A composite index is a synthesis of several sources of information evaluated in or on a system to describe the system that is not explicitly observable. Taking into account both qualitative and quantitative characteristics of the index, judgments from a diverse expert panel were gathered via questionnaire surveys at various stages of index development as illustrated in Figure. 2.
The formulation primarily entails four phases
- Selection of Parameters
- Determination of weights of selected parameters
- Development of sub-index curves for the parameters
- Selection of the appropriate aggregation function
2.1. Selection of Parameters
A comprehensive list of 62 parameters that have the potential to contaminate the leachate was put together based on the literature review and is specified in Table 1. The concentration of the parameters in the leachate in the available literature, as well as the effect of the parameters on the receiving environment and human health, were critical factors in parameter selection. The parameters were divided into two categories. Group 1 consisted of critical parameters that are either found in high concentrations in landfills or have the potential to cause an adverse effect on human health. Group 2 consisted of parameters that are present in leachate but not in such high concentrations to cause an adverse effect on human health.
The fuzzy Delphi method (FDM) was used to select the parameters to be included in the r-LPI via an expert questionnaire survey. The FDM (Ishikawa 1993) incorporates fuzzy set theory (Zadeh 1965) into the standard Delphi method (Dalkey and Helmer 1962). The standard Delphi method is incapable of handling the fuzziness and ambiguity inherent in expert opinions (Chang 2013). To address the shortcomings of the conventional Delphi method, FDM was used for the screening of parameters.
Table1: List of Parameters Proposed for Inclusion in r-LPI
LIST A PARAMETERS
|
LIST B PARAMETERS
|
Aluminum
|
Total Organic Carbon
|
Cadmium
|
Lead
|
Chemical Oxygen Demand
|
Phosphate
|
Cobalt
|
Biological Oxygen Demand
|
Ortho Phosphorus
|
Zinc
|
Benzene
|
Nitrate
|
Nickel
|
Toluene
|
Organic Nitrogen
|
Copper
|
1,2 Dichloroethane
|
Dissolved Methane
|
Arsenic
|
Dichloromethane
|
Total Volatile Acids
|
Mercury
|
Naphthalene
|
Total Coliform Bacteria
|
Chromium
|
Phenolic Compound
|
Fixed Solids
|
Selenium
|
Ethyl Benzene
|
Hardness
|
Chlorides
|
Delta BHC
|
Total Solids
|
Fluoride
|
Xylenes
|
Volatile Suspended Solids
|
Sulphate
|
Phthalate Esters
|
Total Suspended Solids
|
Potassium
|
Chloroform
|
Turbidity
|
Calcium
|
Acetone
|
Pesticides
|
Magnesium
|
Cyanide
|
Perfluorinate Compounds
|
Total Iron
|
Methyl Ethyl Ketone
|
Pharmaceuticals & Personal Care Products (PPCPs)
|
Sodium
|
Vinyl Chloride
|
|
Total Phosphorus
|
Fecal Coliform Bacteria
|
|
Manganese
|
pH
|
|
Ammoniacal Nitrogen
|
Conductivity
|
|
Total Kjeldahl Nitrogen
|
Total Dissolved Solids
|
|
Alkalinity
|
|
|
In the preliminary questionnaire, the panelists were briefed regarding the development of r-LPI. The range of the concentration of parameters present in the landfill leachate, as well as their potential impact on human health and the environment were discussed. They were subsequently asked to rate all the parameters on a 9-point linguistic scale, as shown in Table 2., based on their potential to cause an adverse effect on human health and the environment.
Table 2
Triangular fuzzy numbers for nine-point scale
Linguistic Expressions
|
Fuzzy Number
|
Extremely important
|
(8,9,9)
|
Between very and extremely important
|
(7,8,9)
|
Very Important
|
(6,7,8)
|
Between moderate and Very important
|
(5,6,7)
|
Moderately important
|
(4,5,6)
|
Between very unimportant and Moderately important
|
(3,4,5)
|
Very unimportant
|
(2,3,4)
|
Between extremely and Very unimportant
|
(1,2,3)
|
Extremely unimportant
|
(1,1,1)
|
For the preliminary questionnaire, a panel of 100 environmental experts were contacted in several phases over the course of two months. All the panelists were experts in the field of environmental engineering, predominantly in the field of waste management.
