Introduction

Rapid urbanization has led to the migration of people from villages to cities, which generate thousands of tons of municipal solid waste (MSW) daily. After waste is disposed at landfills, it undergoes a number of physical, chemical, and microbiological changes that lead to the release of a toxic liquid known as leachate. The leachate will continuously migrate through the soil strata, eventually contaminating the groundwater system if no action is taken to prevent this phenomenon (Kanmani and Gandhimathi 2013; Liao et al. 2020). Contamination of groundwater from leachate is a potential environmental problem and needs to be addressed and many studies have been conducted (Locosselli et al. 2019; Molinos-Senante et al. 2015). Physicochemical analysis and indexing method are the most common in assessment the impact of a landfill on groundwater quality.

Physicochemical analysis is commonly used in assessment of the impact of a landfill on groundwater quality (Cumar and Nagaraja 2011; Zaharia and Suteu 2011; Grisey and Aleya 2016; Teta and Hikwa 2017; D Browska et al. 2018). Researchers can select specific physicochemical parameters to analyze according to the unique situation of study area, such as heavy metal concentration (Kanmani and Gandhimathi 2013; D Browska et al. 2018), pH (Zaharia and Suteu 2011; Grisey and Aleya 2016; Teta and Hikwa 2017), or chlorides (Cumar and Nagaraja 2011). However, the physicochemical analysis has two disadvantages. Firstly, it cannot analyze multiple inter-relationships among selected physicochemical parameters. Secondly, there is no unified index for selecting physicochemical parameters, which means the method lacks robustness and normalization. In order to explore inter-relationships among all the physicochemical parameters, some researchers conducted statistical analyses based on the physicochemical parameters (Biswas et al. 2010; Nagarajan et al. 2012; Mor et al. 2006; Smahi et al. 2013; Magda et al. 2014; Maitia et al. 2016; Nathan et al. 2018; Jeykumar and Chandran 2018; Cipranic et al. 2019; Aderemi et al. 2011). The main statistical analysis methods contain factor analysis and Person’s correlation (Biswas et al. 2010; Maitia et al. 2016), correlation matrix for parameters, principal component analysis (PCA), descriptive statistics analysis and multivariate statistical analyses, etc. (Nagarajan et al. 2012; Mor et al. 2006; Aderemi et al. 2011; Smahi et al. 2013). The statistical analysis can easily explore the relation of different physicochemical parameters, but the lack of robustness and normalization still cannot be solved (Magda et al. 2014; Jeykumar and Chandran 2018; Nathan et al. 2018).

To make the assessment more robust and normal, many researchers used standard indices in assessing pollution impact of a landfill. Environmental indices such as the Water Quality Index (WQI) (Deshmukh and Aher 2016; Reymond et al. 2019), Nemerow index (expressed as “Pi”) (Krčmar et al. 2018; Kapelewska et al. 2019; Tenodi et al. 2020), and Leachate Pollution Index (LPI) (Rana et al. 2018; Han et al. 2016) have been developed to determine the extent of pollution. WQI is one of the simplest and widely used methods which can provide a comprehensive model of the groundwater quality and be used to transform large quantities of water quality data into a single number (Wang et al. 2013). To improve the effectiveness of assessing the negative impact of the pollution emitter on water quality, Nemerow index (Pi) was developed which is a comprehensive parameter based on national and international standards (Kapelewska et al. 2019). Leachate from landfill is one of the main pollutions on groundwater (Chonattu et al. 2016; Tenodi et al. 2020). In an effort to develop a method for comparing the leachate pollution potential of various landfill sites in a given geographical area, LPI was developed to represent the level of leachate contamination potential of a given landfill (Rana et al. 2018). In order to improve accuracy and reliability of assessment, some researchers combined two indices to assess the impact of leachate percolation on the groundwater quality (Singh et al. 2016; Rana et al. 2018; Krčmar et al. 2018; Kapelewska et al. 2019; Tenodi et al. 2020). However, with the improvement of reliability, assessment complex was also raised.

In addition to the WQI, PI, and LPI, many groundwater quality assessment methods can be used in assessment of the impact of a landfill on groundwater quality. EWQI, NSFWQI, VFEM, FSEVFS, ICAUCA, and VFSPA have been widely used in groundwater assessment but less used in assessing the impact of leachate on the groundwater quality (Wu et al. 2020; Ocampoduque et al. 2006; Vasanthavigar et al. 2010; Mohebbi et al. 2013; Hoseinzadeh et al. 2015; Xi Chen and Wang 2019). Besides, few above efforts have been made for optimizing both the double judgment approach and the “Pi” equation of N.L. Nemerow index for improving assessment accuracy. With the improvement of accurate and rigorous requirements for groundwater pollution research, practitioners and researchers have not only paid more attention to the regional and complex characteristics of groundwater in various domestic landfills, but also put more efforts into a great deal of the perspective of study area location, domestic landfill type, and the dimension of groundwater pollution component.

Hence, this study includes two main objectives. The first main goal of this study is to develop a new model of optimized fuzzy set pair (OFSP) according to variable fuzzy set (VFS) pair and optimized N.L. Nemerow index for accurate and rigorous evaluation of groundwater under domestic landfills (Shuang et al. 2016;Wang et al. 2011;Zou et al. 2006), and devise the OFSP model by five key elements of optimized synthesis operator “C,” relative difference\( ``{u}_i^{\prime }," \) connection degree“ui, ” optimized N.L. Nemerow index “Pi,” and pollution load ratio “Ji”, so as to achieve the reasonable groundwater quality assessment model, the stable groundwater quality evaluation process, and the convincing groundwater quality evaluation results. The second main purpose of this study is to compare the performance of OFSP model with some mainstream models by their assessment results of groundwater quality to various domestic waste landfills in China.

Study area

In this section, we elaborate the natural characteristics of the study area including the local soil conditions, geography, climate, and hydrogeological setting, to obtain more rigorous research results of groundwater quality assessment in domestic waste landfills.

Based on the Report on the National General Survey of Soil Contamination published by the Ministry of L and Resources of the People’s Republic of China in 2014, the state of soil environment across the country is not optimistic (National Bulletin on Soil Pollution Status 2014). The country’s total soil over-standard rate was 16.1%, of which the proportions of slightly, lightly, moderately, and severely polluted points were 11.2%, 2.3%, 1.5%, and 1.1% respectively. Inorganic pollution is the main type of pollution, followed by organic pollution. The level of inorganic pollutants exceeding standard points accounts for 82.8% of all exceeding standard points, in which eight kinds of inorganic pollutants containing cadmium, mercury, arsenic, copper, lead, chromium, zinc, and nickel exceed the standard rates 7.0%, 1.6%, 2.7%, 2.1%, 1.5%, 1.1%, 0.9%, and 4.8%. In terms of pollution distribution, soil pollution in the South is more serious than that in the North.

