large amount of noise generated during landfill operations can interfere with the residents’ normal life. Therefore, SWLSs should keep a certain distance from residential areas [54], nature reserves and scenic spots, and historical sites [55]. In addition, the SWLSS process should also be guided by the urban master plans and meet the requirements of an urban development plan. Since waste piles attract birds, SWLSS will be safer away from airports [47].

Public acceptance ($Z_6$): The odor and pollution generated when waste is deposited in landfills drive the public to avoid them. An unreasonable SWLS can cause resentment and even conflict when residents resist too much [56]. The authors of [57] showed that a lower-value landfill site is more acceptable to the surrounding residents. If the site is built in compliance with local laws and regulations while increasing the likelihood of employment, the public acceptance of the site will be higher [43].

6. An MCDM Framework Based on SVNSSWPBM Aggregation Method

From the above analysis, it can be seen that the SVNSSWPBM AO combines the advantages of the PA and the BM AOs, which can effectively lessen the negative effects of abnormal information on the final choice in the actual decision process, and also fully considers the interrelatedness of attributes. It also has considerable computational flexibility. We propose an MCDM method based on the SVNSSWPBM AO below.

For the SVN fuzzy MCDM problems, assuming $\mathsf{\Upsilon }=\left\{{\mathsf{\Upsilon }}_i|i=1,2,\dots ,m\right\}$ is a set of alternatives, ${\rm Z}=\left\{{{\rm Z}}_j|j=1,2,\dots ,\wp \right\}$ is a set of evaluation criteria and the weight vector is $ƛ={\left({ƛ}_1,{ƛ}_2,\dots ,{ƛ}_{\wp }\right)}^T$, where ${ƛ}_j\in \left[0,1\right]$ and ${\displaystyle \sum _{j=1}^{\wp }{ƛ}_j}=1$. The expert panel is invited to provide SVN evaluation information. An SVN decision matrix $E={\left(e_{ij}\right)}_{m\times \wp }$ can be obtained after expert consultation. The SVNN $e_{ij}=\left({\varsigma }_{e_{ij}},{\iota }_{e_{ij}},{\upsilon }_{e_{ij}}\right)$ is an evaluation value of the alternative ${\mathsf{\Upsilon }}_i$ given by DMs under evaluation criterion $Z_j$. ${\varsigma }_{e_{ij}}$, ${\iota }_{e_{ij}}$, and ${\upsilon }_{e_{ij}}$ are the true-, indeterminacy-, and falsity-membership degrees of alternative ${\mathsf{\Upsilon }}_i$ with respect to evaluation criterion $Z_j$, respectively.

Figure 1 shows the main steps of the MCDM framework based on the SVNSSWPBM AO.

Step 1. Construct the initial SVN comprehensive evaluation matrix $M={\left(e_{ij}\right)}_{m\times \wp }$.

Step 2. Normalize the comprehensive evaluation matrix $\overline{M}={\left({\tilde{e}}_{ij}\right)}_{m\times \wp }$, where ${\tilde{e}}_{ij}$ is the normalized SVN evaluation values. The comprehensive evaluation matrix $M$ is transformed into a normalized matrix $\overline{M}$ according to the conversion guidelines given in Equation (18):

$${\tilde{e}}_{ij}=\left({\overline{\varsigma }}_{e_{ij}},{\overline{\iota }}_{e_{ij}},{\overline{\upsilon }}_{e_{ij}}\right)=\{\begin{array}{cc}\left({\varsigma }_{e_{ij}},{\iota }_{e_{ij}},{\upsilon }_{e_{ij}}\right)& Z_j\mathrm{is}\mathrm{a}\mathrm{benefit}\mathrm{attribute}\\ \left({\varsigma }_{e_{ij}},1-{\iota }_{e_{ij}},{\upsilon }_{e_{ij}}\right)& Z_j\mathrm{is}\mathrm{a}\cos \mathrm{t}\mathrm{attribute}\end{array}$$

(18)

Step 3. The support degree $\Xi \left({\tilde{e}}_{lk},{\tilde{e}}_{lj}\right)$ between the variables is calculated according to Equation (19) which in turn gives the support degree matrix $S\left(l\right)={\left(\Xi \left({\tilde{e}}_{lk},{\tilde{e}}_{lj}\right)\right)}_{\wp \times \wp }$.

$$\Xi \left({\tilde{e}}_{lk},{\tilde{e}}_{lj}\right)=1-HD\left({\tilde{e}}_{lk},{\tilde{e}}_{lj}\right)$$

(19)

where $l=1,2,\dots ,m$, and $k,j=1,2,\dots ,\wp$.

