Literature DB >> 34945942

Intuitionistic Fuzzy Synthetic Measure on the Basis of Survey Responses and Aggregated Ordinal Data.

Bartłomiej Jefmański1, Ewa Roszkowska2, Marta Kusterka-Jefmańska3.   

Abstract

The paper addresses the problem of complex socio-economic phenomena assessment using questionnaire surveys. The data are represented on an ordinal scale; the object assessments may contain positive, negative, no answers, a "difficult to say" or "no opinion" answers. The general framework for Intuitionistic Fuzzy Synthetic Measure (IFSM) based on distances to the pattern object (ideal solution) is used to analyze the survey data. First, Euclidean and Hamming distances are applied in the procedure. Second, two pattern object constructions are proposed in the procedure: one based on maximum values from the survey data, and the second on maximum intuitionistic values. Third, the method for criteria comparison with the Intuitionistic Fuzzy Synthetic Measure is presented. Finally, a case study solving the problem of rank-ordering of the cities in terms of satisfaction from local public administration obtained using different variants of the proposed method is discussed. Additionally, the comparative analysis results using the Intuitionistic Fuzzy Synthetic Measure and the Intuitionistic Fuzzy TOPSIS (IFT) framework are presented.

Entities:  

Keywords:  Hellwig’s method; decision making; fuzzy measurement; fuzzy multi-criteria method; intuitionistic fuzzy set; ordinal data; synthetic measure; uncertainly

Year:  2021        PMID： 34945942      PMCID： PMC8699974          DOI： 10.3390/e23121636

Source DB:  PubMed          Journal:  Entropy (Basel)        ISSN： 1099-4300            Impact factor:   2.524


1. Introduction

Multiple criteria decision making (MCDM) has been an important research discipline of decision science applied in many areas such as business, management, engineering, and social science [1,2,3]. Nowadays, a lot of new MCDM methods have been introduced to address several practical problems and real-life applications. MCDA methods are widely used in constructing synthetic measures (or composite indicators) for the evaluation of complex socio-economic phenomena [4,5,6]. One of the problems is the assessment of complex socio-economic phenomena using questionnaire surveys when data are represented on an ordinal scale, especially if the object assessments contain positive, negative opinions and an element of uncertainty expressed in the form of no answer, “difficult to say” answer, “no opinion”, etc. In previous studies, some proposals of TOPSIS and Hellwig’s methods based on intuitionistic fuzzy numbers to solve the presented problems were discussed. The classical Hellwig’s method was presented in 1968 by a Polish researcher as a taxonomic method for international comparison of economic development of countries [7]. This method allows ranking multidimensional objects in terms of a complex phenomenon that cannot be described using a single criterion. The method is based on the concept of distance from the pattern object, which was also used in the well-known and popular TOPSIS method. The difference between both methods is that TOPSIS, apart from the distance from the pattern object, also takes into account the distance from the anti-pattern object. The methods based on Hellwig’s and TOPSIS methodology have many features in common, whereas the main difference concerns the method for calculating the synthetic variable value. The methods are characterized by the simplicity of calculations and software options (e.g., available free R packages). Both methods allow including quantitative and qualitative criteria in the assessment of objects. In the case of quantitative criteria, their normalization is required. Fuzzy modifications of both methods were proposed for the qualitative criteria. They consist in replacing the qualitative criteria values with fuzzy sets (most frequently fuzzy numbers). In the vast majority of cases the parameters of fuzzy numbers are determined subjectively by the researchers, which does not always allow reflecting the respondents’ preferences in this matter. It is also worth noting that both methods do not suggest how to determine the weighting factors for the criteria. In addition, they do not take into account the potential correlations between the criteria. Hellwig’s method was promoted in the world literature through the UNESCO research project on the human resources indicators for less developed countries [8,9]. Another application of this method can be found, e.g., in the studies [10,11,12,13,14]. Hellwig’s method was also extended for the fuzzy environment [15,16,17], the intuitionistic fuzzy environment [18,19,20] and the interval-valued intuitionistic fuzzy environment [21]. The idea of using MCDM methods in measuring complex phenomena based on survey data is quite new, therefore the source literature offers only a few scientific publications and research studies addressing this area. The paper [18] presents the concept of the IFSM using Hellwig’s approach for the intuitionistic fuzzy sets. The IFSM allows measuring complex phenomena based on the respondents’ opinions. The IFSM adopts that the respondents assess objects in terms of the adopted criteria using ordinal measurement scales. The respondents’ opinion measurement results are later transformed into intuitionistic fuzzy sets. In another paper [21] a synthetic measure based on Hellwig’s approach and the interval-valued intuitionistic fuzzy set theory is presented. Also, the optimism coefficient is defined, which allows setting the limits of intervals for the proposed parameters. The common feature of both methods is using the transformation of ordinal data to the form of intuitionistic fuzzy sets. The assessment criteria are thus expressed in the form of three parameters of the intuitionistic fuzzy set: membership, non-membership and uncertainty. The difference in these methods consists in determining values of these parameters. In the first case (IFSM method) they are presented as numbers in the interval [0, 1], while in the second case (I-VIFSM method) they take the form of intervals. Finally, [19] proposed the Intuitionistic Fuzzy TOPSIS (IF-TOPSIS) method which can be applied for assessing socio-economic phenomena on the basis of survey data. Motivated by the above-mentioned works, the present paper proposes the general framework for intuitionistic fuzzy multi-criteria procedure, namely the Intuitionistic Fuzzy Synthetic Measure (IFSM) based on distance to the pattern object. The IFSM method has been inspired by Hellwig’s approach of developing a coefficient adapted to an intuitionistic fuzzy environment. The Intuitionistic Fuzzy Synthetic Measure was proposed to address the problem of survey data. It consists of seven main steps: (1) representation of the survey data in the form of intuitionistic fuzzy values; (2) determination of the Intuitionistic Fuzzy Decision Matrix; (3) determination of the intuitionistic fuzzy pattern object; (4) calculation of the distance measures; (5) calculation of the intuitionistic fuzzy coefficient; (6) rank ordering of objects by maximizing the coefficient; and (7) comparing the criteria with the Intuitionistic Fuzzy Synthetic Measure. Two concepts for determining intuitionistic fuzzy pattern objects are discussed. The first one is based on max values from the survey data and the second on max intuitionistic values in general. Next, two measures of distances, i.e., Euclidean and Hamming distance implemented in the coefficient procedure are considered which, additionally, can be based on two or three parameters. This provides eight variants of the proposed IFSM. The usefulness of the proposed approach was examined in the evaluation of satisfaction from local public administration in the context of quality of life in cities using survey data. As was pointed out by [22] the purpose of constructing synthetic measure, among other things, “to condense and summarise the information contained in a number of underlying indicators, in a way that accurately reflects the underlying concept”. Thus finally, the Spearman coefficient for comparison criteria with respect to information transferred for the IFSM is proposed. The objectives and contributions of this study are presented below: to develop a general IFSM based on Hellwig’s approach for the evaluation of socio-economic phenomena with survey data; to study the IFSM based on Hellwig’s approach taking into account two types of Euclidean and Hamming distance implemented in the procedure; to study the IFSM based on Hellwig’s approach considering two or three parameters used in distance measure applied in the procedure; to study the IFSM based on Hellwig’s approach examining different pattern object construction used in the procedure; to propose the method for criteria comparison with the Intuitionistic Fuzzy Synthetic Measure; to demonstrate different variants of the Intuitionistic Synthetic Measure based on Hellwig’s approach through a comparative analysis; to compare different variants of the IFSM based on Hellwig’s approach with the Intuitionistic Fuzzy TOPSIS (IFT) procedure to examine its relevance and effectiveness. The proposed framework, based on the extended Hellwig’s method, has been applied to analyse its relevance. The rest of this article is organized as follows. In Section 2 the basic concepts related to intuitionistic fuzzy sets (IFS) and distances on IFSs are presented. The general framework of the IFSM based on Hellwig’s approach is provided in Section 3. Section 4 discusses a case study solving the problem of rank-ordering of the cities in terms of satisfaction from local public administration using the proposed approach. The comparison results obtained using the IFSM with the IFT framework are also presented. The conclusion and indications for future research are formulated in Section 4.

