A novel hybrid approach based on intuitionistic fuzzy multi criteria group-decision making for environmental pollution problem

3.1 Preliminaries

Let a set E be fixed and let A ⊂ E be a fixed set. An IF set, denoted as A^* in E, is a set that has a form as A * = { 〈 x , μ A ( x ) , v A ( x ) 〉 : x ∈ E }(1) where functions μ _A  : E → [0, 1] and v _A  : E → [0, 1] shows the degree of membership and the degree of non-membership of the element x ∈ E respectively, for every x ∈ E.

It is known that 0 ≤ μ _A  (x) + v _A  (x) ≤1 and thus, 0 ≤ μ _A  (x) ≤1 and 0 ≤ v _A  (x) ≤1 obviously.

As every ordinary fuzzy set has the form {〈 x : μ _A  (x) , 1 - μ _A  (x) 〉 : x ∈ E } and 0 ≤ μ _A  (x) + v _A  (x) ≤1, it can be described π _A  (x) called as “the degree of non-determinacy (uncertainty) of the membership” of element x ∈ E which is denoted by: π A ( x ) = 1 - μ A ( x ) - v A ( x ) .(2)

In the case of π _A  (x) =0, IF set turns into classical fuzzy set that satisfies v _A  (x) =1 - μ _A  (x).

By aid of the expressions above, we can also write an IF set more comprehensive as: A * = { 〈 x , μ A ( x ) , v A ( x ) 〉 : x ∈ E & 0 ≤ μ A ( x ) + v A ( x ) ≤ 1 } where we possess μ _A  (x) → [0, 1]; v _A  (x) → [0, 1] and π _A  (x) → [0, 1].

For being more obvious and elucidative, we briefly define an IF number as 〈μ _A , v _A 〉 in overall study.

In this section, we introduce four basic arithmetic operations which we use in our methodology [17, 20].

Let A ={ 〈 x, μ _A  (x) , v _A  (x) 〉 : x ∈ E } and B ={ 〈 x, μ _B  (x) , v _B  (x) 〉 : x ∈ E } two IF sets, then we have:

Summation: A + B = { 〈 x , μ A + B ( x ) , v A + B ( x ) 〉 : x ∈ E }(3) where μ_A+B (x) = μ _A  (x) + μ _B  (x) - μ _A  (x) . μ _B  (x) and v_A+B (x) = v _A  (x) . v _B  (x)

Subtraction: A - B = { 〈 x , μ A - B ( x ) , v A - B ( x ) 〉 : x ∈ E }(4) where μ A - B ( x ) = { μ A ( x ) - μ A ( x ) 1 - μ B ( x ) , if μ A ( x ) ≥ μ B ( x ) and v A ( x ) ≤ v B ( x ) and v B ( x ) > 0 and v A ( x ) . π B ( x ) ≤ π A ( x ) . v B ( x ) 0 , otherwise

and v A - B ( x ) = { v A ( x ) v B ( x ) , if μ A ( x ) ≥ μ B ( x ) and v A ( x ) ≤ v B ( x ) and v B ( x ) > 0 and v A ( x ) . π B ( x ) ≤ π A ( x ) . v B ( x ) 1 , otherwise

Multiplication: A × B = { 〈 x , μ A × B ( x ) , v A × B ( x ) 〉 : x ∈ E }(5) where μ_A×B (x) = μ _A  (x) . μ _B  (x) and v_A×B (x) = v _A  (x) + v _B  (x) - v _A  (x) . v _B  (x)

Multiplication by a number: Let n a real number n . A = { 〈 x , μ n . A ( x ) , v n . A ( x ) 〉 : x ∈ E }(6) where μ _nA  (x) =1 - [1 - μ _A  (x)]  ⁿ and v _nA  (x) = [v _A  (x)]  ⁿ

n is a positive integer or n = 1/k with k a positive integer. Former is used for multiplication of IF number by a positive integer, while latter is used for a division of IF number by a positive integer.