Step 4. Calculate the weight vector $ƛ={\left({ƛ}_1,{ƛ}_2,\dots ,{ƛ}_{\wp }\right)}^T$ of the evaluation criteria based on the entropy weight method proposed in the literature [58]. The entropy measure ${\Im }_j$ and weight of each attribute ${ƛ}_j$ are calculated according to Equations (20) and (21):

$${\Im }_j=\frac{1}{m}{\displaystyle \sum _{i=1}^m\left(1-\left|2{\overline{\iota }}_{e_{ij}}-1\right|\times \left({\overline{\varsigma }}_{e_{ij}}+{\overline{\upsilon }}_{e_{ij}}\right)\right)},$$

(20)

$${ƛ}_j=\frac{1-{\Im }_j}{\wp -{\displaystyle \sum _{j=1}^{\wp }{\Im }_j}},$$

(21)

where ${ƛ}_j\in \left[0,1\right]$ and ${\displaystyle \sum _{j=1}^{\wp }{ƛ}_j}=1,j=(1,2,\dots ,\wp )$.

Step 5. Calculate the support degree $\Psi \left({\tilde{e}}_{lk}\right)$ between the variables and the rest according to Equation (22), and then obtain the support index ${\xi }_{lk}$ of the variables by Equation (23).

$$\Psi \left({\tilde{e}}_{lk}\right)={\displaystyle \sum _{j=1,j\ne k}^{\wp }{ƛ}_j\Xi \left({\tilde{e}}_{lk},{\tilde{e}}_{lj}\right)},$$

(22)

$${\xi }_{lk}=\frac{\wp {ƛ}_k(1+\Psi \left({\tilde{e}}_{lk}\right))}{{\displaystyle \sum _{t=1}^{\wp }{ƛ}_t(1+\Psi \left({\tilde{e}}_{lt}\right))}},$$

(23)

where $k=1,2,\dots ,\wp$ and $l=1,2,\dots ,m$.

Step 6. Assemble the evaluation values ${\tilde{e}}_{ij}\left(j=1,2,\dots ,\wp \right)$ corresponding to each alternative ${\mathsf{\Upsilon }}_i\left(i=1,2,\dots ,m\right)$ according to the SVNSSWPBM AO in Definition 13, and then obtained the comprehensive evaluation value $e_i$ of each alternative ${\mathsf{\Upsilon }}_i$.

Step 7. Calculate the score value of the comprehensive evaluation value $e_i$ by Equation (2) and the alternatives ${\mathsf{\Upsilon }}_i$ are ranked with reference to the comparison principle in Definition 4 to determine the optimal solution. In general, a higher score value implies a better solution.

7. Case Study

To check the accuracy and viability of the suggested framework, the decision model based on the SVNSSWPBM AO is applied to SWLSS in this part.

Assume that the government plans to choose the best SWLS from four alternatives $\mathsf{\Upsilon }=\left({\mathsf{\Upsilon }}_1,{\mathsf{\Upsilon }}_2,{\mathsf{\Upsilon }}_3,{\mathsf{\Upsilon }}_4\right)$. For a fair and reasonable evaluation, a system of six evaluation criteria is established, namely $Z_1$ (total cost), $Z_2$ (elevation and slope), $Z_3$ (distance from surface and ground water), $Z_4$ (land use), $Z_5$ (public acceptance), and $Z_6$ (safe distance). Obviously, $Z_1$ and $Z_2$ are cost-type, and the rest are benefit-type attributes. University scholars, representatives of research institutes, and environmental policymakers in the government in the field of waste management are invited to create an expert panel of five numbers. They will take part in this SWLSS to guarantee the accuracy and dependability of the evaluation data. There are precise steps.