2. Preliminaries

To start with, the presentation of some basic concepts related to IFS and distances on IFSs are presented. The Intuitionistic Fuzzy Set theory, proposed by Atanassov [23], is an extension of the Fuzzy Set (FS) theory introduced by Zadeh [24] to address uncertainty. ([23,25]). Let be a universe of discourse of objects. An intuitionistic fuzzy set in is given by: where are functions with the condition for every The numbers and denote, respectively, the degrees of membership and non-membership of the element to the set ; the intuitionistic fuzzy index (hesitation margin) of the element in the set . Greater indicates more vagueness. It should be noticed that when for every then intuitionistic fuzzy set is an ordinary fuzzy set. If the universe contains only one element , then the IFS over is denoted as and called an intuitionistic fuzzy value (IFV) [26,27]. Let be the set of all IFVs. The intuitionistic value is the largest, while is the smallest. One of the applications of intuitionistic fuzzy sets in multiple criteria decision making is the possibility of taking into consideration the decision maker’s approval, rejection, and hesitations regarding the evaluated alternatives with respect to criteria. This is the main motivation for using the intuitionistic fuzzy sets in developing the multi-criteria procedure. Euclidean and Hamming distances represent the widely used distances for the intuitionistic fuzzy sets [28]. ([28]). Let us consider two with membership functions , , and non-membership functions , , respectively. The normalized Euclidean between two intuitionistic fuzzy sets A and B is defined as: The normalized Hamming distance between two intuitionistic fuzzy sets A and B is defined as: To compare two IFVs the following score function defined by Chen & Tan [29] was used: and accuracy function defined by Hong and Choi [30]: It can be easily observed that and . ([31]). Let us consider two intuitionistic fuzzy values , , respectively: if if

3. Classical and Intuitionistic Variant of Hellwig’s Method

3.1. Classical Variant of Hellwig’s Method

The classical Hellwig’s method was proposed for quantitative criteria. It adopts the calculation of Euclidean distance from the pattern of development for each assessed object. Most often the pattern of development is an abstract unit presenting the most favorable assessments of the individual criteria. Let   be the set of objects subject to assessment and   the set of criteria constituting a complex phenomenon. It should also be adopted that and are the sets of stimulating (positive) and destimulating (negative) criteria, respectively, influencing the complex phenomenon . The classical variant of Hellwig’s method consists of the following steps: Step 1. Defining the decision matrix: where is the value of the -th object with respect to the -th criterion. Step 2. Determining the normalized decision matrix: using the formula for standardization: where: , . Step 3. Defining the pattern of development (pattern object) in accordance with the principle: Step 4. Calculating the distance of the -th object from the pattern of development using the Euclidean distance: Step 5. Calculating the synthetic measure of development for the -th object: where: , , . Step 6. Ranking the objects according to the decreasing values of . The measure most often takes values from the interval . The higher values of the measure the less the object is away from the pattern of development.