The proposed method is illustrated, validated and compared against two different IF MCDM methods in terms of its ranking performance. In a ranking phase executed by TOPSIS method, we employed two IF operations described as below:

Division: A : B = { 〈 x , μ A : B ( x ) , v A : B ( x ) 〉 : x ∈ E }(7)

where μ A : B ( x ) = { μ A ( x ) μ B ( x ) , if μ A ( x ) ≥ μ B ( x ) and v A ( x ) ≤ v B ( x ) and v B ( x ) > 0 and v A ( x ) . π B ( x ) ≤ π A ( x ) . v B ( x ) 0 , otherwise

and v A : B ( x ) = { v A ( x ) - v A ( x ) 1 - v B ( x ) , if μ A ( x ) ≥ μ B ( x ) and v A ( x ) ≤ v B ( x ) and v B ( x ) > 0 and v A ( x ) . π B ( x ) ≤ π A ( x ) . v B ( x ) 1 , otherwise

Power by a number: Let n a real number A n = { 〈 x , μ A n ( x ) , v A n ( x ) 〉 : x ∈ E }(8) where μ _{A ⁿ}  (x) = [μ _A  (x)]  ⁿ and v _nA  (x) =1 - [1- v _A  (x)]  ⁿ

The following tables represent linguistic terms that we employed. It should be noted that Table 1 [17] is used in determining weights of criteria referring to the question: “How important are the following types of pollution in terms of total environmental pollution?” while Table 2 [19] is used in assessing values for both Noise Pollution and Light Pollution referring to the question: “How high is the noise (light) pollution of each city?” The responses are obtained from experts according to their judgements and they are represented in fifth section.

4.1 SAW

For a given set of m alternatives, i.e. A ={ a₁, a₂, …, a _m  }, and a given set of the n criteria, i.e. C ={ c₁, c₂, …, c _n  }, we briefly revise steps of classical SAW as below:

Step 1. Let a decision matrix that contains m option and n criteria, denoted by [D] _i×j whose r _ij are inputs for i = 1, 2, …, m and j = 1, 2, …, n. Calculate the normalized decision matrix, denoted by [N] _i×j, whose n _ij are inputs for i = 1,  2, …, m and j = 1,  2, …, n, as follows: n ij = r ij ∑ i = 1 m r ij(9)

Note that all the n _i j values lie between 0 and 1 as a consequence of normalization.

Step 2. Compute the weighted normalized decision matrix, denoted by [WN] _i×j, whose s _ij are inputs for i = 1, 2, …, m and j = 1, 2, …, n, by the following formula: s ij = w j . n ij(10)

Step 3. For each alternative, evaluate the sum of weighted normalized values in [WN] _i×j as follows and rank the alternatives in descending order.

m × n matrix is constructed where the number m stands for criteria and the number n stands for decision makers’ opinions.

For step 1, there is no need to calculate the normalized matrix, as the components of IF numbers are positive real numbers that are lower than or equal to 1.

For the next step, using the Equation (6) in IF environment, we calculate weighted IF decision matrix, i.e. W ^T  = [W₁, W₂, …, W _n ], where W _j  =〈 wμ _j , wv _j  〉. For final step, aggregation of IF numbers performed via the Equation (3) where the IF numbers are recursively summed. Note that all operations with IF numbers are properly done in fuzzy environment without defuzzifying them. As we minded to make all operations in fuzzy environment, we developed a hybrid algorithm by persisting in IF numbers while passing from weighting phase to ranking phase.

4.3 PROMETHEE

Among various MCDM approaches, PROMETHEE, developed by Brans [22], further extended by Brans and Vincke [23] and Brans and Mareschal [24], has been widely used and applied in various areas of science, management, engineering, marketing, finance etc.

In this study, we use PROMETHEE II as it enables a complete preorder which is needed more than the partial one even if the authors claimed that the partial preorder contains more realistic information. For better and detailed information about all types of PROMETHEE methods, we encourage the readers to refer [16, 23].