Step 1. Establish the initial SVN SWLS evaluation matrix $M$. The SVN evaluation values of each option according to the above six attributes are listed in

Table 1. Initial SVN evaluation matrix $M$.

	${\mathbf{Z}}_1$	${\mathbf{Z}}_2$	${\mathbf{Z}}_3$	${\mathbf{Z}}_4$	${\mathbf{Z}}_5$	${\mathbf{Z}}_6$
${\mathsf{\Upsilon }}_1$	(0.1, 0.9, 0.9)	(0.2, 0.7, 0.7)	(0.8, 0.2, 0.2)	(0.9, 0.1, 0.1)	(0.7, 0.3, 0.2)	(0.7, 0.3, 0.2)
${\mathsf{\Upsilon }}_2$	(0.4, 0.7, 0.5)	(0.2, 0.7, 0.6)	(0.7, 0.3, 0.2)	(0.8, 0.2, 0.2)	(0.5, 0.3, 0.4)	(0.8, 0.2, 0.2)
${\mathsf{\Upsilon }}_3$	(0.2, 0.7, 0.4)	(0.4, 0.7, 0.5)	(0.5, 0.3, 0.4)	(0.6, 0.3, 0.2)	(0.6, 0.3, 0.2)	(0.4, 0.4, 0.3)
${\mathsf{\Upsilon }}_4$	(0.2, 0.7, 0.4)	(0.2, 0.8, 0.8)	(0.6, 0.3, 0.3)	(0.9, 0.1, 0.1)	(0.6, 0.3, 0.2)	(0.6, 0.3, 0.2)

after the expert consultation.

Step 2. Compute the normalized SVN SWLS evaluation matrix $\overline{M}$.

Table 2. Normalized SVN evaluation matrix $\overline{M}$.

	${\mathbf{Z}}_1$	${\mathbf{Z}}_2$	${\mathbf{Z}}_3$	${\mathbf{Z}}_4$	${\mathbf{Z}}_5$	${\mathbf{Z}}_6$
${\mathsf{\Upsilon }}_1$	(0.9, 0.1, 0.1)	(0.7, 0.3, 0.2)	(0.8, 0.2, 0.2)	(0.9, 0.1, 0.1)	(0.7, 0.3, 0.2)	(0.7, 0.3, 0.2)
${\mathsf{\Upsilon }}_2$	(0.5, 0.3, 0.4)	(0.6, 0.3, 0.2)	(0.7, 0.3, 0.2)	(0.8, 0.2, 0.2)	(0.5, 0.3, 0.4)	(0.8, 0.2, 0.2)
${\mathsf{\Upsilon }}_3$	(0.6, 0.3, 0.2)	(0.5, 0.3, 0.4)	(0.5, 0.3, 0.4)	(0.6, 0.3, 0.2)	(0.6, 0.3, 0.2)	(0.4, 0.4, 0.3)
${\mathsf{\Upsilon }}_4$	(0.6, 0.3, 0.2)	(0.8, 0.2, 0.2)	(0.6, 0.3, 0.3)	(0.9, 0.1, 0.1)	(0.6, 0.3, 0.2)	(0.6, 0.3, 0.2)

shows the normalized SVN evaluation matrix.

Step 3. Calculate the support degree of variables to each other and obtain the support degree matrix $S\left(l\right)(l=1,2,3,4)$:

$$S\left(1\right)=\left(\begin{array}{cccccc}1& 0.8333& 0.9000& 1& 0.8333& 0.8333\\ 0.8333& 1& 0.9333& 0.8333& 1& 1\\ 0.9000& 0.9333& 1& 0.9000& 0.9333& 0.9333\\ 1& 0.8333& 0.9000& 1& 0.8333& 0.8333\\ 0.8333& 1& 0.9333& 0.8333& 1& 1\\ 0.8333& 1& 0.9333& 0.8333& 1& 1\end{array}\right),$$