China located in the eastern part of Asia and on the west coast of the Pacific Ocean is selected for this study (Jingyun et al. 2018; Chen et al. 2019). The land area is about 9.6 million square kilometers. The terrain characteristic, which is high in the west and low in the east, is descended in three ladders. The geographic location of it is shown in Fig. 1. The mountains, plateaus, and hills account for about 67% of the land area, and basins and plains account for about 33% of the land area (Zheng et al. 2015). Western China has the largest Qinghai-Tibet Plateau in the world, with an average elevation of over 4,000 m. The terrain is high, and the snowy mountain glaciers are large in size, which is called the first ladder of China’s topography. The eastern and northern parts of the Qinghai-Tibet Plateau are the second ladder of China’s topography. It includes areas such as Inner Mongolia, Xinjiang, Loess Plateau, Sichuan Basin, and Yunnan-Guizhou Plateau, with an average elevation of 1000–2000 m. In the region, there are many grasslands, Gobi, deserts, plateaus, and the surface, which are covered with deep loess. Their surface is broken, the vegetation is small, the soil and water loss are serious, the terrain is rugged, and the limestone is widely distributed. The rest of the area is known as the third ladder of China’s topography, which is mostly plains and hills. It ranges from east to the coastline. That terrain has dropped to between 500 and 1000 m. From the north to the south, the northeast plain, the North China Plain, and the middle and lower reaches of the Yangtze River are distributed. The region is rich in soil, and abundant in rain, and the river network is the most important farming area in China. And the hilly areas of the third ladder of China’s topography are generally densely forested and rich in minerals.

Fig. 1
figure 1

Geographic location of domestic waste landfills

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The monsoon climate in China is very significant, most of China being affected by the monsoon climate (Fang et al. 2015; Jingyun et al. 2018). And most of the inland areas north of 40° N are temperate continental climate. The precipitation in these areas is sparse, and the annual precipitation is between 300 and 500 mm. The temperature varies greatly between different times of the day as well by year. The northwestern region is a temperate continental climate with cold winters and hot summers with less precipitation. The southwest region belongs to the plateau mountain climate. The average daily temperature is lower than 10 °C, and the hottest temperature is lower than 5 °C, even lower than 0 °C. Different from the year, the temperature varies greatly between different times of the day, but the solar radiation is strong and the sunshine is sufficient. The North China region is a temperate monsoon climate with high temperatures and rains in summer and cold and dry winters. The coldest month average temperature is − 28–0 °C, and the average summer temperature is about 22 °C. Central China has a temperate monsoon climate and a subtropical monsoon climate. The rainfall is concentrated in the summer and there is often heavy snow in the north of the winter. Generally, the climate is mild, the heat is sufficient, the precipitation is concentrated, the spring temperature is changeable, the summer and autumn are dry, the cold period is short, and the heat cycle is long. East China and most of the regions in South China have subtropical monsoon climates, with high temperatures and rains in summer and low temperatures and low rainfall in winter. That is a transition zone between the subtropical zone and the temperate zone. The temperature in summer is quite high and the temperature in winter is quite low. Hainan Island, the southern part of Taiwan Island, the Leizhou Peninsula, and Xishuangbanna belong to the tropical monsoon climate. The average temperature of the coldest month is not lower than 15 °C, and the annual minimum temperature is not less than 5 °C on average, and there is no frost all year round.

China has a vast territory, and the geologic, climate, and hydrogeological setting features of different regions vary greatly (Chen et al. 2019). As a whole, it shows that the southern region has developed water systems and abundant rainfall, while the northern region is cold and dry with less rain. The groundwater quality is closely related to multiple factors such as geography, geomorphology, climate, and hydrology. Therefore, the selection of the sampling area in this study should consider multiple factors including geomorphology, climate, and hydrology to reflect the representativeness and comprehensiveness of the collected samples, and finally ensure the rationality and authority of the research results. The sample collection points (locations) of this study area (Fig. 1) are shown as follows.

As we know, groundwater is widely used in China’s domestic water, agricultural irrigation, and industrial production. It is closely related to people’s life and production. It is also directly related to the high-quality development of the national economy and its positive contribution to world economic security and stability. In recent years, due to problems such as chemical pollution of domestic waste landfill, the quality of groundwater has been seriously affected. Therefore, excavating the impact of domestic waste landfill on groundwater quality and clarifying the relationship between the two are necessary and important for regulating waste sorting and targeted treatment of groundwater. In order to obtain convincing research results, in this study, we randomly collect the various representative domestic landfills of China as groundwater sample collection points (Table 1) to select sample data to fully reflect the overall situation of groundwater quality under domestic waste landfills in China. By collecting the groundwater quality samples in monitoring well, dive well, or agricultural irrigation well, etc. from different regions in China, and testing the collected samples with standardized and uniform precision, the sample data that meet the requirements of this study can be obtained.

Table 1 Background of regulated and unregulated domestic landfills
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Materials and methods

Sample collection and preparation

Based on the data collection method and detailed process above, for the selected research areas such as “Asowei#,” “Shandong Yidi#, and “Fuxin city landfill,” we collect the corresponding measured data of groundwater pollution components under domestic waste landfills (Table 2) such as “MPN,” “CFU,” and “NH3–N,” respectively (Wu et al. 2020; Xi Chen and Wang 2019; Ocampoduque et al. 2006; Vasanthavigar et al. 2010; Mohebbi et al. 2013; Hoseinzadeh et al. 2015). For example, the measured values of “MPN,” “CFU,” “NH3–N,” “As,” “Hg,” “Na,” “CaCO3,” “COD,” “Mn,” and “Pb” of Asowei# are 7.212, 8.339, 10.887, 8.359, 10.126, 12.338, 5.562, 15.347, 9.935, and 14.551, respectively. In the same way, all relevant measured values of groundwater pollution components in the study are present in the table below.

Table 2 The measured values of groundwater pollution components
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Development of approach

Variable fuzzy set

To evaluate groundwater quality, the basic operational principles of variable fuzzy set (called VFS), which is a necessary part of optimized fuzzy set pair (OFSP) model, is required in this paper (Wang et al. 2011; Xu et al. 2015; Li et al. 2020). And the unstandardized relative membership variable (\( {u}_i^{\prime } \)) of variable fuzzy set, which is not affected by assessment system error, is crucial for assessment of groundwater quality in domestic waste landfills (Chen et al. 2012;Yan et al. 2016). Therefore, a key equation of \( {u}_i^{\prime } \) of variable fuzzy set is constructed as follows.