Before expressing IF-PROMETHEE, it is worth to revise the classical PROMETHEE’s steps and its general framework. Thus, given the data and the weights of each criterion, here are the steps of the algorithm:

Step 1. Among six types of functions, we assess a function f _k for the general preference p _k  (d) of each criterion. In this study, we prefer to employ V. type function, named as “Criterion with linear preference and indifference area”, that is best fitted to the problems that contain both quantitative and qualitative data. For more information about other types of functions, pioneer work [23] can be seen.

Moreover, p value, i.e. a preference threshold, and q value, i.e. an indifference threshold, are determined. Some small deviations in the determination of p and q values do not often draw important variation in the obtained ranking and thus, the authors claimed that decision makers are able to determine these values easily in accordance with their judgments. In order to prevent the variability of p and q values, we standardize these values as follows: q j = a j max - a j min 4 , for j = 1 , 2 , … , n(11) p j = 3 ( a j max - a j min ) 4 , for j = 1 , 2 , … , n(12)

Namely, it is rational to assign one-fourth of the range of i th criteria and three-fourth of the range i th criteria for q and p values respectively.

Step 2. For a given set of m alternatives, i.e. A ={ a₁, a₂, …, a _m  }, and a given set of the n criteria, i.e. C ={ c₁, c₂, …, c _n  }, the relative importance of each criterion is measured as follows: d k ( a ik , a jk ) = a ik - a jk for k = 1 , 2 , … , n(13) where n is the number of criteria; i = 1, 2, …, m and j = 1, 2, …, m, m is the number of alternatives.

Hence, at the end of this step, we obtain n matrix with m x m size.

Step 3. We calculate the general preference function P _k  (d) of each criterion using V. type function (See Fig. 1) as indicated below:

OPEN IN VIEWER

Fig. 1

Type V Function: Criterion with linear preference and indifference area.

P k ( a ik , a jk ) = { 0 , if d k ( a ik , a jk ) ≤ q k d k ( a ik , a jk ) - q k p k - q k , if q k < d k ( a ik , a jk ) ≤ p k 1 , if d k ( a ik , a jk ) > p k(14)

It should be mentioned that the general preference function p (x, y) represents the intensity of option x compared to option y such that p (x, y) = 0 means an indifference between x and y, p (x, y) = 1 means strict preference of x over y where (x, y) can be denoted as d = f (x) - f (y). We note that this function has to be a non-decreasing and all d numbers must be equal to zero for negative values.

Step 4. For a given or previously determined weighted vector, i.e. W ^T  = [W₁, W₂, …, W _n ], outranking relation π for every alternative a _ik , a _jk  ∈ A can be computed as follows: π : A × A → [ 0 , 1 ] π ( a i , a j ) = ∑ k = 1 n W k . P k ( a ik , a jk )(15)

Preferences indices π (a _i , a _j ) are employed to reveal preference of the decision maker on any alternative a _i simultaneously comparing with alternative a _j .

Hence, via n matrix previously calculated, we reduce to one matrix with m x m size.

Step 5. In order to elucidate the strength and weakness of alternatives, “leaving flow”, denoted as Φ⁺ (x) for an option 0, and “entering flow”, denoted as Φ^- (x) for an option x are calculated as follows: Φ + ( a i ) = 1 n - 1 ∑ i = 1 i ≠ j m π ( a i , a j )(16)

and Φ - ( a i ) = 1 n - 1 ∑ j = 1 i ≠ j m π ( a i , a j )(17)

for every i = 1, 2, …, m

These values assure to possess partial preorder, named as “PROMETHEE I” The highest leaving flow and the lowest entering flow show the best alternative performance via partial ranking.

Step 6. To get overall ranking, new flow must be considered as: Φ net = Φ + ( a i ) - Φ - ( a i ) for i = 1 , 2 , … , m(18)

In this case, the high value of new flow for alternative a as compared to alternative b means that the alternative a outranks alternative b.