(24)

$$S\left(2\right)=\left(\begin{array}{cccccc}1& 0.9000& 0.8667& 0.8000& 1& 0.8000\\ 0.9000& 1& 0.9667& 0.9000& 0.9000& 0.9000\\ 0.8667& 0.9667& 1& 0.9333& 0.8667& 0.9333\\ 0.8000& 0.9000& 0.9333& 1& 0.8000& 1\\ 1& 0.9000& 0.8667& 0.8667& 1& 0.8000\\ 0.8000& 0.9000& 0.9333& 0.9333& 0.8000& 1\end{array}\right),$$

(25)

$$S\left(3\right)=\left(\begin{array}{cccccc}1& 0.9000& 0.9000& 1& 1& 0.8667\\ 0.9000& 1& 1& 0.9000& 0.9000& 0.9000\\ 0.9000& 1& 1& 0.9000& 0.9000& 0.9000\\ 1& 0.9000& 0.9000& 1& 1& 0.8667\\ 1& 0.9000& 0.9000& 1& 1& 0.8667\\ 0.8667& 0.9000& 0.9000& 0.8667& 0.8667& 1\end{array}\right),$$

(26)

$$S\left(4\right)=\left(\begin{array}{cccccc}1& 0.9000& 0.9667& 0.8000& 1& 1\\ 0.9000& 1& 0.8667& 0.9000& 0.9000& 0.9000\\ 0.9667& 0.8667& 1& 0.7667& 0.9667& 0.9667\\ 0.8000& 0.9000& 0.7667& 1& 0.8000& 0.8000\\ 1& 0.9000& 0.9667& 0.8000& 1& 1\\ 1& 0.9000& 0.9667& 0.8000& 1& 1\end{array}\right),$$

(27)

Step 4. The weight vector $ƛ={\left(0.173,0.157,0.161,0.242,0.131,0.136\right)}^T$ of the evaluation criteria is determined according to Equations (22) and (23).

Step 5. Calculate the support degree matrix $\Psi ={\left(\Psi \left({\tilde{e}}_{lk}\right)\right)}_{4\times 6}$ between the variables and the remaining ones as a whole and the support degree index matrix $\xi ={\left({\xi }_{lk}\right)}_{4\times 6}$.

$$\Psi =\left(\begin{array}{cccccc}0.7402& 0.7631& 0.7692& 0.6712& 0.7891& 0.7841\\ 0.7142& 0.7694& 0.7680& 0.6708& 0.7562& 0.7768\\ 0.7771& 0.7748& 0.7708& 0.7081& 0.8191& 0.7594\\ 0.7575& 0.7533& 0.7469& 0.6167& 0.7995& 0.7945\end{array}\right),$$

(28)

$$\xi =\left(\begin{array}{cccccc}1.0357& 0.9522& 0.9799& 1.3913& 0.8063& 0.8347\\ 1.0255& 0.9607& 0.9844& 1.3982& 0.7956& 0.8356\\ 1.0468& 0.9488& 0.9708& 1.4075& 0.8114& 0.8147\\ 1.0535& 0.9538& 0.9745& 1.3557& 0.8168& 0.8456\end{array}\right).$$

(29)

Step 6. Assemble the evaluation values $e_i(i=1,2,3,4)$ of the alternatives by the SVNSSWPBM AO proposed in this paper. For the convenience of calculation, we choose $c=1$, $d=1$, and $\kappa =-1$. The results are calculated as follows.

$$e_1=\left(0.7903,0.2097,0.1669\right),$$

(30)

$$e_2=\left(0.6507,0.2704,0.2627\right),$$

(31)

$$e_3=\left(0.5261,0.3209,0.2830\right),$$

(32)

$$e_4=\left(0.6815,0.2532,0.2015\right).$$

(33)

Step 7. Calculate the score values $D(e_i)(i=1,2,3,4)$ of the alternatives by Equation (2) and rank them according to Definition 5 to select the optimal solution.

$$D\left(e_1\right)=0.8046$$

(34)

$$D(e_2)=0.7058$$

(35)

$$D\left(e_3\right)=0.6407$$

(36)

$$D\left(e_4\right)=0.7423$$

(37)