3.2. The Intuitionistic Fuzzy Synthetic Measure Based on Hellwig’s Approach for the Evaluation of Socio-Economic Phenomena Using Survey Data

In this section, a general framework for Intuitionistic Fuzzy Synthetic Measures is proposed. Let   be the set of objects under the survey evaluation,   the set of criteria for the objects assessed by the respondents using an ordinal measurement scale. The respondents’ answers are collected in a questionnaire survey. It was adopted that the respondents answered the questions using different scales, which can be aggregated into three groups: “a positive opinion about the object”, “a negative opinion about the object”, “no opinion or no answer”. The same importance was adopted in the evaluation of objects to the criteria. i.e., the weights of criteria are equal [32]. The procedure to evaluate the socio-economic phenomena is as follows: Step 1. Representation of the survey data in the form of intuitionistic fuzzy values. The respondents’ opinions about the object for each criterion are represented by IFVs , where: —the fraction of positive opinions about i-th object with respect to j-th criterion, —the fraction of negative opinions about i-th object with respect to j-th criterion, —the fraction of opinion type “don’t know”, “no answers” for i-th object with respect to j-th criterion, and . The following has been adopted: where: —the total number of respondents who positively evaluated the i-th object with respect to j-th criterion; —the total number of respondents who negatively evaluated the i-th object with respect to j-th criterion; —the total number of respondents with hesitancy opinion about the i-th object for j-th criterion; —the total number of respondents who evaluated the i-th object with respect to the j-th criterion. It has been noted that . Clearly, instead of the total number of responses, the percentage of relevant responses common for the secondary survey data can be used. In this way i-th object is represented by the vector: where . Step 2. Determination of the Intuitionistic Fuzzy Decision Matrix. Based on the survey data representation in the form of intuitionistic fuzzy values obtained in step 1 the intuitionistic fuzzy decision matrix is given in the form: Step 3. Determination of the intuitionistic fuzzy pattern object. The intuitionistic fuzzy pattern object can be determined twofold: is based on maximum IFV and takes the form of: is based on maximum and minimum values and takes the form of: where , denote the evaluation information of i-th object with respect to j-th criterion and . Step 4. Calculation of the distance measures. After selecting the distance measure, the distance measures between the objects and the intuitionistic fuzzy pattern object selected in step 3 are calculated using one of the Formulas (3)–(6). The distance measure from the pattern object takes the form of: where , . Step 5. Calculation of the Intuitionistic Fuzzy Synthetic Measure. The Intuitionistic Fuzzy Synthetic Measure (IFSM) coefficient is defined as follows: where: , , . Step 6. Rank ordering of objects by maximizing the coefficient . The highest value of then the highest position of the object . Step 7. Comparing the individual criteria with the Intuitionistic Fuzzy Synthetic Measure using Information Transfer Measure (ITM). The important two problems should be addressed while building the IFSM, condensing information and accurately representing the underlying concept. The criteria should capture the most important properties of the analyzed phenomena, represent them accurately and provide a large amount of information. There should be a positive correlation between the criteria and the synthetic measure, and also each criterion should contribute to the decision-maker(s)’ views on its importance regarding the concept [22]. Now the measure of the information transferred from each criterion to the IFSM is defined. The criteria should capture the most important properties of the analyzed phenomena, represent them accurately and provide a large amount of information. First, the individual criteria represented by the intuitionistic fuzzy values are ordered using accuracy function and score function (see Definition 3). Then the Spearman coefficient between the ranking criteria and the ranking obtained by the IFSM measure is calculated. The Spearman coefficient is a nonparametric measure of dependence for the variables measured at least on an ordinal scale. The measure is normalized in the range [–1, 1]. It allows measuring the power and determining the direction of the correlations. Formally, the Information Transfer Measure for j-th criterion is defined as follows: The shows the power and direction between the criterion and the synthetic measure . It should be observed that taking into account the way of survey data representation in the form of intuitionistic fuzzy values this coefficient should be positive. In the case where the importance of the criterion is the same, the measures for should be similar. The procedure of analyzing the survey data for IFSM is presented in Figure 1.
Figure 1

Procedure for the analysis of survey data for IFSM.

Classification of variants of the IFSM based on an intuitionistic fuzzy framework with respect to pattern objects and distance measures is presented in Table 1.
Table 1

Classification of variants IFSM methods based on an intuitionistic fuzzy framework with respect to pattern objects and distance measures.

MethodsPattern ObjectsThe Distance MeasureNumber of Parameters in the Distance Measure
IFSM_me2based on max and min valuesEuclidean distancedistance based on two parameters
IFSM_me3based on max and min valuesEuclidean distancedistance based on three parameters
IFSM_mh2based on max and min valuesHamming distancedistance based on two parameters
IFSM_mh3based on max and min valuesHamming distancedistance based on three parameters
IFSM_ae2based on (1,0) valuesEuclidean distancedistance based on two parameters
IFSM_ae3based on (1,0) valuesEuclidean distancedistance based on three parameters
IFSM_ah2based on (1,0) valuesHamming distancedistance based on two parameters
IFSM_ah3based on (1,0) valuesHamming distancedistance based on three parameters

4. Empirical Example

4.1. Problem Description and Data Source

The approach to the analysis of survey data proposed in the article, applying the presented procedure and the IFSM method, was used in the analysis of the results from the fifth survey on quality of life in European cities. The survey provides a unique insight into city life. It gathers the experience and opinions of city dwellers. The fifth survey on quality of life in European cities was conducted for the European Commission by the IPSOS company. The survey covered the inhabitants of 83 cities in the EU, the EFTA countries, the UK, the Western Balkans, and Turkey. The survey was conducted between 12 June and 27 September 2019, with a break between 15 July and 1 September. A total of 700 interviews were completed in each surveyed city. This means that a total of 58,100 inhabitants of 83 cities participated in the survey. The survey covers eight fields of the quality of life in cities: overall satisfaction, services and amenities, environmental quality, economic well-being, public transport, the inclusive city, local public administration, as well as safety and crime. For the first time, the fifth round of the survey includes questions about the quality of the city administration. The high-quality, efficient, and transparent local public administration is very important for improving the quality of life in European cities. In addition, improving the quality of institutions at the local level is the heart of the EU and the EU Cohesion Policy. In the empirical example, European cities would be ranked only in the field of local public administration. Thus, only five questions of the questionnaire concerning satisfaction from the local public administration were used [33]: “I will read you a few statements about the local public administration in your city. Please tell me whether you strongly disagree, somewhat disagree,… Q1: I am satisfied with the amount of time it takes to get a request solved by my local public administration; Q2: The procedures used by my local public administration are straightforward to understand; Q3: The fees charged by my local public administration are reasonable; Q4: Information and services of my local public administration can be easily accessed online; Q5: There is corruption in my local public administration.” Our study aims at measuring and benchmarking inhabitants’ satisfaction with local public administration using the IFSM approach. Inhabitants’ satisfaction, as a complex phenomenon, was characterized using five criteria described by five questions Q1–Q5: C1—time for request, C2—procedures; C3—fees charged, C4—information and services, C5—corruption. In the assessment of criteria, a five-point measurement scale was used: strongly disagree, somewhat disagree, somewhat agree, strongly agree, don’t know/no answer. The characteristics of the research sample in terms of gender, age, and level of education are presented in Table 2.
Table 2

Sociodemographic characteristics of the respondents.