After the collection of fuzzified expert’s opinions, equation 1 was used to aggregate expert’s opinions.
|
\({l}_{ij}={\left(\prod _{k =1 }^{k}{l}_{ijk}\right)}^{1/k}, {m}_{ij}={\left(\prod _{k =1 }^{k}{m}_{ijk}\right)}^{1/k}, {u}_{ij}={\left(\prod _{k =1 }^{k}{u}_{ijk}\right)}^{1/k}\)
|
(1)
|
After fuzzy aggregation of expert’s opinion, defuzzification of fuzzified values is accomplished using equation 2 (Hsu et al. 2010; Wu and Fang 2011).
|
\(F= \frac{L + M + U}{3}\)
|
(2)
|
After defuzzification of the expert’s opinion, the screening criteria for the parameters to be included in the r-LPI were set at 7.0 based on the expert’s opinion. Table 3 summarizes the preliminary questionnaire findings.
Table 3
Defuzzified results of FDM
Leachate Parameter
|
Defuzzified Values
|
Mercury
|
7.984
|
Lead
|
7.762
|
Arsenic
|
7.844
|
Total Chromium
|
7.427
|
BOD
|
7.025
|
COD
|
7.053
|
pH
|
7.034
|
FCB
|
7.097
|
Cyanide
|
7.400
|
Phenolic Compound
|
7.025
|
Pesticides
|
7.043
|
2.2. Determination of Weights
In this step, the relative value or contribution of an indicator to an index is reflected in the form of weight assigned to it in the index. There are a multitude of weighting techniques available, each of which can generate a unique set of overall results (OECD 2008). Although several composite indicators with equal weighting parameters have been reported in the literature (Babcock 1970; Dojlido et al. 1994; Ott and Thorn 1976). Assigning equal weights to all the parameters may result in an incoherent index structure during the grouping and aggregation process (OECD 2008). The statistical weighting method, like principal component analysis, may result in irrational weighing, with insignificant parameters securing higher relative weights. Methods entailing expert opinions like AHP should make it easier to prioritize criteria based on their importance.
Accounting for subjectivity in such dynamic decision-making necessitates the use of multi-criteria decision-making techniques. AHP (Saaty 1977) is one of the most extensively used multi-criteria decision making (MCDM) techniques in MSW management (Ekmekçioĝlu et al. 2010; Goulart Coelho et al. 2017; Soltani et al. 2015; Yap and Nixon 2015). Although AHP is designed to elicit expert knowledge, it is incapable of representing human thoughts as it involves human subjectivity, which induces a vagueness type of uncertainty and necessitates the use of decision-making under uncertainty (Kahraman et al. 2003). The standard AHP methodology is flawed because it seeks an exact value to articulate the decision maker’s judgment in comparison to the alternative (Wang and Chen 2007). The AHP approach is often admonished because it employs an unbalanced scale of judgment and fails to account for the inherent ambiguity and uncertainty in the pairwise comparison (Deng 1999). A fuzzy AHP, synthesis of AHP, and fuzzy theory (Zadeh 1965) were introduced to resolve the shortcomings of traditional AHP (Van Laarhoven and Pedrycz 1983). It has been discovered that decision-makers are more precise and consistent in making interval judgments than when making fixed value judgments (Bozbura et al. 2007; Wang et al. 2016). This is due to their inability to express the fuzzy essence of the comparison process (Kahraman et al. 2003). Thus, in this study, relative weights of the parameters of the r-LPI were determined using FAHP.
There are various FAHP methods that can be used to calculate the weights of the r-LPI parameters. In order to obtain crisp weights from the fuzzy pairwise comparison matrices, there are three FAHP methods, namely, the extent analysis (Chang 1996), the fuzzy preference programming (FPP) based nonlinear method (Mikhailov 2003), and the logarithmic fuzzy preference programming (LFPP) (Wang and Chin 2011). All three FAHP methods were used to calculate and compare the weights of the r-LPI parameters and the results were reported elsewhere. From the comparative analysis, the LFPP method was chosen as its results were the most accurate (Bisht et al. 2022a).