Since $D\left(a_1\right)\succ D\left(e_4\right)\succ D\left(e_2\right)\succ D\left(e_3\right)$, it is observed that the priority relationship of the alternatives is ${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$, and the optimal site among the four alternative SWLSs is ${\mathsf{\Upsilon }}_1$. On balance, it has the best soil conditions, location, and the highest acceptance by the people and minimizes the impact on the surrounding residents and environment. From the raw evaluation data, site ${\mathsf{\Upsilon }}_4$ is, among other alternatives, the closest option to the ideal solution. However, the inherent slope and height conditions are weaker compared to site ${\mathsf{\Upsilon }}_1$. The main reason that limits site ${\mathsf{\Upsilon }}_2$ from being a suitable site is its poor performance in terms of costs and safety distances. The evaluation values of site ${\mathsf{\Upsilon }}_3$ under economic attributes are lower, which does not meet the requirements of building an environmentally friendly society. At the same time, site ${\mathsf{\Upsilon }}_3$ has low social acceptance.

As observed from the above calculation, the environmental indicators (56.0%) have the highest importance, followed by social indicators (26.7%) and economic indicators (17.3%). Among the criteria layers of economic factors, “land use” and “distance from surface and ground water” are the most important, followed by “elevation and slope”. This is because soil conditions determine the risk of wastewater infiltration while slope and height determine the speed of underwater infiltration and surface water pollution. The pollution of surface and ground water caused by leachate and rainwater in the SWLS is the most important issue in SWLSS. If the public is not consulted and a safe distance is not maintained from the SWLS, the construction process will be hindered and public health and the surrounding environment will be affected. Hence, social indicators are also important to SWLSS. Since economic efficiency is not the main purpose of building an SWLS, it can be found that an economic indicator is considered the least important factor in this study.

7.1. Sensitivity Analysis

The above result of our discussion on SWLSS is achieved with $c=d=1$ and $\kappa =-1$. Whether different values of c, d, and $\kappa$ will affect the results of aggregation and decision in any way needs to be discussed further. Next, we first keep $\kappa =-1$ constant and explore how the aggregation results and final choice are altered when the parameters such as c and d take different values.

Table 3. Aggregation results under different values of c and d.

c, d	${\mathit{e}}_1$	${\mathit{e}}_2$
c = 2, d = 1	(0.7947, 0.2053, 0.1645)	(0.6556, 0.2687, 0.2602)
c = 4, d = 3	(0.7911, 0.2089, 0.1665)	(0.6516, 0.2701, 0.2623)
c = 6, d = 5	(0.7906, 0.2094, 0.1667)	(0.6510, 0.2703, 0.2625)
c = 8, d = 7	(0.7905, 0.2095, 0.1668)	(0.6509, 0.2704, 0.2626)
c = 10, d = 9	(0.7904, 0.2096, 0.1668)	(0.6817, 0.2704, 0.2627)
c, d	$e_3$	$e_4$
c = 2, d = 1	(0.5285, 0.3199, 0.2800)	(0.6891, 0.2485, 0.1987)
c = 4, d = 3	(0.5266, 0.3208, 0.2824)	(0.6829, 0.2524, 0.2010)
c = 6, d = 5	(0.5263, 0.3209, 0.2828)	(0.6821, 0.2529, 0.2013)
c = 8, d = 7	(0.5262, 0.3209, 0.2829)	(0.6818, 0.2530, 0.2014)
c = 10, d = 9	(0.5262, 0.3209, 0.2829)	(0.6817, 0.2513, 0.2015)

and

Table 4. Score values and ranking under different c and d.

c, d	Score Value	Ranking
c = 2, d = 1	D = (0.8083, 0.7089, 0.6429, 0.7473)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$
c = 4, d = 3	D = (0.8053, 0.7064, 0.6411, 0.7432)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$
c = 6, d = 5	D = (0.8049, 0.7061, 0.6409, 0.7426)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$
c = 8, d = 7	D = (0.8047, 0.7060, 0.6408, 0.7425)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$
c = 10, d = 9	D = (0.8047, 0.7059, 0.6408, 0.7424)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$

display the aggregation values, score values, and ranking outcomes based on the SVNSSWPBM AO with various values for c and d.