FeatureCategoryPercentage
GenderMale47.739%
Female52.261%
Age15–194.965%
20–249.288%
25–3418.683%
35–4417.592%
45–5415.872%
55–6413.920%
65–7411.407%
75+8.273%
EducationLess than Primary education 0.173%
Primary education1.308%
Lower secondary education10.389%
Upper secondary education35.257%
Post-secondary non-tertiary education8.056%
Short-cycle tertiary education12.886%
Bachelor or equivalent18.269%
Master or equivalent10.986%
Doctoral or equivalent2.221%
Don’t know/No Answer/Refuses0.455%

Source: [33].

4.2. Analysis of the Results

In this part, the empirical results concerning the evaluation of the satisfaction with local public administration in European cities using the IFSMes are presented. Due to the large number of cities covered by the survey individual steps of the proposed procedure were presented based on the example of the selected 2 cities: Zurich (the best in all rankings) and Palermo (the worst in all rankings). The selected cities were the first and the last in the ranking obtained using all IFSM methods. The assessment of the selected cities in terms of 5 criteria using the 5 categories is presented in Table 3.
Table 3

The assessment of cities.

CityCategory *C1C2C3C4C5
Palermo145.85%28.70%36.42%15.81%6.24%
2 37.74%40.18%40.07%29.24%14.32%
3 10.94%23.62%19.53%42.20%41.56%
4 1.88%4.41%2.55%8.79%30.36%
99 3.59%3.10%1.43%3.96%7.52%
Zurich11.70%2.68%1.90%0.49%33.15%
210.67%17.88%16.88%7.82%34.05%
3 45.59%46.67%51.07%33.17%14.56%
4 27.74%25.63%26.04%46.91%2.58%
99 14.30%7.13%4.11%11.61%15.65%
Total700700700700700

* 1—Strongly disagree, 2—Somewhat disagree, 3—Somewhat agree, 4—Strongly agree, 99—Don’t know/No Answer/Refuses. Source: [33].

According to Formula (15), the respondents’ assessments were transformed into IFVs (Table 4).
Table 4

The assessment of cities using the IFVs.

CityParameterC1C2C3C4C5
Zurich ν 0.1240.2060.1880.0830.171
μ 0.7330.7230.7710.8010.672
π 0.1430.0710.0410.1160.157
Palermo ν 0.8360.6890.7650.4510.719
μ 0.1280.2800.2210.5100.206
π 0.0360.0310.0140.0400.075

Source: [33].

It has been observed that for the criteria C1, C2, C3, C4 the is obtained by summing up the categories 1, 2, and by summing up the categories 3, 4. Taking into account the form of question Q5 for the criterion C5 the value is obtained by summing up the categories 3, 4 while by summing up the categories 1, 2. The assessment criteria in the form of IFVs for all cities are listed in Table A1, Table A2 and Table A3 in the Appendix A.
Table A1

Degrees of non-membership to IFVs for cities.

CityC1C2C3C4C5
Aalborg0.1640.2930.2460.1010.166
Amsterdam0.3270.3670.4130.1270.281
Ankara0.4090.3350.4250.2420.422
Antalya0.3490.2130.3440.1340.337
Antwerpen0.2600.1990.3710.3580.264
Athina0.6190.5800.7240.3520.584
Barcelona0.5400.3530.5820.3020.435
Belfast0.3070.3020.3230.1560.370
Beograd0.6240.6170.5580.2800.749
Berlin0.5790.6000.2930.2690.371
Białystok0.2900.3480.2920.1270.301
Bologna0.4420.4590.4870.1920.386
Bordeaux0.3820.3570.3210.2170.265
Braga0.4560.3030.4020.2500.535
Bratislava0.3330.4650.2670.2470.498
Bruxelles0.3630.2110.3440.2480.284
Bucharest0.4580.4900.3210.2800.639
Budapest0.3950.3370.3760.1310.373
Burgas0.4390.3920.4760.1410.535
Cardiff0.2380.2690.2950.1390.196
Cluj-Napoca0.2790.3160.2490.1790.580
Diyarbakir0.5720.4940.3810.3940.507
Dortmund0.4600.5270.4500.2480.396
Dublin0.2840.2490.2640.1320.339
Essen0.3680.5300.3230.2120.236
Gdańsk0.3160.3440.2800.1550.356
Genève0.1590.2000.3620.2320.384
Glasgow0.3810.3180.3680.1720.330
Graz0.3160.2540.2770.1080.225
Groningen0.1880.1990.4070.0800.167
Hamburg0.3020.4490.3000.1960.251
Helsinki0.3730.5050.2970.2180.341
Irakleio0.6510.5480.7160.1930.627
Istanbul0.4630.3310.4280.2750.564
København0.2990.3230.2060.1000.167
Košice0.3460.3570.2840.1730.481
Kraków0.3110.4790.3930.1610.298
Lefkosia0.4610.2060.4250.1350.593
Leipzig0.2560.3290.3670.1570.203
Liège0.3090.2200.3600.2280.312
Lille0.4340.3320.3850.2790.255
Lisboa0.5740.4760.5010.2620.572
Ljubljana0.3470.3230.2070.1750.563
London0.3740.3400.3420.1440.259
Luxembourg0.2620.2890.2090.1470.310
Madrid0.4180.3820.4460.2640.398
Málaga0.4020.3350.3860.2290.417
Malmö0.3340.4640.2350.1730.252
Manchester0.2430.2610.2720.1150.387
Marseille0.4290.3970.5030.3610.457
Miskolc0.2320.2710.3620.0830.439
Munich0.3540.4010.2730.1560.191
Naples0.7190.6260.7110.4110.607
Oslo0.4520.4600.3300.2220.279
Ostrava0.3630.4360.2660.1230.534
Oulu0.3490.4550.3720.2400.286
Oviedo0.4660.4440.5270.3010.472
Palermo0.8360.6890.7650.4500.719
Paris0.4230.3790.4010.2560.305
Piatra Neamt0.3270.3600.2830.1940.562
Podgorica0.5150.4270.3170.3100.670
Praha0.3960.4390.1900.1490.471
Rennes0.3680.3110.3150.2010.164
Reykjavík0.5160.4960.5060.2110.570
Riga0.4430.4710.7120.2680.681
Rome0.8270.7100.7270.4020.772
Rostock0.1990.3860.2540.1750.234
Rotterdam0.3800.3340.3110.1850.223
Skopje0.6750.4370.4230.3280.764
Sofia0.5240.5200.4610.2480.503
Stockholm0.3680.4270.2170.1740.225
Strasbourg0.3220.3040.3590.2460.252
Tallinn0.2510.3130.1470.1270.563
Tirana0.4820.3960.5100.2290.756
Turin0.6420.6110.6480.2990.492
Tyneside conurbation0.2800.2630.3310.1770.270
Valletta0.3190.2520.2000.1410.185
Verona0.5300.5410.3950.2350.596
Vilnius0.3930.3420.3850.2510.423
Warszawa0.3910.4850.4350.1910.331
Wien0.2320.3100.2660.1190.226
Zagreb0.6540.6010.5880.2200.753
Zurich0.1240.2060.1880.0830.171
Table A2