When using FAHP to rank alternatives, there are four key stages: goal identification, hierarchy development, creation of pairwise comparison matrices, and relative weight calculation. The hierarchical structure of the problem for ranking the parameters by FAHP is illustrated in Figure. 3. In the second questionnaire, the panelists were asked to give their responses on a linguistic scale for the development of a fuzzy pairwise comparison matrix. All the experts that responded to the first questionnaire were consulted.
A linguistic scale was used to collect the responses of the experts. The concept of linguistic variables allows for the approximate representation of phenomena that are too complex or ill-defined to be expressed in a conventional, quantifiable form. Table 4 shows how the assessment of weights is represented by a linguistic component.
The parameters of the r-LPI were divided into 3 main criteria, namely:
1. Basic Pollutants
2. Heavy Metals
3. Toxicants
Table 4
Linguistic Variable for Pairwise Comparison
Linguistic Scale
|
Fuzzy Number
|
Linguistic Scale
|
Fuzzy Reciprocal Scale
|
Equally Important
|
1 = (1,1,1)
|
Equally Unimportant
|
1 = (1,1,1)
|
Equal to Moderately Important
|
2 = (1,2,3)
|
Equal to Moderately Unimportant
|
1/2 = (1/3, 1/2, 1)
|
Moderately Important
|
3 = (2,3,4)
|
Moderately Unimportant
|
1/3 = (1/4, 1/3, 1/2)
|
Moderately to Strongly Important
|
4 = (3,4,5)
|
Moderately to Strongly Unimportant
|
1/4 = (1/5, 1/4, 1/3)
|
Strongly Important
|
5 = (4,5,6)
|
Strongly Unimportant
|
1/5 = (1/6, 1/5, 1/4)
|
Strongly to Very Strongly Important
|
6 = (5,6,7)
|
Strongly to Very Strongly Unimportant
|
1/6 = (1/7, 1/6, 1/5)
|
Very Strongly Important
|
7 = (6,7,8)
|
Very Strongly Unimportant
|
1/7 = (1/8, 1/7, 1/6)
|
Very Strongly Important to Extremely Important
|
8 = (7,8,9)
|
Very Strongly Important to Extremely Unimportant
|
1/8 = (1/9, 1/8, 1/7)
|
Extremely Important
|
9 = (8,9,9)
|
Extremely Unimportant
|
1/9 = (1/9, 1/9, 1/8)
|
Firstly, the criteria were ranked relative to their importance to the goal, i.e. Pollution potential of landfill leachate. After that, a pairwise comparison of the parameters resulting from the preliminary survey was done based on the criteria in which they are categorized. The pairwise comparison matrix to record the responses of the experts is shown in Table 5. The experts were given four such pairwise comparison matrices to capture their responses.
Table 5
Pairwise comparison of the criteria based on their Pollution potential
Pollution Potential
|
Toxicants
|
Metals
|
Basic Pollutants
|
Toxicants
|
1
|
A12
|
A13
|
Metals
|
X
|
1
|
A21
|
Basic Pollutants
|
X
|
X
|
1
|
In the subsequent steps, the parameters within the criteria were compared with each other based on their potential to contaminate the landfill leachate. After the creation of the pairwise comparison matrix, the responses of the experts were checked for consistency using the consistency ratio (CR), which was computed using the consistency index (RI) and the random index (RI). The consistency ratio (CR), which was calculated using equation (4).
|
Consistency Index,\(CI = \frac{{}_{max}-n}{n - 1}\) | (3) |
Where n denotes the number of parameters being compared.
| Consistency Ratio,\(CR = \frac{CI}{RI}\) | (4) |
RI is dependent on the value of n. Responses with a CR up to 0.1 can be considered consistent, although the value of 0 is considered optimal (Saaty 1977). Responses with a CR exceeding 0.1 were returned to the panelists for revision attributable to logical discrepancies and inconsistent judgments in the pairwise comparisons. The details of the responses received are depicted in Figure 4.
The relative weight of the criteria and sub-criteria was estimated using the LFPP. The LFPP method is summarized below.