From Table 3, it can be found that the different values of c and d do lead to different aggregation results and score values. The values of c and d are larger when the score values are smaller. However, the variation of c and d does not pose an impact on the ranking results and the optimal solution. Consistent with the above, site ${\mathsf{\Upsilon }}_1$ is always our optimal choice. Table 4 shows that although the parameters c and d are changing, the score values obtained in each case are very close to the situation when $c=1$ and $d=1$. This indicates that c and d do not have a large impact on the decision results within a certain range of values, reflecting that the proposed AO has computational flexibility and strong stability. Parameters c and d represent the degree of the input evaluation values. Since the economic, environmental, and social factors interact with each other, the parameters can be adjusted to meet the different decision problems with interlinked indicators.

Next, we will discuss the effects of the parameter $\kappa$ in the SVNSSWPBM AO on the score values, ranking results, and the optimal choice. Keeping $c=d=1$ constant and assigning different numbers to the parameter $\kappa$, the aggregation results, score values, and ranking results of each solution are shown in

Table 5. Aggregation results under different values of $\kappa$.

$\mathit{\kappa }$	${\mathit{e}}_1$	${\mathit{e}}_2$
−2	(0.8001, 0.1999, 0.1625)	(0.6553, 0.2683, 0.2576)
−3	(0.8100, 0.1900, 0.1592)	(0.6622, 0.2669, 0.2526)
−4	(0.8199, 0.1801, 0.1561)	(0.6711, 0.2658, 0.2474)
−5	(0.8296, 0.1704, 0.1530)	(0.6813, 0.2647, 0.2424)
−6	(0.8385, 0.1615, 0.1496)	(0.6920, 0.2636, 0.2376)
−7	(0.8462, 0.1538, 0.1461)	(0.7025, 0.2623, 0.2333)
−8	(0.8527, 0.1473, 0.1426)	(0.7121, 0.2607, 0.2295)
$\kappa$	$e_3$	$e_4$
−2	(0.5252, 0.3194, 0.2786)	(0.6870, 0.2499, 0.1980)
−3	(0.5260, 0.3181, 0.2741)	(0.6947, 0.2474, 0.1957)
−4	(0.5277, 0.3170, 0.2694)	(0.7050, 0.2452, 0.1941)
−5	(0.5300, 0.3161, 0.2643)	(0.7171, 0.2428, 0.1928)
−6	(0.5325, 0.3152, 0.2590)	(0.7299, 0.2400, 0.1917)
−7	(0.5353, 0.3143, 0.2538)	(0.7420, 0.2369, 0.1908)
−8	(0.5381, 0.3134, 0.2489)	(0.7524, 0.2334, 0.1900)

and

Table 6. Score values and ranking under different values of $\kappa$.

κ	Score Value	Ranking
−2	D = (0.8126, 0.7098, 0.6424, 0.7464)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$
−3	D = (0.8202, 0.7142, 0.6446, 0.7505)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$
−4	D = (0.8279, 0.7193, 0.6471, 0.7552)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$
−5	D = (0.8354, 0.7247, 0.6499, 0.7605)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$
−6	D = (0.8425, 0.7303, 0.6528, 0.7661)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$
−7	D = (0.8488, 0.7356, 0.6557, 0.7714)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$
−8	D = (0.8543, 0.7406, 0.6586, 0.7763)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$

. Figure 2 helps us to observe more clearly the dynamic trend of the alternatives’ score values with the changing $\kappa$.

From Table 5 and Table 6 and Figure 2, we can clearly find that when the parameter $\kappa$ in the SVNSSWPBM AO takes different values, we obtain different aggregation values for each alternative. As $\kappa$ decreases, the score value of each alternative steadily increases, with ${\mathsf{\Upsilon }}_1$ increasing the fastest and the other alternatives increasing at the same rate overall. However, one can see that although the aggregation results and the score values are changing with the parameter $\kappa$, the priority relationship of the score value of the alternatives is always ${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$, and ${\mathsf{\Upsilon }}_1$ always tends to be the site with the best overall performance in terms of economic, environmental, and social aspects. Therefore, the parameter $\kappa$ can be flexibly adjusted to obtain different score values to meet the preference of different types of DMs without affecting the decision outcomes.