All operations are performed in the IF environment by using the Equations 6).

Let AC _ij  =〈 acμ _ij , acv _ij  〉, A and C signify “Alternative” and “Criteria” respectively, an IF matrix for i = 1, 2, …, m where m is the is the number of the alternatives and for j = 1, 2, …, n where n is the number of criteria.

For the step 1, the process of standardization of threshold p and q values into the IF environment is calculated as below:

Let S j = ∑ i = 1 m AC ij for every j = 1, 2, …, n where the summation of two IF number, i.e. AC_1j + AC_2j, (AC_1j + AC_2j) + AC_3j  . . .   etc., is performed recursively by using (3). For example AC_1j+ AC_2j = 〈 acμ_1j + acμ_2j - acμ_1j × acμ_2j ; acv_1j × acv_2j 〉 In case we let S j = 〈 sum μ j , sumv j 〉 , using Equation (6) we obtain M _j  =〈 meanμ _j , meanv _j  〉 where meanμ _j  = 1 - (1 - sumμ _j ) ^1/m and meanv j = ( sumv j ) 1 / m .

Thus, using again Equation (6), q _j  =〈 qμ _j , qv _j  〉 is obtained where qμ _j  = 1 - (1 - meanμ _j ) ^1/2 and qv j = meanv j 1 / 2 with p _j  =〈 pμ _j , pv _j  〉 where pμ _j  = 1 - (1 - qμ _j ) ³ and pv _j  = (qv _j ) ³.

For the step 2, in order to subtract two IF number, we use Equation (4) to calculate the distance IF matrix where the elements of IF matrix d ^k can be noted as 〈 d μ ij k , dv ij k 〉 . d k ( AC ik , AC jk ) = AC ik - AC jk(19)

According to the conditions of the subtraction in IF environment, the followings are observed:

For all n matrix, the diagonals must be equal 〈0, 1〉.

For the third step where the general preference function matrix P ^k  (d) of each criterion must be calculated, we assess the new formula which is appropriate in IF environment, depending on V. type function, as indicated below: P k ( d k ) = { 〈 0 , 1 〉 d μ i j k ≤ q μ j 〈 d μ i j k − q μ j p μ j − q μ j ; β ( 1 − d μ i j k − q μ j p μ j − q μ j ) 〉 q μ j < d μ i j k ≤ p μ j 〈 1 , 0 〉 d μ i j k > p μ j(20) for every k = 1, 2, …, m

Note that the elements of IF matrix P ^k  (d ^k ) can be noted as 〈 pd μ ij k , pdv ij k 〉 .

For step 4, with respect to the weighted vector W ^T  = [W₁, W₂, …, W _n ] obtained by IF-SAW where the elements of W ^T can be noted as W _k  =〈 wμ _k , wv _k  〉, outranking relation matrix π (WP ^k ) can be calculated in two level.

In first level for all k = 1, 2, …, m, that the elements of IF matrix WP ^k can be noted as 〈 wp μ ij k , wpv ij k 〉 , each element of general preference function P ^k  (d ^k ) matrix is multiplied by the weighted vector W ^T as below: WP k = W T . P k ( d k )(21) where wp μ ij k = pd μ ij k × w μ k and wpv ij k = pdv ij k + wv k - pdv ij k × wv k by using the formula (5).

In the second level, we finish building the outranking relation matrix π (WP ^k ) as follows:

π (WP ^k ) =〈 πμ _ij , πv _ij  〉 for every i = 1, 2, …, m where π μ ij = ∑ k = 1 n wp μ ij k and π v ij = ∑ k = 1 n wpv ij k that the summation must be done by using operation (3).