In the above method, we take the logarithmic of the fuzzy pairwise comparison matrix using the approximate equation:
| ln ãij ≈ (ln lij, ln mij, ln uij), I,j = 1, ……, n. | (5) |
As a result, the membership function of a triangular fuzzy opinion can be defined as
| \({\mu }_{ij} \left(ln\left(\frac{{w}_{i}}{{w}_{j}}\right)\right)= \left\{\begin{array}{c}\frac{ln \left({w}_{i}/{w}_{j}\right)-{ln l}_{ij}}{ln {m}_{ij}-{l}_{ij}}, \\ \frac{{ln u}_{ij }- ln \left({w}_{i}/{w}_{j}\right)}{{ln u}_{ij}-{ln u}_{ij}}, \end{array}\begin{array}{c}ln \left(\frac{{w}_{i}}{{w}_{j}}\le ln {m}_{ij}\right),\\ ln \left(\frac{{w}_{i}}{{w}_{j}}\ge ln {m}_{ij}\right),\end{array}\right.\)
|
(6)
|
Where \({\mu }_{ij}\) (ln (wi/wj)) denotes the degree of membership of ln (wi/wj) in the approximate fuzzy judgment ln ãij = (ln lij, ln mij, ln uij). The crisp priority vector λ = min { \({\mu }_{ij}\) (ln (wi/wj)) | I = 1, ……., n – 1; j = i+1, ……, n} can be used to optimize the minimum membership degree. The resulting model can be constructed as follows:
Maximizeλ
|
\(\text{S}\text{u}\text{b}\text{j}\text{e}\text{c}\text{t}\text{e}\text{d} \text{t}\text{o}\left\{\begin{array}{c}{\mu }_{ij} \left(ln\left(\frac{{w}_{i}}{{w}_{j}}\right)\right)\ge , \\ {w}_{i}\ge 0,\end{array} \begin{array}{c}i = 1, ........, n-1; j= i+1, ........., n,\\ i = 1, ......, n,\end{array}\right.\)
|
(7)
|
Or as
Maximize 1 -λ
Subjected to
|
\(\left\{\begin{array}{c}ln{w}_{i}-ln{w}_{j}-\text{l}\text{n}\left({m}_{i\text{j}}/{l}_{i\text{j}}\right)\ge \text{l}\text{n}{\text{l}}_{\text{i}\text{j}},\\ -ln{w}_{i}+ln{w}_{j}-\text{l}\text{n}\left({u}_{i\text{j}}/{m}_{i\text{j}}\right)\ge - \text{l}\text{n}{\text{u}}_{\text{i}\text{j}},\\ {w}_{i}\ge 0,\end{array} \right.\begin{array}{c}i = 1, ........, n-1; j= i+1, ........., n,\\ i = 1, ........, n-1; j= i+1, ........., n,\\ i = 1, ........, n\end{array}\)
|
(8)
|
The above two equivalent models do not incorporate the normalization constraint \(\sum _{i=1}^{n}{w}_{i}\). This is because if the normalization constraint is used, the model would become computationally intensive. After the model’s priority is obtained; the normalization process can be done using the equation (8). Before normalization, without sacrificing generality, we can assume \({w}_{i}\ge 1\) for all the i = 1, ……., n such that \({ln w}_{i}\ge 0\) for i = 1, …..., n. The non-negative assumption for \({ln w}_{i}\ge 0\) (i = 1, ……., n) is not essential.
In general, the above model does not guarantee that the membership degree λ will have a positive value. This is because no weight exists within their support interval that can satisfy all the fuzzy judgments Ã. That is, not all the inequalities \(ln{w}_{i}-ln{w}_{j}-ln\left({m}_{i\text{j}}/{l}_{i\text{j}}\right)\ge ln{l}_{i\text{j}}\) or \(-ln{w}_{i}+ln{w}_{j}-ln\left({u}_{i\text{j}}/{m}_{i\text{j}}\right)\ge - ln{\text{u}}_{i\text{j}}\) may exist at the same time.