From the above discussions, it is concluded that the SWLS ranking and final choice always remain consistent regardless of the values of c, d, and $\kappa$. The evaluation results obtained from the above sensitivity and comparative analysis reflect the flexibility and accuracy of the proposed method for SWLSS.

7.2. Comparative Analysis

Next, we select MCDM methods based on SVN weighted average (SVNWA) and SVN weighted geometric average (SVNWGA) AOs [59], SVN normalized weighted BM (SVNNWBM) AO [30], SVN power weighted average (SVNPWA) AO [29], and SVN weighted Bonferroni power (SVNWBP) AO [31] to solve the practical problem of SWLSS presented in this paper. To ensure the impartiality of the computational results, we assign a value of 1 to all the parameters. The outcomes of the comparison are displayed in

Table 7. Aggregation results under different AOs.

Aggregation Operators	${\mathit{e}}_1$	${\mathit{e}}_2$
SVNWA [59]	(0.8219, 0.1781, 0.1500)	(0.6855, 0.2574, 0.2469)
SVNWGA [59]	(0.7938, 0.2062, 0.1599)	(0.6488, 0.2638, 0.2670)
SVNNWBM [30]	(0.8037, 0.1963, 0.1573)	(0.6649, 0.2620, 0.2570)
SVNPWA [29]	(0.8200, 0.1800, 0.1511)	(0.6846, 0.2580, 0.2468)
SVNWBPM [31]	(0.7562, 0.2438, 0.2010)	(0.6170, 0.3092, 0.3037)
Aggregation Operators	$e_3$	$e_4$
SVNWA [59]	(0.5462, 0.3120, 0.2635)	(0.7435, 0.2158, 0.1805)
SVNWGA [59]	(0.5358, 0.3149, 0.2831)	(0.6924, 0.2403, 0.1944)
SVNNWBM [30]	(0.5406, 0.3135, 0.2744)	(0.7065, 0.2350, 0.1907)
SVNPWA [29]	(0.5461, 0.3120, 0.2638)	(0.7381, 0.2194, 0.1826)
SVNWBPM [31]	(0.4978, 0.3602, 0.3219)	(0.6554, 0.2840, 0.2368)

and

Table 8. Scores and ranking under different AOs.

Aggregation Operators	Score Values	Ranking
SVNWA [59]	D = (0.8312, 0.7271, 0.6569, 0.7824)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$
SVNWGA [59]	D = (0.8092, 0.7060, 0.6461, 0.7526)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$
SVNNWBM [30]	D = (0.8167, 0.7152, 0.6509, 0.7603)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$
SVNPWA [29]	D = (0.8296, 0.7266, 0.6568, 0.7787)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$
SVNWBPM [31]	D = (0.7705, 0.6680, 0.66052, 0.7115)	${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$

. Figure 3 graphically displays the findings of the scheme ranking comparison and the aggregation based on each AO.

Although the aggregation results and score values obtained by the AOs differ slightly from the SVNSSWPBM AO, Table 8 demonstrates that the final ranking results of developed AO are in accordance with some extensive AOs for SWLSS, which is always ${\mathsf{\Upsilon }}_1\succ {\mathsf{\Upsilon }}_4\succ {\mathsf{\Upsilon }}_2\succ {\mathsf{\Upsilon }}_3$. The optimal SWLS obtained by different AOs is always ${\mathsf{\Upsilon }}_1$, while site ${\mathsf{\Upsilon }}_3$ is considered as the poorly performing option overall. This implies that the SVNSSWPBM AO has a very high reasonableness.

The following are the discussions about differences in score values resulting from various existing AOs mentioned above:

(1)

The SVNWA AO [59] focuses on group effects, while the SVNWGA AO [59] emphasizes individual characteristics. At the same time, both results obtained from the above AOs are simple summations of evaluation information under ordinary algebraic paradigms. They cannot take into account the interconnectedness of SWLS evaluation indicators and the calculation processes are also rigid.