The assumption is that the number of both groundwater pollution assessment indexes and assessment levels are “t” and “n,” respectively. Hence, the “\( {u}_i^{\prime } \)” equation (Eq. 1) and its sub-equations (Eqs. 2 and 3) of VFS for groundwater quality assessment of OFSP are respectively expressed below.

$$ {u}_i^{\prime }=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left[1+{\left(\raisebox{1ex}{${d}_{hg}$}\!\left/ \!\raisebox{-1ex}{${d}_{hb}$}\right.\right)}^{\alpha}\right]$}\right.;\mathrm{i}=\mathrm{h} $$
(1)
$$ {d}_{hg}={\left\{{\sum}_{i=1}^t{\left[{w}_i\left(1-{\mu}_{\overset{\sim }{A}}\left({u}_{ih}\right)\right)\right]}^p\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.} $$
(2)
$$ {d}_{hb}={\sum}_{i=1}^t{\left[{\mu}_{\overset{\sim }{A}}{w}_i-{\left({u}_{ih}\right)}^p\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.} $$
(3)

where “α,” “p,” and “h” respectively represent various parameter, and “wi” represents the corresponding weight of the ith evaluation metrics.

In addition, the location map is also crucial for assessment of groundwater quality in domestic waste landfill. It is easy to find that the location map, which is demonstrated in Fig. 2, could intuitively reflect the attraction and exclusion domains of the VFS of groundwater pollution assessment.

Fig. 2
figure 2

Location map

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Based on the location map, whether the point of “x” is to the left or right of “M” in the X interval, the value of \( {D}_{\overset{\sim }{A}}(u) \) can be obtained by the following equations (Eqs. 4 and 5).

$$ {D}_{\overset{\sim }{A}}(u)={\left(\frac{x-a}{M-a}\right)}^{\beta },x\in \left[a,M\right];{D}_{\overset{\sim }{A}}(u)={\left(\frac{x-a}{c-a}\right)}^{\beta },x\in \left[c,a\right] $$
(4)
$$ {D}_{\overset{\sim }{A}}(u)={\left(\frac{x-b}{M-b}\right)}^{\beta },x\in \left[M,b\right];{D}_{\overset{\sim }{A}}(u)=-{\left(\frac{x-b}{d-b}\right)}^{\beta },x\in \left[b,d\right] $$
(5)

where “β” represents the non-negative parameter.

Set pair analysis

For the aim of the OFSP model is to optimize groundwater assessment, set pair analysis devotes the connection degree matrix of “Q” to the calculation of “Pi” of groundwater quality in various domestic waste landfills of China (Chen et al. 2012; Wang et al. 2013). The main steps of employing set pair analysis (SPA) to determine the connection degree matrix of “Q” are formed in the following manner.

Solution steps of SPA

  1. Step 1:

    Design a new equation of “\( {\mathrm{u}}_{P_n-{Q}_m} \),” in which the number of logic category is increased to k+2. The structure of the equation is constructed as follows.

$$ {\mathrm{u}}_{P_n-{Q}_m}={\sum}_{n=1}^L{w}_n{a}_n+{\sum}_{t=1}^K{\sum}_{n=1}^L{w}_n{b}_{n,t}{i}_t+{\sum}_{n=1}^L{w}_n{c}_nj $$
(6)

where bt =1, 2, …, k.

  1. Step 2:

    With the help of a measurement schematic chart (Fig. 3), the result of “uik” can be computed by the following function equation.

Fig. 3
figure 3

The measurement schematic chart

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The equation (Eq. 7) of function relationship between the measure metric “a” and the connection are written as follows.

$$ {u}_{ik}=1-\left|\cos {a}_{ik}\right|=1-\left|\frac{S_{ik}-{x}_i}{\sqrt{1+{\left({x}_i-{S}_{ik}\right)}^2}}\right| $$
(7)

Note that “k” is the grade; “aik” is the angle of the metric “i”; “xi” is actual measured result; “Sik” is one of criterion values.

  1. Step 3:

    Devise the connection degree matrix of “Q” for groundwater quality in various domestic landfills. The connection degree matrix of “Q” is consisted by the connection degrees of “u,” which include the connection degree values of p grades corresponding to n evaluation indexes of groundwater samples. The corresponding matrix of “Q” (Eq. 8) is displayed below.

$$ \mathrm{Q}={\left[{u}_{ik}\right]}_{n\times p}=\left[\begin{array}{cccc}{u}_{11}& {u}_{12}& \dots & {u}_{1p}\\ {}{u}_{21}& {u}_{21}& \dots & {u}_{2p}\\ {}\dots & \dots & \dots & \dots \\ {}{u}_{n1}& {u}_{n2}& \dots & {u}_{np}\end{array}\right] $$
(8)

Optimized assignment of weights

Game theory, which is one of the measure to optimizing the weight assignment, has been adopted to determine “w” of each metric to acquire the equilibrium weight in this paper (Amiri et al. 2014). Thus, the game theory is used to compute the weights assignment of various metrics by optimized assignment of weights (OAW).

For the purpose of objective assessment to groundwater quality of domestic waste landfills, OAW dedicates the optimized assignment method, which can reasonably assign weight to each assessment index, to OFSP. According to the calculation of OAW, the corresponding equilibrium weights of various evaluation metrics can be obtained. The core algorithm of OAW to calculate weights are constructed (Algorithm 1) as follows.

figure a

Optimized N.L. Nemerow index

In the following, this study designs an optimized N.L. Nemerow index to form the mathematics equation of \( ``{P}_i=\sqrt{a\left({\overline{P}}_{\max 1}^2+{\overline{P}}_{\max 2}^2+{\overline{P}}_{\max 3}^2+{\overline{P}}_{\max 4}^2+{\overline{P}}^2\right)}" \), so as to make OFSP more balanced and robust for asseement of contaminated groundwater quality (Yan et al. 2014; Xu et al. 2015; Nambiar et al. 2020).

Groundwater pollution study of regulated and unregulated domestic waste landfills is closely related to the drinking water safety of groundwater. Therefore, the evaluation of groundwater quality of domestic waste landfill is very important, and performance metrics requirements such as accuracy and stability of the evaluation are very high. This study considers the diversity of groundwater pollution components under domestic waste landfill, and their different levels of pollution and efficiency for groundwater, so, it is necessary to create a novel groundwater quality assessment method to reflect the actual effect of various chemical elements on groundwater pollution. For example, the pollution components including “Hg,” “As,” and “Pb,” which have a great impact on water quality, are not reasonable to use only their weights to indicate the pollution effect.

Based on the N.L. Nemerow index idea (Eq. 14) and the equilibrium idea of

synthetic operator (Eq. 16), this study develops the method of N.L. Nemerow index (Xu et al. 2015; Wu et al. 2018).

$$ p=\sqrt{\left({P}_{\mathrm{max}}^2+{\overline{P}}^2\right)/2} $$
(14)
$$ {P}_i=\frac{c_i}{s_{ij}};i=1,2,\dots, n;j=1,2,\dots, m $$
(15)
$$ {b}_j=\underset{i=1}{\overset{m}{\vee }}\left({a}_i\cdot {r}_{ij}\right)=\underset{1\le i\le m}{\max}\left\{{a}_i\cdot {r}_{ij}\right\},i=1,2\dots n $$
(16)

where Eq. 16 uses the method of basic real multiplication.