To prevent I from taking negative value, two non-negative deviation variables \({\delta }_{ij} and{ \eta }_{ij}\)for I = 1, ……, n-1 and j = i+1, ……, n are used, and the following objective function and constraints LFPP are achieved:
|
\(Mininize J={\left(1 - \right)}^{2}+M\bullet \sum _{i = 1}^{n-1}\sum _{j= i+1}^{n}\left({\delta }_{ij}^{2}+ {\eta }_{ij}^{2}\right)\)
|
(9)
|
|
\(subjected to \left\{\begin{array}{c}{x}_{i} - {x}_{j}-\left({\text{m}}_{\text{i}\text{j}}/{\text{l}}_{\text{i}\text{j}}\right)+{\delta }_{i\text{j}}\le ln {l}_{i\text{j}},\\ {-x}_{i} + {x}_{j}-\left({\text{u}}_{\text{i}\text{j}}/{\text{m}}_{\text{i}\text{j}}\right)+{\eta }_{i\text{j}}\le -ln {u}_{i\text{j}},\\ ,{\text{x}}_{\text{i}}\ge 0,\\ {\delta }_{i\text{j}}, {\eta }_{i\text{j}}\ge 0,\end{array} \begin{array}{c}i = 1, ........, n-1; j= i+1, ........., n,\\ i = 1, ........, n-1; j= i+1, ........., n,\\ i = 1, ........, n,\\ i = 1, ........, n-1; j= i+1, ........., n,\end{array}\right.\)
|
(10)
|
Let xi (I = 1,2, ……., n) be the optimal solution to the model. The normalized priorities for fuzzy pairwise comparison matrix à =\({\left(\text{ã}\text{i}\text{j}\right)}_{n\times n}\) can be obtained as
|
\({w}_{i}=\frac{exp\left({x}_{i}\right)}{\sum _{j-1}^{n}exp\left({x}_{j}\right)}, \text{i} = 1, \dots \dots , \text{n}\)
|
(11)
|
The relative weights of the criteria and sub-criteria thus obtained are tabulated in Table 6
Table 6
Weights of the parameters of the r-LPI
Criteria
|
Criteria Weight
|
Sub-Criteria
|
Sub Criteria Local Weights
|
Global Weights
|
Toxicants
|
0.380
|
Cyanide
|
0.451
|
0.171
|
Pesticides
|
0.299
|
0.114
|
Phenolic Compounds
|
0.251
|
0.095
|
Metals
|
0.363
|
Mercury
|
0.374
|
0.136
|
Lead
|
0.255
|
0.093
|
Arsenic
|
0.231
|
0.084
|
Total Chromium
|
0.140
|
0.051
|
Basic Pollutants
|
0.257
|
FCB
|
0.305
|
0.078
|
BOD
|
0.278
|
0.071
|
COD
|
0.240
|
0.062
|
pH
|
0.176
|
0.045
|
2.3. Development of Normalized Curves
Composite indicators such as r-LPI is a unique index developed by the coalescence of chosen parameters with varying relative weights. In this step, the r-LPI parameters were transformed into a uniform scale. Only then can the parameters be aggregated. Normalization is a crucial step in the formulation of r-LPI, as it transforms potentially incomparable parameters to a scale that can be compared. Ranking, standardization, and categorical scaling are some of the recommended normalization methods (OECD 2008). There are various functions used for the normalization of sub-index curves. The commonly used functions are the implicit function, which is inexpressible by a mathematical equation but can be plotted on a graph, or an explicit function, which can be represented via a mathematical equation. A multitude of environmental indices has used these functions, like the water quality indices (Almeida et al. 2012; Brown et al. 1970; House and Newsome 1989), the Leachate pollution index (Kumar and Alappat 2003), and the i-index (Sebastian et al. 2019a).
The rating curves were drawn for each of the 11 parameters contributing to the development of r-LPI. The curves were engineered to reflect the contribution of the parameters to leachate Pollution as a function of their concentration. Consequently, the abscissa bounds were set in accordance with the concentration range of individual parameters. The equivalent normalized value, i.e. the level of leachate pollution which varied between 5-100 was indicated on the ordinate of the curve. The rating curves were so developed that at no point did they generate a null value, opening avenues for multiplicative aggregation techniques in the subsequent stages.