Degrees of membership to IFVs for cities.

CityC1C2C3C4C5
Aalborg0.6700.6080.5680.8460.788
Amsterdam0.4870.5710.5240.8050.447
Ankara0.5620.6420.5620.7250.492
Antalya0.6440.7800.6390.8170.518
Antwerpen0.5570.7240.6190.5260.468
Athina0.3740.4050.2640.5260.174
Barcelona0.4460.6280.4080.6680.481
Belfast0.5000.6300.5810.6900.418
Beograd0.3280.3590.4050.5960.088
Berlin0.3260.3150.6020.5530.293
Białystok0.6870.6120.6780.7640.442
Bologna0.5180.5080.5000.7440.485
Bordeaux0.5750.5870.5980.7240.484
Braga0.4850.6600.5640.6770.281
Bratislava0.5090.4620.6590.6670.231
Bruxelles0.6270.7890.6190.7000.498
Bucharest0.4580.4910.6370.5520.118
Budapest0.4590.5670.5230.6770.304
Burgas0.5190.5670.4800.8020.305
Cardiff0.5720.6510.6370.7540.573
Cluj-Napoca0.6280.6010.6960.6300.131
Diyarbakir0.3750.4950.5990.5740.428
Dortmund0.5040.4350.4790.6540.341
Dublin0.5890.6590.6370.7770.483
Essen0.5660.3640.5740.6580.382
Gdańsk0.6320.6130.6830.7810.378
Genève0.7170.7460.5730.6840.384
Glasgow0.4770.5580.5290.6690.492
Graz0.6280.7110.6970.8310.591
Groningen0.5510.6070.5430.8270.600
Hamburg0.6290.4760.5790.7140.418
Helsinki0.3910.3980.6090.7230.598
Irakleio0.3240.4240.2650.6560.315
Istanbul0.5080.6300.5170.6810.323
København0.5890.5800.6160.8330.789
Košice0.5690.5790.6740.7320.228
Kraków0.5930.4780.5640.7460.369
Lefkosia0.5310.7710.5340.7360.362
Leipzig0.6310.5960.5510.6340.378
Liège0.5690.7470.6130.6060.396
Lille0.5200.6370.5510.6150.432
Lisboa0.3400.4920.4600.6470.238
Ljubljana0.5440.5970.6750.6990.235
London0.4960.5850.5790.7640.519
Luxembourg0.6810.6900.7640.8380.612
Madrid0.5400.5740.5080.6320.412
Málaga0.5020.5850.5170.6600.407
Malmö0.5070.4380.5740.7480.672
Manchester0.6110.6950.6420.7540.537
Marseille0.5000.5490.4270.5690.248
Miskolc0.5110.5820.5660.7060.271
Munich0.5000.5160.6440.7310.488
Naples0.2460.3540.2700.5010.250
Oslo0.2910.3330.4910.6400.548
Ostrava0.4780.4800.6740.8100.248
Oulu0.5490.4680.5610.6880.605
Oviedo0.4710.5110.4240.5950.379
Palermo0.1280.2800.2210.5100.206
Paris0.5440.5970.5320.7090.445
Piatra Neamt0.5940.5630.6820.5400.179
Podgorica0.4370.5240.6130.6260.146
Praha0.4150.4160.6880.7180.241
Rennes0.6160.6540.6480.7470.624
Reykjavík0.2730.4100.4530.6430.353
Riga0.4140.4830.2520.6340.205
Rome0.1550.2710.2610.5220.156
Rostock0.6490.5270.6980.6930.449
Rotterdam0.5090.6110.6290.7230.420
Skopje0.3150.5310.5510.5840.108
Sofia0.3230.3700.5230.6080.175
Stockholm0.4260.4500.6240.6750.538
Strasbourg0.6440.6700.5940.6700.490
Tallinn0.5350.5470.6310.7930.282
Tirana0.4850.5880.4740.7150.220
Turin0.2990.3610.3320.5580.362
Tyneside conurbation0.5400.6690.5930.6200.488
Valletta0.5420.6520.6170.7230.489
Verona0.4110.4230.5680.6550.281
Vilnius0.4520.5610.4960.6790.359
Warszawa0.5360.4540.5200.7790.363
Wien0.6780.6690.7070.8190.615
Zagreb0.3130.3400.3770.5570.092
Zurich0.7330.7230.7710.8010.672
Table A3

Degrees of hesitancy for IFVs for cities.