By analyzing the principle of the above formula for groundwater assessment, it can be seen that Formula 14 overemphasizes the impact of the maximum pollution factor on water pollution, and does not highlight the pollution effects of other heavily polluted chemical elements. Therefore, in order to balance the actual effect of each chemical element and approximate the real pollution effect, this study devises OFSP for assessment of contaminated groundwater quality. The innovatives and improvements of the OFSP are carried out below: (1) Add 3 variable parameters, such as \( {\overline{P}}_{\max 2}^2 \), \( {\overline{P}}_{\max 3}^2\ and\ {\overline{P}}_{\max 4}^2 \), so as to highlight the large actual pollution effect of the heavily polluted elements while taking into account the comprehensive calculation of multiple variable indicators; (2) employ \( {\overline{P}}_{\mathrm{max}}^2 \)instead of \( {P}_{\mathrm{max}}^2 \) to restore and highlight the actual pollution effect of the heavy pollution elements as much as possible; (3) compute the arithmetic mean of Pi with Pmax1, Pmax2, Pmax3, and Pmax4, respectively, so that the conversion values of \( {\overline{P}}_{\max 1}^2 \),\( {\overline{P}}_{\max 2}^2 \), \( {\overline{P}}_{\max 3}^2 \), and\( {\overline{P}}_{\max 4}^2 \) are achieved.

Based on the above ideas and method improvements, the mathematics equation (Eq. 17) of OFSP is obtained as follows.

$$ {\displaystyle \begin{array}{c}p=\sqrt{a\left({\overline{P}}_{\max 1}^2+{\overline{P}}_{\max 2}^2+{\overline{P}}_{\max 3}^2+{\overline{P}}_{\max 4}^2+{\overline{P}}^2\right)}\\ {};{P}_i=\frac{c_i}{s_{ij}}\kern0.5em ;\mathrm{i}=1,2,\dots, \mathrm{n};\mathrm{j}=1,2,\dots, \mathrm{m};a=\frac{1}{k+1}\end{array}} $$
(17)

where “C,” “S,” and “\( \overline{\mathrm{P}} \)” represent measured element concentration, criterion element concentration, and mean, respectively. \( \overline{\mathrm{P}}\max 1 \) is the mean of Pi and Pmax1; “k” is the number of values of the maximum first k-bit weighting factors.

It is not difficult to find that our designed OFSP compensates for the disadvantages such as the invalidation of the weight of contaminated pollution components and the distortion of the information of contaminated pollution components in the groundwater assessment. The weight of each pollution factor in water quality evaluation was further weighed, and the effect of the largest amount of pollution factors on groundwater quality was diluted. It shows the significant advantages of balance and robustness.

OFSP model

In this section, based on the theory of variable fuzzy set pair and optimized N.L. Nemerow index (Wu et al. 2018; Su et al. 2009), we develop an optimized fuzzy set pair (OFSP) model for groundwater quality assessment of domestic waste landfill. The OFSP consists of seven specific operation steps, the advantages and reasons for improvement of each operation link, which are explained in Algorithm 2.

In order to achieve the reasonable assessment model, the stable experimental process, and the convincing experiment results, we devise the OFSP model by five key elements of optimized synthesis operator “C,” relative difference\( ``{u}_i^{\prime }," \) connection degree “ui, ” optimized N.L. Nemerow index “Pi,” and pollution load ratio “Ji” (Dadzie 2020; Yan et al. 2014; Su et al. 2009). The core algorithm of optimized fuzzy set pair model is compiled below.

figure c

It is noted that compared with the single judgment principle of “pj,” the proposed algorithm adds a sub-algorithm of the double judgment to evaluate the groundwater quality, so as not to reduce the accuracy of the assessment level.

The research proposes a novel OFSP groundwater assessment model by improving some basic theories such as variable fuzzy sets model, set pair analysis theory, entropy method, Nemerow index method, and cumulative pollution load ratio method (Wang et al. 2011; Li 2020; Chen et al. 2012; Wang et al. 2013; Amiri et al. 2014; Yan et al. 2014; Xu et al. 2015; Nambiar et al. 2020; Wu et al. 2018). Firstly, based on the characteristics and assessment requirements of groundwater pollution in the north and south of China, we improve the calculation method of the variable fuzzy set. Secondly, using the refined evaluation index and evaluation grade in the improved variable fuzzy set module to calculate the raw data, we get the result of the comprehensive relative membership degree (\( {u}_i^{\prime } \)) of IVFS. Thirdly, integrating the sub-modules of optimized set pair analysis model, we calculate the connection degree matrix Q with the optimized set pair analysis module. Fourthly, based on the game theory, we create an optimized weight assignment method, which determine balance weight “w” corresponding the value of each measurement parameter. Fifth, we propose the optimized Nemerow index method, and develop an improved algorithm and its function corresponding to the optimized Nemerow index method, which is as follows: \( {P}_i=\sqrt{a\left({\overline{P}}_{\max 1}^2+{\overline{P}}_{\max 2}^2+{\overline{P}}_{\max 3}^2+{\overline{P}}_{\max 4}^2+{\overline{P}}^2\right)} \). Finally, based on the ideas of traditional cumulative pollution load index, we propose an improved cumulative pollution load ratio method module. By logically fusing the above theoretical methods and their corresponding sub-modules, combined with the comprehensive fuzzy operator “C,” relative difference degree \( ``{u}_i^{\prime }," \) relative connection degree “ui,” Nemerow index value “Pi,” and cumulative pollution load ratio “Ji” and other core elements, we achieve accurate assessment and classification analysis on regional groundwater pollution status, significantly improving the accuracy of pollution evaluation and other multiple performance. In addition, using randomly collected groundwater quality data in China, we compare our method with some state-of-art groundwater pollution assessment models such as IVFSPA, PCA, and ICAUCA to test the assessment performance of OFSP. Based on the numerical results of performance indicators from the perspectives of accuracy, relevance, stability, and rationality, it can fully prove that the proposed OFSP evaluation model performs the best. At the same time, based on the numerical results of performance indicators such as difference and universality, it can also prove that the OFSP model has multiple performance advantages for groundwater pollution evaluation.

Assessment metrics and level criteria

According to the characteristics of groundwater pollution under domestic landfills in southern and northern China, this study specifically selects “MPN,” “CFU,” and “Hg,” etc. (Table 3) as the key metrics (Wu et al. 2020; Ocampoduque et al. 2006; Vasanthavigar et al. 2010; Mohebbi et al. 2013; Hoseinzadeh et al. 2015; Xi Chen and Wang 2019) of groundwater quality assessment to evaluate corresponding groundwater, and to objectively reflect the overall situation of groundwater quality in domestic waste landfills (Verma 2020; Osborne and Kovacic 2010; Beck et al. 1988).

Table 3 Assessment metrics and level criteria of groundwater
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The various criteria values of “I,” “II,” “III,” “IV,” and “V” of the assessment metrics are displayed in the following table.