The leachate disposal standards and the concentration range of the parameters reported in landfill leachate were considered. Since all of the r-LPI parameters, except for pH, indicate increased pollution with an increase in the concentration of the parameters, the graph exhibited a continually increasing trend. In the case of pH, the graph was divided into three parts: as pH increases from 2 to 5, the curve had a sharp negative slope, since higher pH values in this range correspond to less pollution potential, resulting in a lower normalized score. When the pH range was 5 – 9, the curve was flat, correlating to a low normalized score, as it is the optimal range of pH for leachate. When pH varied from 9 – 14, a sharply ascending curve was drawn because a higher pH value in this range correlates to high pollution potential, resulting in an increased normalized value. The curves were implicitly drawn because of their non-linearity. Therefore, a mathematical equation cannot uniformly represent them. Even though mathematical functions have been set for uniform and non-uniform normalization curves (Swamee and Tyagi 2007), the behavior of different parameters cannot precisely be established, eventually leading to inconsistencies (Singh et al. 2008).
The curves thus developed were sent to a panel of 35 experts in the form of a third questionnaire. The panelists were then asked to develop the rating curves that represented the leachate pollution produced by various strengths or concentrations of the individual r-LPI pollutants. The panelists were provided information pertaining to the leachate disposal standards, the average concentration, and the range of the concentration of the pollutants to facilitate the development of the rating curves. In the third questionnaire, a 70% response rate was received. Although the panelist's views were generally agreed upon, a few panelists proposed slight changes. Almost 22% of the experts on the panel decided to modify the graph. An average curve was therefore developed, which incorporated all the changes that the panelists proposed for the final normalized curves.
The final curves, as shown in figure 5, can be used to retrieve the sub-index values of the r-LPI parameters.
2.4. Aggregation of Sub-indices
Aggregation is the final and one of the most important steps in the development of a composite indicator. It is a process that involves the integration of the sub-indices to form a single composite index, like r-LPI, to quantify the Pollution potential of landfill leachate. During aggregation, there may be a loss of some information. However, the information lost should not lead to misinterpretation of the result. Otherwise, the utility of the indices will decline.
Several aggregation functions have been used for the development of environmental indices (OECD 2008; Ott 1978). Additive aggregation methods (Brown et al. 1970; Kumar and Alappat 2004; Sebastian et al. 2019b) and multiplicative aggregation methods (Almeida et al. n.d.; Dinius 1987;) are commonly used aggregation methods. Although there are no rules for the selection of an aggregation function, however, the chosen aggregation function can have an impact on the usefulness of the indicator being developed.
Table 7
Sub-Index Values of the r-LPI Parameters
Parameter
|
Concentration (mg/L)
|
Sub-index value
|
Cyanide
|
0.03
|
8
|
Pesticidesa
|
20
|
52
|
Phenolic Compounds
|
0.25
|
6
|
Mercury
|
0.87
|
99
|
Lead
|
0.6
|
31
|
Arsenic
|
0.03
|
7
|
Total Chromium
|
3.22
|
40
|
FCBa
|
13
|
64
|
COD
|
5653
|
63
|
BOD
|
2641
|
68
|
pH
|
8.2
|
15
|
Note: All values are u=in mg/L except, pH and FCB. |
a Assumed concentration values. |
Most aggregation models encounter ambiguity, eclipsing, transparency, and rigidity as issues and problems caused by the abstraction of information and data (Jollands et al. 2003). Ambiguity or overestimation occurs if the aggregated value, even if the sub-indices are within limits, exceeds the permissible limits. In contrast, eclipsing occurs when, despite the fact that one or more sub-indices exceed the permissible value, the aggregated value is still within the permissible limits. Rigidity occurs when the addition of supplementary variables leads to inconsistencies in the aggregated value due to weakness in the aggregation function. The problem of transparency arises when information is lost during the process of disintegration of the index and when the aggregation function is insensitive and does not recognize the importance of the contributing sub-indices. All of these issues will eventually