CityC1C2C3C4C5
Aalborg0.1660.0990.1860.0530.045
Amsterdam0.1860.0620.0630.0670.272
Ankara0.0290.0230.0130.0330.086
Antalya0.0070.0070.0170.0500.145
Antwerpen0.1830.0770.0100.1160.267
Athina0.0070.0150.0120.1220.242
Barcelona0.0140.0190.0090.0300.084
Belfast0.1920.0680.0960.1540.211
Beograd0.0480.0240.0370.1240.163
Berlin0.0950.0850.1040.1780.336
Białystok0.0230.0400.0300.1090.256
Bologna0.0390.0330.0130.0640.130
Bordeaux0.0430.0560.0810.0590.252
Braga0.0590.0370.0340.0730.184
Bratislava0.1580.0730.0740.0850.271
Bruxelles0.0100.0000.0370.0520.218
Bucharest0.084 0.0190.0420.1690.243
Budapest0.1460.0960.1010.1920.322
Burgas0.0420.0410.0440.0560.160
Cardiff0.1900.0790.0680.1070.231
Cluj-Napoca0.0930.0820.0550.1920.289
Diyarbakir0.0530.0110.0200.0320.065
Dortmund0.0360.0370.0700.0990.263
Dublin0.1270.0920.0990.0910.178
Essen0.0660.1060.1030.1300.382
Gdańsk0.0520.0430.0370.0630.266
Genève0.1240.0540.0650.0840.232
Glasgow0.1420.1240.1030.1590.178
Graz0.0560.0350.0250.0610.183
Groningen0.2610.1940.0500.0930.233
Hamburg0.0690.0750.1200.0900.331
Helsinki0.2360.0970.0940.0590.061
Irakleio0.0260.0280.0190.1520.058
Istanbul0.0290.0390.0550.0440.113
København0.1120.0960.1770.0680.044
Košice0.0840.0640.0420.0960.291
Kraków0.0970.0430.0430.0940.333
Lefkosia0.0080.0230.0410.1290.045
Leipzig0.1130.0740.0820.2090.419
Liège0.1210.0330.0270.1660.292
Lille0.0470.0310.0640.1060.313
Lisboa0.0860.0310.0390.0910.190
Ljubljana0.1090.0800.1180.1260.202
London0.1300.0750.0790.0910.222
Luxembourg0.0570.0210.0270.0150.078
Madrid0.0420.0440.0460.1050.190
Málaga0.0950.0800.0960.1110.176
Malmö0.1590.0980.1900.0790.076
Manchester0.1460.0450.0860.1320.076
Marseille0.0700.0530.0700.0700.295
Miskolc0.2570.1470.0720.2100.290
Munich0.1460.0820.0830.1120.321
Naples0.0360.0200.0200.0880.142
Oslo0.2570.2070.1790.1380.174
Ostrava0.1580.0850.0600.0680.218
Oulu0.1020.0780.0680.0720.109
Oviedo0.0620.0460.0490.1040.149
Palermo0.0360.0310.0140.0400.075
Paris0.0330.0230.0670.0340.250
Piatra Neamt0.0790.0770.0350.2660.259
Podgorica0.0480.0500.0700.0630.184
Praha0.1900.1450.1210.1330.288
Rennes0.0160.0350.0370.0520.212
Reykjavík0.2110.0940.0410.1470.077
Riga0.1440.0460.0360.0980.114
Rome0.0180.0190.0120.0760.072
Rostock0.1530.0870.0470.1320.317
Rotterdam0.1110.0550.0600.0920.357
Skopje0.0100.0320.0250.0890.129
Sofia0.1530.1110.0160.1440.322
Stockholm0.2060.1230.1580.1510.237
Strasbourg0.0350.0260.0470.0840.259
Tallinn0.2150.1410.2220.0800.155
Tirana0.0340.0150.0150.0550.024
Turin0.0590.0280.0200.1430.145
Tyneside conurbation0.1800.0680.0760.2040.242
Valletta0.1390.0960.1830.1360.326
Verona0.0590.0360.0370.1100.123
Vilnius0.1540.0980.1180.0700.218
Warszawa0.0740.0610.0460.0300.306
Wien0.0900.0210.0270.0620.159
Zagreb0.0340.0590.0340.2230.154
Zurich0.1430.0710.0410.1160.157
The assessments of cities in terms of five criteria in the form of IFVs were used to construct an intuitionistic fuzzy decision matrix, a fragment of which is presented below for the three selected cities: Weights have not been assigned to individual criteria. In our opinion, all the aspects (e.g., time for request, procedures, fees charged, information and services, corruption) should be balanced, i.e., they are equally important in evaluating satisfaction from the local administration. The coordinates of intuitionistic fuzzy pattern objects were determined twofold: based on (1,0) values and second for maximum and minimum IFVs, respectively (Table 5 and Table 6).
Table 5

The coordinates of an intuitionistic fuzzy pattern object based on (1,0) values.

ParameterC1C2C3C4C5
ν 00000
μ 11111
π 00000
Table 6

The coordinates of an intuitionistic fuzzy pattern object based on max and min values.

ParameterC1C2C3C4C5
ν 0.1240.1990.1470.0800.164
μ 0.7330.7890.7710.8460.789
π 0.1430.0130.0820.0740.047
Using the normalized Euclidean or Hamming distance in accordance with the Formulas (3)–(6) the distances of each city from the intuitionistic fuzzy pattern objects and values were calculated. Finally, the IFSM coefficients were calculated (Table 7).
Table 7

Distances and IFSM values.