Results and discussion

Evaluation results of OFSP model

In this section, this study takes random sample data collected from every landfill in Table 1 as sample to compute the Pi values and groundwater quality assessment level by the model of OFSP, and to analyze groundwater pollution characteristics in regulated and unregulated domestic waste landfills (Shuang et al. 2016; Chen et al. 2012; Yan et al. 2016; Amiri et al. 2014; Yan et al. 2014; Su et al. 2009; Zhang et al. 2011; Su et al. 2010). The brief simulation calculations of the key parts of the OFSP model and their corresponding results are explained below.

As stated, based on the groundwater sample data of Asowei# in Beijing, we employ Eqs. 7 and 8 to compute the parametric values of u11. Then the evaluation result of class “I” of the parameters (u11) are acquired. Similarly, we get the other four evaluation results of class “II”, “III”, “IV,” and “V” of the corresponding parameters. Then, all evaluation results of groundwater pollutants can be regulated to the matrix form of Q. According to the variable values of w1 and w2, which were respectively computed by Eqs. 9 and 10, we obtain the “w” value of various groundwater pollutants using Eqs. 11 to 13. In the following, the results of “C” and their “P” matrix are acquired by using the corresponding equations in Algorithm 2. At last, we obtain the Pi values (0.0616, 0.0632, 0.0522, 0.0418, and 0.0337) and groundwater quality assessment level (“II”) of Asowei# in Beijing.

In the same way, based on the same groundwater pollutant data of the rest sampling landfills of different regions such as “Daxing District” in Beijing, “Heping District: Shandong Yidi#” and “Dadong District North: Paoziyan” in Shenyang, Liaoning Province, etc., we calculate their groundwater evaluation level and Pi values through the OFSP model. By categorization, the Pi values and groundwater quality assessment levels (Table 4) of all sampling landfills in China could be presented as follows.

Table 4 The evaluation results of OFSP
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It is not difficult to find that there are 7 domestic waste landfills with the highest level (IV) of groundwater pollution in this study, which are “Laogang landfill” in Shanghai; “Ruian Dongshan garbage dump” in Wenzhou, Zhejiang Province; and “A domestic waste landfill” in Luoyang, Henan Province, etc. And there are 13 domestic waste landfills with the lowest level (II) of groundwater pollution, which are “Asowei#” in Beijing; “A domestic waste landfill#” in Shijiazhuang, Hebei Province; and “Maiji District: laogougou landfill#,” in Tianshui, Gansu Province etc. Where in southern China, the Pi values of “I,” “II,” “III,” “IV,” and “V” of “Laogang landfill” in Shanghai (class “IV”) are 0.0474, 0.0573, 0.0701, 0.0783, and 0.0684, respectively. The Pi values of “I,” “II,” “III,” “IV,” and “V” of “A domestic waste landfill#” in Shanghai (class “II”) are 0.0530, 0.0639, 0.0527, 0.0403, and 0.0401, respectively. In northern China, the Pi values of “I,” “II,” “III,” “IV,” and “V” of “Daxing District” in Beijing (class “III”) are 0.0437, 0.0546, 0.0747, 0.0537, and 0.0318, respectively. The Pi values of “I,” “II,” “III,” “IV,” and “V” of “Laizhou municipal solid waste landfill#” in Laizhou, Shandong Province (class “II”), are 0.0637, 0.0738, 0.0614, 0.0455, and 0.0433 respectively. Obviously, the groundwater pollution levels of domestic waste landfills in China range from “II” to “IV” (Fig. 4), which is a relatively serious pollution situation.

Fig. 4
figure 4

a Pollution degree of study area (Northern China). b Pollution degree of study area (Southern China)

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Analysis of various evaluation results

In this section, in order to get more persuasive analysis conclusions of the various evaluation results, we randomly select the same sample values as a method of the proposed OFSP model, and employ eight various assessment models, such as the models of VFSPA, VFEM, LPI, FSEVFS, NSFWQI, EWQI, PCA, and ICAUCA, to achieve the assessment computation separately (Singh et al. 2016;Rana et al. 2018;Manimekalai 2012;Bahroz 2015;Chonattu et al. 2016;Mishra et al. 2016;Ocampoduque et al. 2006;Vasanthavigar et al. 2010;Mohebbi et al. 2013;Hoseinzadeh et al. 2015;Xi Chen and Wang 2019). In the following, according to the above computations of various models, the corresponding assessment results (Table 5) of groundwater under the domestic waste landfills are acquired.

Table 5 The evaluation results of various models
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The groundwater quality assessment results of domestic waste landfills in different regions by various assessment methods in Table 5 show that, based on OFSP, the groundwater pollution level of “Beishenshu, Liulitun” in Beijing is grade “IV.” The groundwater pollution level of “Heping District: Shandong Yidi#” in Shenyang, Liaoning Province, is grade “III.” And the groundwater pollution level of “A domestic waste landfill#” in Shanghai is grade “II,” etc. Where, in all regulated domestic landfills, there are 0 landfills with the groundwater pollution level of IV. The number of landfills of groundwater pollution level “III” is 5, which include the landfills of “Heping District: Shandong Yidi#,” “Shenhe District: Stadium#,” and “Huanggu District: Light Industry Machinery Plant#.” And the landfills such as “Asowei#,” “Anding#,” and “Chaoyang District#,” which are the groundwater pollution level “II,” have 29. Meanwhile, in all unregulated domestic landfills, the number of landfills of groundwater pollution level “IV” is 17, which include the landfills of “Liulitun,” “Gaoantun,” and “Fengtai District.” The number of groundwater pollution level “III” is 18, including “Beishenshu,” “Jiaojiapo,” and “Daxing District” etc. And there are 0 landfills with the groundwater pollution level of II.

Dual perspective of area location and landfill type

Analyzing from the dual perspective of regional orientation and landfill type, we find that although the percentage of regulated domestic landfills in the southern region is higher than that in the northern region, the pollution degree of their groundwater is on the whole higher than that in the northern region (Xin et al. 2017; Shao et al. 2018). Then, in terms of the dimension of regulated domestic landfill, the level of groundwater pollution in the southern region is roughly the same as that in the northern region of China, while on the basis of the dimension of unregulated domestic landfill, the level of groundwater pollution in the southern region is also higher than that in the northern region of China.

Furthermore, based on the corresponding data in the above Table 5, the results of their “Ji” values can be respectively calculated by Eq. 25. For example, the “Ji” values of regulated domestic landfills in southern China are JI =0, JII =0.786, JIII =0.214, JIV =0, and JV =0. Then, according to the analysis of the above calculation results of Ji, the same quantitative conclusions can be obtained as follows.