For step 5, leaving flow, i.e. Φ⁺ (π), and entering flow, i.e. Φ^- (π) measures are calculated in two level. First, we calculate sum Φ i + ( π ) = 〈 ϕ μ i , ϕ v i 〉 and sum Φ i - ( π ) = 〈 φ μ i , ϕ v i 〉 where ϕ μ i = ∑ j = 1 m π μ ij , ϕ v i = ∑ j = 1 m π v ij for every i = 1, 2, …, m and φ μ i = ∑ i = 1 m π μ ij ϕ v i = ∑ i = 1 m π v ij for every j = 1, 2, …, m that the summation must be done by using (3) in IF environment.

Second, using by (6), we compute Φ i + ( π ) = 〈 1 - ( 1 - φ μ i ) 1 / m , ( ϕ v i ) 1 / m 〉 and Φ i - ( π ) = 〈 1 - ( 1 - φ μ i ) 1 / m , ( ϕ v i ) 1 / m 〉 for every i = 1, 2, …, m.

Within this step, we can obtain the partial order, i.e. PROMETHEE I order, and we possess the Φ⁺ (π) and Φ^- (π) values for every alternative in IF environment where the higher μ value in IF numbers is preferential in a partial ranking.

From the beginning, all calculations are performed in IF environment. However, in order to obtain the final rankings, i.e. Φ ^net values, we need to defuzzify the IF numbers. For the last step, to remedy this, we propose a new formula as below: Φ i net = ( ϕ μ i - ϕ v i ) - ( φ μ i - φ v i )(22) for every alternative i = 1, 2, …, m.

The proposed formula is significant since differences of (ϕμ _i  - ϕv _i ) and (φμ _i  - φv _i ) imply the positive outrank and the negative outrank in a crisp value respectively. We proposed Equation (22) as a new defuzzification formula to be used in IF-PROMETHEE.

Since all criteria are considered as “benefit”, the highest Φ ^net value shows the most polluted city while the lowest one signifies the least polluted city. Note that these Φ ^net values are crisp numbers which vary between –2 and 2.

Among several types of pollution coming from different sources and having different consequences, air pollution has a vital role for a clean environment. It is formed by the presence of pollutants in the atmosphere. These pollutants can exist in the form of particulate matter such as dust or excessive gases like carbon monoxide or other vapors that cannot be effectively removed through natural cycles. Depending on the concentration of air pollutants, various effects can appear as smog increases, higher rain acidity, reduction in the supply of oxygen to the heart especially in people suffering from heart disease, pollen allergies, higher rates of asthma, crop depletion etc. According to the European Environment Agency (EEA), more than 97 percent of Turkey’s urban population is exposed to unsafe amounts of particulate matter pollution. From the evidence available, Turkey emerges as a country with one of the highest rates of premature deaths in Europe [25].

Water pollution involves any contaminated water, whether from chemical, particulate, or bacterial matter that degrades the water’s quality and purity. Water pollution can occur in oceans, rivers, lakes, and underground reservoirs, and as different water sources flow together through the water cycle the pollution can spread. The effects of water pollution include decreasing the quantity of drinkable water available, lowering water supplies for crop irrigation, and affecting fish and wildlife populations that require water of a certain purity for survival.

Soil, or land pollution, is contamination of the soil that prevents natural growth and balance in the land whether it is used for cultivation, habitation, or a wildlife preserve. Some soil pollution, such as the creation of landfills, is deliberate, while much more is accidental and can have widespread effects. Thus, it can lead to poor growth and reduced crop yields, loss of wildlife habitat, water and visual pollution, soil erosion, and desertification.

Noise, or sound pollution, is an excessive noise with harmful impact on the activity of human life. It is one of the biggest health risks in urban life that it can cause hypertension, high stress levels, tinnitus, hearing loss, sleep disturbances and other hazardous effects. Despite being related to increased health problems, it is often overlooked.

Radioactive pollution refers to physical pollution as a result of release of radioactive substances into the environment during nuclear activities such as nuclear explosions, nuclear weapons tests, mining of radioactive areas, disposal of radioactive waste, accidents at nuclear power plants as well as use of radiation in medicine. Exposing to radiation can cause not only harmful diseases, also cancer or human death.