Table 8
r-LPI values for the study area using different aggregation functions
Aggregation Function
|
Mathematical Form
|
r-LPI Values
|
Unweighted Arithmetic
|
\(\sum _{i = 1}^{n}{P}_{i}\)
|
41.18
|
Weighted Arithmetic
|
\(\frac{\sum _{1}^{n}{W}_{i}{P}_{i}}{\sum _{1}^{n}{W}_{i}}\)
|
41.19
|
Root Sum Power Function (10)
|
\({\left(\sum _{i = 1}^{n}{P}_{i}^{10}\right)}^{1/10}\)
|
99.48
|
Weighted root sum power (4)
|
\({\left(\sum _{i = 1}^{n}{{W}_{i}P}_{i}^{4}\right)}^{1/4}\) | 65.01 |
Weighted root sum power (10) | \({\left(\sum _{i = 1}^{n}{{W}_{i}P}_{i}^{10}\right)}^{1/10}\) | 81.30 |
Root Mean Square Function | \({\left(\frac{1}{n}\sum _{i = 1}^{n}{P}_{i}^{2}\right)}^{1/2}\) | 50.56 |
Weighted root sum square function | \(\frac{{\left(\sum _{i = 1}^{n}{W}_{i}{P}_{i}^{2}\right)}^{0.5}}{\sum _{i = 1}^{n}{W}_{i}}\)
|
52.23
|
Maximum Operator
|
\(= max ({P}_{1},{P}_{2},{P}_{3}-{P}_{n})\)
|
99
|
Minimum Operator
|
\(= min ({P}_{1},{P}_{2},{P}_{3}-{P}_{n})\)
|
7
|
Weighted ambiguity and eclipsity free function
|
\({\left(\sum _{i = 1}^{n}{{W}_{i}P}_{i}^{2.5}\right)}^{0.4}\) | 56.26 |
Subindex powered weight function | \(\sum _{i = 1}^{n}{P}_{i}^{{W}_{i}}\)
|
14.96
|
Unweighted Multiplicative Function
|
\({\left(\prod _{i = 1}^{n}{P}_{i}\right)}^{1/n}\)
|
28.16
|
Weighted Multiplicative function
|
\(\prod _{i = 1}^{n}{P}_{i}^{{W}_{i}}\)
|
26.42
|
Square root unweighted harmonic mean square function
|
\(\sqrt{\frac{n}{\sum _{i = 1}^{n}\frac{1}{{P}_{i}^{2}}}}\)
|
12.43
|
lead to a misinterpretation of leachate's pollution potential. The r-LPI will not suffer from the issue of transparency and rigidity as expert opinions have been used to select the attributes. However, the issue of ambiguity and eclipsing may persist. Thus, the selection of the aggregation function is crucial. However, the selection of the same lacks scientific evidence. To redress this, sensitivity analysis was done, and the most sensitive aggregation function was selected.
To determine the optimal aggregation function for r-LPI, a multitude of possible aggregation functions were applied to an active landfill leachate characteristic. The analysis took into account leachate from an active landfill site (Dhapa landfill) in Kolkata, India, as reported by De et al. (2016). The normalized parameter value was deduced from the sub-index curves and is illustrated in Table 7.
Different weighted and unweighted functions of aggregating the r-LPI were investigated to ascertain an eclipsing and ambiguity-free function. The r-LPI values resulting from the different aggregation functions are shown in Table 8.
All unweighted aggregation functions were discarded based on the result obtained, as equal weighting implies that all the sub-indices have the same weight. This can mask the lack of a statistical and analytical basis for deciding weights. Furthermore, equal weighing may imply unequal weighting for the sub-indices, since the sub-index with the most indicators would be given more weight in the overall index. Thus, the unweighted aggregation will be ineffective in this analysis. Further, all the aggregation functions resulting in the r-LPI value of more than 100 were also discarded as the practical range of r-LPI is 0-100. Furthermore, the majority of these functions show ambiguity. The sensitivity analysis was therefore carried out with weighted arithmetic, Weighted root sum (power 4, 10), weighted root sum square function, weighted ambiguity and eclipsity free function, and weighted multiplicative function since they exhibit comparatively less ambiguity and eclipsing. Sensitivity analysis is a necessary step to gauge the robustness and the transparency of the composite indicator (Ott 1978). It enables us to understand if the variance in the output can be attributed to variation in the input, either qualitatively or quantitively. A thorough investigation into the selection of appropriate aggregation functions was carried out and reported elsewhere (Bisht et al. 2022b). As a result, the weighted arithmetic aggregation function was found to exhibit comparatively less eclipsing than the weighted multiplicative and is also sensitive to variations in the sub-index values and was thus used in the analysis (Bisht et al. 2022b).
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\(r-LPI= \frac{\sum _{i=1}^{n}{w}_{i}{P}_{i}}{\sum _{i= 1}^{n}{w}_{i}}\)
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(12)
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Where Pi = Normalized value of the parameters
Wi = Corresponding weights