That is, the overall pollution level of the regulated domestic landfills in southern China is at class “II” (JII =0.786), which belongs to the “II” major pollution area; and the overall pollution level of the regulated domestic landfills in northern China is also at class “II” (JII =0.900), which belongs to the “II” major pollution area. And the overall pollution levels of the unregulated domestic landfills in southern and northern China are class “IV” (JIV =0.813) and class “III” (JIII =0.789), respectively. The corresponding numerical results figure produced by the OFSP model is intuitively shown in Fig. 5.

Fig. 5
figure 5

Ratio results of area location and landfill type. Note that “S.R” (“N.R”) means regulated domestic landfills in southern (northern) China; “S.UNR” (“N.UNR”) means unregulated domestic landfills in southern (northern) China

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Perspective of domestic landfill type

Based on the analysis of perspective of domestic landfill type, the pollution degree of regulated domestic waste landfills, which has 29 “II” landfills, 5 “III” landfills, and 0 “IV” landfill, etc., is significantly lower than the unregulated domestic waste landfills, which has 0 “II” landfill, 18 “III” landfills, and 17 “IV” landfills, etc. Taking the landfills in East China as an example, according to the corresponding results in Table 5, the pollution level of regulated domestic waste landfills in this area is much lower than that of unregulated landfills. In addition, in line with the data in Table 5, the “Ji” values of corresponding regulated (or unregulated) domestic landfills such as JII, JIII, and JIV can be respectively computed by Eq. 25.

In the following, according to the discussion of the above numerical results of Ji, we use the double judgment mode to judge overall pollution level (San et al. 2018). Then, the judgment conclusion can be formed as follows. On the whole, the main pollution level of the unregulated domestic landfills is the class “IV,” and on the whole, the main pollution level of the regulated domestic waste landfill is the class “II.” The above numerical results figure (Fig. 6) produced by the OFSP model is displayed.

Fig. 6
figure 6

Number results of domestic landfill type

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Dimension of groundwater pollution component

Based on the level of pollution component categories, we analyze the distribution of all chemical elements (such as “MPN,” “CFU,” and “NH3–N”, etc.) in groundwater, and their distribution characteristics in the groundwater of various types of domestic waste landfills (Wu et al. 2020; Hoseinzadeh et al. 2015; Yan et al. 2019). From the perspective of indiscriminate chemical contamination, “CFU,” “MPN,” and “CaCO3” are the chemical elements with the largest proportion of polluting components in the groundwater of domestic waste landfills. Then, from the perspective of heavy pollutant, the proportions of “As,” “Mn,” and “COD” in the groundwater of the unregulated domestic landfills are higher, and their proportions in the groundwater of the regulated domestic landfills are lower. The proportions of pollution components such as “Na,” “NH3–N,” and “Pb” in the groundwater of both the regulated domestic landfills and the unregulated domestic landfills are similar, which are at ordinary levels. And the proportions of “MPN,” “CFU,” “CaCO3,” “Hg,” and “KMnO4” in the groundwater of both the regulated and unregulated domestic waste landfills are relatively high.

It is worth noting that although the percentage of regulated domestic waste landfills in southern China is higher than that in northern China, the proportion of groundwater pollution components such as “Hg” and “Pb” in groundwater in that area is much higher than that in northern China. Similarly, the proportions of contaminations of “MPN” and “CFU” in the groundwater of domestic landfills in the southern region are generally higher than those in the northern region.

Here, in light of the data in Table 2, the results of Ji of various contaminations can be operated using Eq. 25. For example, JNH3–N =0.975, JTDS =0.563, and JMo =0.074. Then, based on the numerical results of “Ji” values, the universal pollution indexes of groundwater (Ji >60%), which include the chemical elements of “NH3–N,” “KMnO4,” and “Hg,” are determined. The pollution indicators of “Mn,” “TDS,” and “Pb,” etc. (60%≥Ji>10%) are local pollution components in groundwater. And the remaining pollution metrics, such as “Mo,” “Cu,” and “Cd” (Ji≤10%), belong to the point source pollution component in groundwater under domestic waste landfills. The percentage histogram (Fig. 7) corresponding to the analysis results is shown.

Fig. 7
figure 7

Analysis results of pollution component dimension. Note that this figure shows the results for only a few representative groundwater pollution components

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Through in-depth analysis of the research results from the abovementioned perspectives, the following comprehensive conclusions and recommendations can be drawn. (1) The measures such as garbage classification and regulatory treatment in domestic landfills are conducive to the improvement of groundwater quality; (2) the current actions of waste sorting and regulatory treatment have effectively curbed heavy pollution components such as “As,” “Mn,” and “COD.” However, these actions are not obvious for the containment of heavy pollution components such as “MPN,” “CFU,” “CaCO3,” “Hg,” and “KMnO4,” especially the elements of “Hg,” “MPN,” and “CFU.” Hence, the further targeted measures are needed. The reason for this is that although their proportion in groundwater is not large, the intensity of pollution to groundwater is very strong. For non-heavy pollution components, these actions of waste sorting and regulatory treatment are obvious for the containment of “Cu,” “Mo,” and “Cd,” etc., but for the other non-heavy pollution components (e.g., “C6H6”) in groundwater, the containment is not more significant; (3) from the dimension of the results of Pi and assessment level to analyze the distribution characteristics of the regional location of groundwater pollution, it is not difficult to define that the pollution level of groundwater in domestic landfills (especially the unregulated domestic landfills) in the southern region, especially in eastern and southern China, is more serious. Thus, it is necessary to take some targeted and efficient measures for groundwater pollution in domestic landfills, and increase the intensity and pertinence of landfill management.

OFSP model performance

Precision performance analysis

Based on the assessment results of groundwater quality in Table 5 for various domestic waste landfills, we compare the groundwater quality levels obtained by various mainstream methods. The comparison result of assessment levels can well prove the versatility and precision of the proposed OFSP method to groundwater quality evaluation in domestic waste landfills.

In this section, we take the assessment results of both item 30 of “A domestic waste landfill/Luoyang.Henan” and item 40 of “Fengcun village garbage dump/Zhanjiang.Guangdong” in Table 4 as an example to demonstrate the superiority of our proposed OFSP model, and further verify that the OFSP model is more universal and precise for groundwater evaluation (Shuang et al. 2016; Su et al. 2010). From Table 4, we can see that only the assessment results of OFSP are class “IV”; the results of other models are “III.” The reason for this is that the “P1–P5” values of “A domestic waste landfill/Luoyang.Henan” are 0.0336, 0.0518, 0.0783, 0.0707, and 0.0492, respectively; and the “P1–P5” values of “Fengcun village garbage dump/Zhanjiang.Guangdong” are 0.0231, 0.0346, 0.0604, 0.0535, and 0.0329, respectively (Fig. 8). It is clear that the assessment results of both “A domestic waste landfill/ Luoyang.Henan” and “Fengcun village garbage dump/Zhanjiang. Guangdong” are initially positioned between class “III” and “IV.” Then, through careful analysis, we find that the “P2” value in the table is significantly lower than “P4”. As such, we adopt the double judgment mode to achieve conclusion that the assessment results of the second judgment to “A domestic waste landfill/ Luoyang.Henan” and “Fengcun village garbage dump/Zhanjiang. Guangdong” are class “IV,” which is more precise and universal. The corresponding judgment diagram of numerical result is shown below.