Light pollution is a pollution occurred by inappropriate, obtrusive or excessive use of artificial light. Some main causes are alteration or degradation of natural light in the environment as a consequence of industrialization and modernization and it results in headache incidence, body fatigue, medically defined stress, or increase in anxiety. It also affects the quality of life and safety of humans. As a side effect, excessive nighttime lighting interestingly releases more than 12 million tons of carbon dioxide into the atmosphere every year.

5.2 Data information

In order to generate air pollution data, we consider typical air pollutants that cause immediate concern. Both PM10 (Particulate Matter) particles, the fraction of particulates in air of very small size (<10μm), and SO₂ are exploited in “mg” by using monthly average values taken from many measurement stations for every city. Hence, it is shortly noted that an air pollution quality relies on SO₂ and concentrations of PM10.

People who use a source of drinking water need to monitor the level of nitrates in their well water. Thus, for Water Pollution data, we present annual average nitrate level which is derived from many potable water sources for every city. Therefore, the water pollution values refer to average nitrate quantity in “mg/L”.

As one of the major source of environmental devastation is caused by modern food production, modern fertilizers that consist of varying amounts of nitrogen (N), phosphorus (P) and potassium (K) are considered for a soil pollution. Total quantity used on the basis of plant nutrient content (N, P, K) and pesticides that are substances including insecticides, herbicides, fungicides, rodenticides, nematocytes, acaricides are figured in “tons”. The higher the value is, the more the agricultural activity is indicative.

Average activity concentrations of radionuclides (Ra-226, Th-232, K-40 and Cs-137) in the soil samples in “bq/kg” (special unit of radioactivity) are considered as a Radiactive Pollution data.

Noise pollution is measured in logarithmic units called decibels while light pollution is measured in photometry. However due to the lack of the data about both pollutions, we resorted to demand the opinions of competent decision makers. Moreover, linguistic scale in Noise criterion is used for the evaluation of alternatives to make the procedure easy for experts. As seen, four quantitative and two qualitative criteria are defined.

In Table 3, ratings of all cities based on quantitative criteria which are aggregated from [25] are shown. The linguistic data belonging to two qualitative criteria are given in the next subsection.

Let a _ij a value of alternative A _j  ∈ A on quantitative criteria x _i  ∈ X. Since the phyiscal dimensions and measurements of quantitative criteria are different, a _ij needs to be normalized. In order to establish an MCDM problem under Atanassov’s IF environment, we specify a new transition rule as below, inspired from both [14, 26]. μ ij = { α i a ij a i max for i ∈ C 1 δ i a i min a ij for i ∈ C 2(23)

and v ij = { β i ( 1 - μ ij ) for i ∈ C 1 γ i ( 1 - μ ij ) for i ∈ C 2(24) where C¹ and C² are the set of quantitative benefit criteria and quantitative cost criteria, respectively, and a i max = max 1 ≤ j ≤ n { a ij } and a i min = min 1 ≤ j ≤ n { a ij } .

Note that, since all criteria can be considered as benefit because of the fact that the aim of this study is to find the most polluted city and then the most cleanest city, we can simply transformed these a _ij values into IF numbers indicated as 〈 α i a ij a i max ; β i ( 1 - μ ij ) 〉 .

Values of parameters α _i  ∈ [0, 1], β _i  ∈ [0, 1], δ _i  ∈ [0, 1] and γ _i  ∈ [0, 1] are determined according to characteristics of the algorithm. Thus, as it is experimentally appropriate to assign these numbers close to 1, we fix as α _i  = 0, 9 and β _i  = 0, 85 for the quantitative criteria i = 1, 2, 3 and 5. We also define α₄ = α₆ = 0, 9 and β₄ = β₆ = 0, 85 where we use them in third step of PROMETHEE where i = 4 and i = 6 stand for qualitative criteria.

In this study, we developed a GDM method in an uncertain environment where DMs’ expressions are aggregated into single group decision opinions.