CityMeasureIFSM_me2IFSM_me3IFSM_mh2IFSM_mh3IFSM_ae2IFSM_ae3IFSM_ah2IFSM_ah3
Zurich d+ 0.0470.0640.0290.0540.2180.2330.2070.260
d0 0.4600.4650.4390.4740.6380.6430.6180.662
IFSM value0.8990.8630.9350.8870.6580.6380.6650.607
Palermo d+ 0.5440.5450.5330.5580.7240.7240.7110.731
d0 0.4600.4650.4390.4740.6380.6430.6180.662
IFSM value−0.183−0.177−0.214−0.178−0.135−0.127−0.152−0.105
The values of IFSM coefficients for all cities are presented in Table A4 in the Appendix A.
Table A4

Values of IFSM coefficients for cities.

CityIFSM_me2IFSM_me3IFSM_mh2IFSM_mh3IFSM_ae2IFSM_ae3IFSM_ah2IFSM_ah3
Aalborg0.7840.7660.8390.8010.5770.5580.5970.541
Amsterdam0.5420.5180.5680.5280.3880.3690.4040.346
Ankara0.5290.5230.5290.5070.3840.3870.3760.390
Antalya0.6750.6550.7280.6790.5070.5050.5170.516
Antwerpen0.5540.5270.5960.5340.4200.3990.4240.364
Athina0.0300.0230.014−0.0080.0200.0190.0100.016
Barcelona0.3560.3520.3630.3500.2630.2670.2580.284
Belfast0.5650.5480.5780.5330.4030.3840.4110.341
Beograd0.0240.0250.0280.0330.0160.0180.0200.026
Berlin0.2400.2160.2620.2040.1740.1550.1870.120
Białystok0.6490.6120.6840.6140.4760.4620.4860.451
Bologna0.4460.4410.4480.4280.3190.3200.3180.322
Bordeaux0.5900.5640.5920.5460.4240.4120.4210.386
Braga0.4000.3940.4320.4120.2980.2960.3070.295
Bratislava0.3920.3760.4310.4070.2890.2750.3070.253
Bruxelles0.6370.6090.6740.6170.4780.4700.4790.466
Bucharest0.2430.2330.2840.2530.1850.1770.2020.171
Budapest0.4490.4140.4770.4110.3230.2960.3390.254
Burgas0.3840.3800.4250.4050.2830.2830.3020.297
Cardiff0.7230.6920.7350.6820.5160.4920.5220.452
Cluj-Napoca0.3990.3740.5150.4440.3180.2990.3660.301
Diyarbakir0.3100.3110.2960.3030.2250.2290.2100.236
Dortmund0.3490.3340.3440.3110.2480.2400.2440.218
Dublin0.6710.6570.6950.6630.4850.4720.4940.439
Essen0.4360.3890.4670.3840.3170.2900.3320.258
Gdańsk0.5990.5690.6410.5840.4420.4290.4550.422
Genève0.5930.5760.6700.6370.4560.4420.4770.427
Glasgow0.5320.5180.5310.4920.3770.3620.3770.312
Graz0.7580.7330.7870.7360.5530.5450.5590.534
Groningen0.6730.6210.7430.6490.4970.4610.5280.434
Hamburg0.5440.5030.5680.5000.3940.3700.4040.340
Helsinki0.4780.4750.4920.4840.3400.3300.3500.311
Irakleio0.0850.0870.0970.1040.0610.0650.0690.088
Istanbul0.3870.3850.4040.3930.2860.2880.2870.293
København0.7440.7310.7940.7630.5460.5330.5650.519
Košice0.4620.4380.5280.4750.3480.3340.3750.330
Kraków0.4930.4580.5200.4670.3560.3360.3700.320
Lefkosia0.4370.4320.5220.5060.3400.3410.3710.376
Leipzig0.5650.4890.6050.5080.4140.3700.4300.332
Liège0.5710.5360.6100.5470.4250.4030.4340.375
Lille0.5060.4720.5110.4570.3680.3510.3630.321
Lisboa0.2190.2180.2200.2190.1560.1550.1570.147
Ljubljana0.4460.4370.5260.4890.3410.3290.3740.320
London0.5940.5780.6060.5780.4240.4100.4310.379
Luxembourg0.7820.7680.8070.7710.5720.5730.5740.572
Madrid0.4500.4410.4400.4100.3230.3190.3130.295
Málaga0.4760.4700.4730.4500.3400.3330.3360.296
Malmö0.5780.5710.6050.5830.4160.4030.4300.377
Manchester0.6940.6930.7140.7090.5010.4920.5080.468
Marseille0.3040.2870.3010.2730.2210.2110.2140.182
Miskolc0.4850.4420.5520.4510.3570.3220.3920.285
Munich0.5840.5440.6100.5580.4200.3940.4340.359
Naples−0.031−0.028−0.063−0.044−0.026−0.022−0.045−0.021
Oslo0.3750.3510.3950.3140.2650.2400.2810.185
Ostrava0.4120.4030.4880.4660.3080.2980.3470.302
Oulu0.5400.5390.5340.5300.3850.3820.3800.357
Oviedo0.3250.3240.3070.3000.2310.2300.2180.209
Palermo−0.183−0.177−0.214−0.178−0.135−0.127−0.152−0.105
Paris0.5110.4900.5100.4670.3680.3600.3630.344
Piatra Neamt0.3850.3560.4580.3760.2970.2760.3250.262
Podgorica0.2500.2470.2920.2830.1920.1910.2080.198
Praha0.3860.3600.4570.3880.2870.2630.3250.238
Rennes0.6930.6620.7070.6490.5050.4970.5030.483
Reykjavík0.2340.2340.2300.2250.1610.1560.1630.133
Riga0.1170.1220.1340.1630.0860.0870.0960.090
Rome−0.178−0.173−0.204−0.163−0.132−0.124−0.145−0.098
Rostock0.6320.5860.6710.6010.4630.4340.4770.401
Rotterdam0.5760.5280.6000.5400.4170.3900.4260.363
Skopje0.1080.1080.1450.1440.0890.0920.1030.120
Sofia0.2020.1820.2090.1580.1420.1270.1490.093
Stockholm0.5370.5100.5640.4880.3850.3580.4010.309
Strasbourg0.6210.5890.6290.5770.4530.4400.4470.416
Tallinn0.4820.4630.5840.5190.3670.3440.4150.331
Tirana0.2450.2460.2930.2990.1890.1940.2080.239
Turin0.1100.1100.0900.0880.0740.0750.0640.067
Tyneside conurbation0.6180.5870.6300.5680.4450.4180.4480.369
Valletta0.6680.6070.7060.6160.4860.4470.5020.402
Verona0.2700.2710.2770.2760.1940.1950.1970.196
Vilnius0.4380.4280.4390.4160.3120.3000.3120.259
Warszawa0.4340.4080.4540.3980.3120.2990.3230.290
Wien0.7860.7690.7990.7640.5660.5580.5680.543
Zagreb0.001−0.0020.009−0.0150.000−0.0020.006−0.004
Zurich0.8990.8630.9350.8870.6580.6380.6650.607
The position of cities in the ranking was determined based on the IFSM coefficient values, following the principle that the higher the value of the IFSM coefficient, the higher the city’s position in the ranking (Table A5; Appendix A).
Table A5