Fig. 8
figure 8

Judgment diagram of numerical result

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Performance indices validation

To compare the proposed OFSP model with some mainstream models, based on groundwater of domestic waste landfill in various regions of the country, we employ the same way using in the study, and select the random data from 69 sampling wells to run validation experiment about 4000 times, so as to further confirm the superior performance of our OFSP model. The validation results in 6 distinct aspects of performance indices of 9 various models, including of OFSP, VFSPA, VFEM, LPI, FSEVFS, NSFWQI, EWQI, PCA, and ICAUCA, are obtained as follows (Singh et al. 2016; Rana et al. 2018; Manimekalai 2012; Bahroz 2015; Chonattu et al. 2016; Mishra et al. 2016; Ocampoduque et al. 2006; Vasanthavigar et al. 2010; Mohebbi et al. 2013; Hoseinzadeh et al. 2015; Xi Chen et al. 2019).

In Fig. 9, the horizontal axis represents various models, and the vertical axis represents the values of precision. Of all experiments, the proposed OFSP model has the best performance of precision, which obtains the highest precision result (0.985) of all evaluation models in groundwater quality evaluation. Meanwhile, the precision results of ICAUCA, PCA, EWQI, NSFWQI, FSEVFS, LPI, VFEM, and VFSPA are 0.803, 0.873, 0.868, 0.794, 0.756, 0.891, 0.917, and 0.952, respectively. Thus, the mean of precision of all evaluation models is 0.871.

Fig. 9
figure 9

Precision values

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In Fig. 10, the horizon axis represents various methods, and the vertical axis represents the values of robustness. Of all experiments, our OFSP model has a good ability of robustness, which achieves the highest robustness result (0.953) of all evaluation models in groundwater quality evaluation. Meanwhile, we could easily get that the other mainstream models like ICAUCA, PCA, EWQI, NSFWQI, FSEVFS, LPI, VFEM, and VFSPA do not achieve a higher level of robustness compared with the proposed OFSP model. Here, the robustness results of them are 0.778, 0.831, 0.829, 0.882, 0.784, 0.921, 0.927, and 0.936, respectively. Therefore, the mean of robustness of all assessment models is 0.872.

Fig. 10
figure 10

Robustness values

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In Fig. 11, the horizon axis represents the various models, and the vertical axis represents the corresponding discrimination results. It is clear that the discrimination results of ICAUCA, PCA, EWQI, NSFWQI, FSEVFS, LPI, VFEM, VFSPA, and OFSP are 0.277, 0.233, 0.284, 0.301, 0.268, 0.211, 0.206, 0.195, and 0.217, respectively. Thus, the mean of discrimination of all assessment models is 0.244. Obviously, in the most experiments, compared with other eight assessment models, we could easily see that the proposed OFSP model achieves the middle level of discrimination for groundwater evaluation. The reason for being inferior to the mainstream models of LPI, VFEM, and VFSPA may be that the proposed OFSP model has a rigorous and comprehensive assessment structure.

Fig. 11
figure 11

Discrimination values

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In Fig. 12, the horizontal axis represents various models, and the vertical axis represents the values of correlation. Of all experiments, the proposed OFSP model has the best performance of correlation in groundwater evaluation, which obtains the highest correlation result (0.934) of all evaluation models in groundwater quality evaluation. Meanwhile, the correlation results of ICAUCA, PCA, EWQI, NSFWQI, FSEVFS, LPI, VFEM, and VFSPA are 0.773, 0.847, 0.821, 0.759, 0.735, 0.872, 0.908, and 0.926, respectively. Hence, the mean of correlation of all evaluation models is 0.842.

Fig. 12
figure 12

Correlation values

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In Fig. 13, the horizon axis represents various methods, and the vertical axis represents the value of rationality. Here, the mean of rationality of all evaluation models is 0.802, and the rationality results of ICAUCA, PCA, EWQI, NSFWQI, FSEVFS, LPI, VFEM, VFSPA, and OFSP are 0.730, 0.805, 0.629, 0.703, 0.702, 0.857, 0.901, 0.943, and 0.946, respectively. Obviously, of all experiments, our designed OFSP model outperforms the other evaluation methods in groundwater quality evaluation.

Fig. 13
figure 13

Rationality values

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In Fig. 14, the horizon axis represents the various models, and the vertical axis represents the results of versatility. It is not difficult to find that the results of versatility of ICAUCA, PCA, EWQI, NSFWQI, FSEVFS, LPI, VFEM, VFSPA, and OFSP are 0.812, 0.776, 0.939, 0.927, 0.787, 0.901, 0.913, 0.824, and 0.837, respectively. Hence, the mean of versatility of all assessment models is 0.858. Obviously, in the most experiments, compared with other eight assessment models, we can easily get that the proposed OFSP model achieves the middle rank of versatility. The rigorous and comprehensive evaluation structure is the main reason for being worse than the mainstream models of EWQI, NSFWQI, LPI, and VFEM in performance metric of versatility.

Fig. 14
figure 14

Versatility values

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Finally, in order to make comparison of the performance indices of precision, robustness, discrimination, correlation, rationality, and versatility more clearly, we summarize the performance metrics values of precision, robustness, discrimination, etc., into the following table (Table 6) for comparison and analysis. All comparison results of performance indexes are present in Table 6.

Table 6 The results of performance indices
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It is easy to find that, in line with precision (0.985), correlation (0.934), robustness (0.953), and rationality (0.946), our designed OFSP model has the best performance. In addition, according to the indexes of discrimination (0.217) and versatility (0.837), the designed OFSP model also has a good ability. And the abovementioned sound conclusions can also be confirmed in Fig. 15.