(2)

The SVNPWA AO [29] can only eliminate the negative influence of abnormal values on the decision results to a certain extent but cannot properly deal with the problem of the interrelationship between indicators. In contrast, the SVNNWBM AO [30] can only reflect the correlation between attributes but cannot overcome the adverse effects of anomalous information in solving complex decision problems.

(3)

The SVNWBPM AO [31] lacks the flexibility of algebraic operations although it focuses on the intercorrelation between attributes and the overall equilibrium type. In fact, the SVNWBPM AO is a general case of the SVNSSWPBM AO proposed in this study, which includes parameter $\kappa$ in addition to parameters c and d.

In summary, we can find that the proposed SVNSSWPBM AO combines the advantages of PA and BM AOs, which can flexibly and effectively deal with the complex SWLSS problem with interrelated indicators and unreasonable evaluation values. The range of common forms of SVNSSWPBM AO that can be acquired by adjusting the parameters in SS operations expands the decision model’s potential applications. DMs can choose the proper parameters based on various decision scenarios to satisfy their decision preferences.

8. Conclusions

For the problem of SWLSS in the SVN environment, this paper proposes a decision method based on SVNSSWPBM AO and multi-criteria analysis. First, taking economic, environmental, and social factors as the starting point, an index system that includes six corresponding sub-criteria for evaluating SWLS is constructed regarding existing studies. Then, the SVNSSPBM and SVNSSWPBM AOs are proposed and all their properties are listed. On this basis, a decision model based on the SVNSSWPBM AO for solving MCDM problems is given. Finally, the proposed framework is applied to solve a practical SWLSS problem as well as deep sensitivity and comparative analysis is performed. After a multidimensional evaluation, the site ${\mathsf{\Upsilon }}_1$ is determined to be the most suitable SWLS. The results of further comparative and sensitivity analysis not only validate the superiority and feasibility of the model proposed but also provide a new decision method for solving practical SWLSS problems. Therefore, the presented method shows valuable theoretical and practical significance in MCDM areas.

From a theoretical point of view, the proposed SVNSSPBM and SVNSSWPBM AOs for aggregating SVN information enrich the theory of SVNSs and AOs. The proposed AOs integrate the advantages of PA and BM AOs, which not only consider the problem of possible intercorrelation between attributes but also solve the difficulty that abnormal extreme information affects the decision result. Moreover, several parameters included in these AOs make it easier for DMs to aggregate the evaluation value flexibly according to their risk preferences. The proposed decision framework based on SVNSSWPBM AO enriches the theory of decision methods. Since its practicality is confirmed in the case study, it can be further applied to solve the problems of material selection [60], online learning evaluation [61], green supplier selection [62], community platform selection [63], etc.

It also offers certain practical significance; the random accumulation of solid waste can cause resource pollution, air quality degradation, and health threats. Therefore, whether an SWLS is reasonable will directly affect the ecological environment. Countries with inadequate waste management systems have mostly focused on cost minimization or profit maximization when selecting SWLSs, ignoring their impact on the ecological environment. We summarize the main factors affecting SWLSS to provide entry points for DMs to target SWLSs. The DMs should take into account the opinions of the group, integrate the economic and environmental benefits, and select sites with intact bedrock conditions as far as possible from natural resources and residential areas.

Although the proposed methodology is effective to determine SWLS preferences in a stable and flexible manner, there are still some shortcomings in this study. The improvements for these limitations point the way for future research. First, the SWLS evaluation index weights are only based on objective evaluation data, while the subjective judgment of the DMs cannot be ignored in the actual decision process. Future research can focus on how to examine the indicator weights more comprehensively. AHP, BWM, and other weight determination methods can be combined with SVNSSWPBM AO to obtain more convincing results. In addition, the panel of experts is constructed to give consistent review information after consultation, but it is difficult to unify the internal opinions of various experts. Group consistency methods can be studied in future research. We can also try to extend the proposed model to the expanded form of SVNSs in the future since it has strong generalizability.