Data collection is provided by a questionnaire through the assessment of five decision makers, who are supposed to express their opinions using seven linguistic terms according to the scale given in Table 1. Each linguistic variable is associated to an IF number. The importance weights of the experts are decided through their experience and skills in the related field. The weight vector for the experts is t _j  = (0.25,  0.25,  0.25,  0.125,  0.125) where the sum of the values is equal to 1.

The linguistic data acquired from the experts and the associated IF numbers for each criterion according to the decision makers are given in Tables 4 and 5.

Let X _ij  =< μ _ij , v _ij  > a criteria-decision maker matrix in IF environment where i = 1, 2, …, 6 and j = 1, 2, …, 5 define number of criteria and number of decision makers respectively.

Since a normalization procedure is not required, we can directly aggregate the evaluation scores of criteria by utilizing the importance weights assigned to the experts derived from the Equation (6) and we obtain t _j  . X _ij matrix. Finally, using the Equation (3) by applying the operation recursively, we obtain the sum of the rows.

In Table 6, last column of the table can be considered as weights of each criterion that are taken into account in second phase.

As the linguistic data is required, similar study is also conducted for two qualitative criteria in order to determine IF values for each alternative by applying IF-SAW method under GDM process where the number of decision makers and their weights are the same. However, unlike the work above, information is provided by using only five linguistic terms according to the scale given in Table 2. As the procedure is based on IF-SAW, it is preferred to present the decision makers’ opinions with linguistic variables as indicated in Tables 7 and 8, for the Noise and Light Pollution, respectively.

The IF results of two tables after using IF-SAW method can be seen in Table 9 in the next subsection.

Considering weights of all criteria, the IF values of two qualitative criteria and the IF values of four quantitative criteria obtaining by the transformation rule from pure data into IF environment, we can employ IF-PROMETHEE to the following inputs in Table 9 in which all numbers are rounded two decimal after comma.

Applying the steps in the IF-PROMETHEE section, a preference threshold p value and an indifference threshold q value are determined as in Table 10.

In the next steps, the distance IF matrix d ^k and the general preference function matrix P ^k  (d) are respectively calculated for every criteria. Tables 11 and 12 represent the distance and general preference function values for the first criterion.

Since the IF weight vector that is calculated is W ^T  = [〈0.9, 0.05〉, 〈0.7, 0.23〉〈0.48, 0.4〉, 〈0.38, 0.5〉〈0.75, 0.18〉, 〈0.22, 0.66〉] (see Table 6, last column), we obtain the weighted general preference function matrix WP ^k  (d) for each criterion as represented for the first criterion in Table 13.

For Step 5, by using six WP ^k  (d ^k ) matrix, we obtain the outranking relation matrix π (WP ^k ) as in Table 14.

Finally, Φ⁺ (π), i.e. positive outrank, Φ^- (π), i.e. negative outrank, and defuzzied net flow values, i.e. net outrank, are represented in Table 15.

The highest net outrank shows the most polluted city among others.

In order to test the robustness and accuracy of our methodology, we applied two other methods for a ranking phase, namely IF-TOPSIS inspired from [27] and IF-SAW. According to values shown in Table 16, results obtained by IF-PROMETHEE are very close to the results conducted by two other methods.

According to our results, Konya is the least polluted city between the major cities in Turkey. Even though there is an agriculture-based industry in the area, the city follows firmly anti-pollution regulations. It also features transportation and transfer stations that sort through garbage and take out recyclable materials. Bursa is the most polluted city. The city’s groundwater has been found to contain some pollutants higher than the World Health Organization’s recommended amount. The extension of the city due to increasing population is leading to further deterioration of the land like deforestation and construction activities.

In order to have an acceptable pollution level, cities should take some measures as reducing industrial smokestack emissions, increasing the use of renewable power sources, prioritising rapid transit, planning effective sewage treatment, preventing deforestation. A series of policy on energy conservation and energy use must be adopted and government responsibility and intensifying penalties must be clarified.