Fig. 15
figure 15

Performance of indices values

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Discussion

The research randomly collects the values of groundwater pollution index parameter from some representative regions of North and South China and uses the OFSP evaluation model to conduct comprehensive calculations (Yan et al. 2016; Shuang et al. 2016; Amiri et al. 2014; Yan et al. 2014; Chen et al. 2012; Zhang et al. 2011; Su et al. 2010; Su et al. 2009). We obtain the pollution levels “I,” “II,” “III,” “IV,” and “V” and the optimized Nemerow index value “Pi” based on the sample evaluation locations corresponding to different locations in the north and south of China. Through the analysis and collection of the groundwater pollution evaluation results, we obtain the overall and local groundwater pollution characteristics in the north and south of China. From the perspective of the whole China, there are 7 domestic waste landfills with the highest level (IV) of groundwater pollution in this study, which are “Laogang landfill” in Shanghai; “Ruian Dongshan garbage dump” in Wenzhou, Zhejiang Province; and “A domestic waste landfill” in Luoyang, Henan Province, etc. And there are 13 domestic waste landfills with the lowest level (II) of groundwater pollution, which are “Asowei#” in Beijing; “A domestic waste landfill#” in Shijiazhuang, Hebei Province; and “Maiji District: laogougou landfill#,” in Tianshui, Gansu Province etc. Therefore, the level of groundwater pollution is serious in the north and south of China. From the perspective of the specific locations of China, in southern China, the Pi values of “I,” “II,” “III,” “IV,” and “V” of “Laogang landfill” in Shanghai (class “IV”) are 0.0474, 0.0573, 0.0701, 0.0783, and 0.0684, respectively. The Pi values of “I,” “II,” “III,” “IV,” and “V” of “A domestic waste landfill#” in Shanghai (class “II”) are 0.0530, 0.0639, 0.0527, 0.0403, and 0.0401, respectively. In northern China, the Pi values of “I,” “II,” “III,” “IV,” and “V” of “Daxing District” in Beijing (class “III”) are 0.0437, 0.0546, 0.0747, 0.0537, and 0.0318, respectively. The Pi values of “I,” “II,” “III,” “IV,” and “V” of “Laizhou municipal solid waste landfill#” in Laizhou, Shandong Province (class “II”), are 0.0637, 0.0738, 0.0614, 0.0455, and 0.0433 respectively. Obviously, the groundwater pollution levels of domestic waste landfills in China range from “II” to “IV,” which is a relatively serious pollution situation. Therefore, whether it is in southern China or northern China, the level of groundwater pollution is significantly serious.

And then, we apply different groundwater pollution evaluation models such as OFSP, VFSPA, and ICAUCA to calculate and judge the corresponding groundwater pollution levels (Mohebbi et al. 2013; Hoseinzadeh et al. 2015; Xi Chen and Wang 2019; Singh et al. 2016; Rana et al. 2018; Manimekalai 2012; Bahroz 2015; Chonattu et al. 2016; Mishra et al. 2016; Ocampoduque et al. 2006; Vasanthavigar et al. 2010). Based on the groundwater pollution evaluation results calculated by the nine different evaluation models, we conduct a multi-level analysis on groundwater pollution under different landfills in the entire north and south regions. Finally, we get the pollution characteristics and regulations of groundwater pollution in the north and south of China after conducting a multi-level analysis from dual perspective of area location and landfill type, perspective of domestic landfill type, and dimension of groundwater pollution component. The results are as follows: (1) dual perspective of area location and landfill type—the overall pollution level of the regulated domestic landfills in southern China is at class “II,” which belongs to the “II” major pollution area; and the overall pollution level of the regulated domestic landfills in northern China is also at class “II,” which belongs to the “II” major pollution area. And the overall pollution levels of the unregulated domestic landfills in southern and northern China are class “IV” and class “III,” respectively. (2) Perspective of domestic landfill type: on the whole, the main pollution level of the unregulated domestic landfills is class “IV,” and on the whole, the main pollution level of the regulated domestic waste landfill is the class “II.” (3) Dimension of groundwater pollution component: from the perspective of indiscriminate chemical contamination, “CFU,” “MPN,” and “CaCO3” are the chemical elements with the largest proportion of polluting components in the groundwater of domestic waste landfills. Then, from the perspective of heavy pollutant, the proportions of “As,” “Mn,” and “COD” in the groundwater of the unregulated domestic landfills are higher, and their proportions in the groundwater of the regulated domestic landfills are lower. The proportions of pollution components such as “Na,” “NH3–N,” and “Pb” in the groundwater of both the regulated domestic landfills and the unregulated domestic landfills are similar, which are at ordinary levels. And the proportions of “MPN,” “CFU,” “CaCO3,” “Hg,” and “KMnO4” in the groundwater of both the regulated and unregulated domestic waste landfills are relatively high.

Due to the significant negative factors such as high pollution, large number of chemical pollution discharge companies, high negative environmental effects on economic aggregation, large discharge of chemical and industrial waste, interaction with some positive factors such as regional groundwater pollutant discharge control, prevention technology, strategy optimization, and financial support, groundwater pollution is generally serious in the north and south of China, especially in the south.

It goes without saying that the key reason for good performance of OFSP model has been its predominant arithmetic module of the optimized N.L. Nemerow index equation and double judgment approach (Shuang et al. 2016; Xu et al. 2015; Wang et al. 2011). The OFSP model balances the actual effect of each chemical element and approximate the real pollution effect through the following methods (Wu et al. 2018; Yan et al. 2014; Aunnop et al. 2019; Tamiris et al. 2019; Beheshti 2019). Firstly, the OFSP model adds three variable parameters, such as \( {\overline{P}}_{\max 2}^2 \), \( {\overline{P}}_{\max 3}^2,\mathrm{and}\ {\overline{P}}_{\max 4}^2 \), so as to highlight the large actual pollution effect of the heavily polluted elements while taking into account the comprehensive calculation of multiple variable indicators; secondly, it employs \( {\overline{P}}_{\mathrm{max}}^2 \) instead of \( {P}_{\mathrm{max}}^2 \) to restore and highlight the actual pollution effect of the heavy pollution elements as much as possible; finally, it computes the arithmetic mean of Pi with Pmax1, Pmax2, Pmax3, and Pmax4, respectively, so that the conversion values of \( {\overline{P}}_{\max 1}^2 \),\( {\overline{P}}_{\max 2}^2 \), \( {\overline{P}}_{\max 3}^2 \), and\( {\overline{P}}_{\max 4}^2 \) are achieved.

However, because the OFSP model incorporates more independent operation modules, its calculation speed is relatively slow, which affects the timeliness of the whole OFSP model for groundwater detection. That is the limitation of this study. Therefore, in further study, we are planning to extend the research from the perspective of timeliness performance. While ensuring the performance of accuracy, we will focus on improving the timeliness performance of the proposed OFSP model.

Conclusion

In this paper, a novel OFSP is presented for accurate and rigorous evaluation of groundwater under domestic landfills. The OFSP model developed in the study in line with VFS and optimized N.L. Nemerow index, which has been equipped with both the optimized double judgment approach and the “Pi” equation of N.L. Nemerow index, can significantly enhance the precision and robustness of the groundwater quality evaluation of domestic waste landfill. Thus, the proposed model can achieve the reasonable groundwater quality assessment model, the stable groundwater quality evaluation process, and the convincing evaluation results. In order to clarify the comprehensive impacts of domestic landfills in different regions and types on groundwater pollution, and demonstrate the effectiveness of OFSP, a case study on groundwater quality assessment of various domestic landfills in China is conducted. The experiment results show that the OFSP model could get a scientific conclusion of comprehensive impacts, and could play a good performance on groundwater quality evaluation in domestic landfills, compared with other mainstream models. As such, the OFSP model proposed in this study would be a reliable tool for environmental managers and engineers to evaluate groundwater in domestic landfills.