Rank of IFSM coefficients for cities.

CityIFSM_me2IFSM_me3IFSM_mh2IFSM_mh3IFSM_ae2IFSM_ae3IFSM_ah2IFSM_ah3
Aalborg34222424
Amsterdam3131313031333131
Ankara3530373334283720
Antalya1011998797
Antwerpen2929272725252727
Athina7879797978787979
Barcelona6160616061596154
Belfast2824302829293033
Beograd7978787879797878
Berlin7072707270717072
Białystok1413141615131412
Bologna4743534848465338
Bordeaux2223282522212821
Braga5352565054535649
Bratislava5556575356585759
Bruxelles1514151414121510
Bucharest6970686969696869
Budapest4549475146524758
Burgas5955585459565847
Cardiff788879811
Cluj-Napoca5457424749504246
Diyarbakir6464656364646562
Dortmund6262626262626263
Dublin1210131013111313
Essen5053495850544957
Gdańsk1922181820191816
Genève2120171317161715
Glasgow3432363635343642
Graz55665565
Groningen11127111114714
Hamburg3034323530313234
Helsinki4137453943434543
Irakleio7777767677777676
Istanbul5654595658555951
København66556656
Košice4345384041413837
Kraków3841414140404140
Lefkosia4947403445394024
Leipzig2736253228322535
Liège2627222421242225
Lille3738434437374339
Lisboa7271727172727270
Ljubljana4646393742443941
London2019232023222322
Luxembourg43333232
Madrid4444545247475450
Málaga4239484644424848
Malmö2421241927232423
Manchester871071010109
Marseille6565646865656468
Miskolc3942344539453453
Munich2325212324262129
Naples8181818181818181
Oslo6061606160616067
Ostrava5251464353514645
Oulu3226352932303530
Oviedo6363636463636364
Palermo8383838383838383
Paris3635444236354432
Piatra Neamt5859505955575055
Podgorica6767676667686765
Praha5758515757605161
Rennes99111298118
Reykjavík7169717071707171
Riga7474757375757575
Rome8282828282828282
Rostock1618161716181619
Rotterdam2528262626272628
Skopje7676747574747473
Sofia7373737473737374
Stockholm3333333833363344
Strasbourg1716202118172017
Tallinn4040293138382936
Tirana6868666568676660
Turin7575777776767777
Tyneside conurbation1817192219201926
Valletta1315121512151218
Verona6666696766666966
Vilnius4848554952485556
Warszawa5150525551495252
Wien22444343
Zagreb8080808080808080
Zurich11111111
Descriptive statistics and box plots for the values of IFSM coefficients are presented in Table 8 and Figure 2.
Table 8

Descriptive statistics for IFSM values.

Descriptive StatisticsSatisfaction with Administration
IFSM_me2IFSM_me3IFSM_mh2IFSM_mh3IFSM_ae2IFSM_ae3IFSM_ah2IFSM_ah3
Min−0.183−0.177−0.214−0.178−0.135−0.127−0.152−0.105
Max0.8990.8630.9350.8870.6580.6380.6650.607
Range1.0811.0391.1491.0640.7920.7650.8170.712
Average0.4440.4260.4690.4330.3250.3130.3340.298
Standard deviation0.4440.4260.4690.4330.3250.3130.3340.298
Figure 2

Box plots for the IFSM values.

Based on Table 8 and Figure 2 three main observations can be made: determining the coordinates of the pattern object based on intuitionistic values (1,0) resulted in a lower value of the IFSM for cities compared with the IFSM when the coordinates of the pattern object are based on max and min values. The IFSM area of variability also decreased; the IFSM similarly differentiates cities in terms of the adopted synthetic criterion, i.e., satisfaction with public administration services; regardless of the method used for determining the coordinates of the pattern object, the number of cities for which the IFS values are below and above the IFSM average remains at a similar level; regardless of the method for determining the coordinates of the pattern object, the IFSM values present a slight response to the choice of the distance measure and the number of parameters that take these distances into account. The introduction of the third uncertainty parameter in measuring the distance between cities and the pattern object slightly lowers the mean values of IFSM and reduces the variability range of these values. This regularity has been observed for two methods used in determining the coordinates of the pattern object. The Spearman coefficients between IFSM values are presented in Table 9.
Table 9

Spearman coefficients between IFSM measures.

Coefficient IFSM_me2IFSM_me3IFSM_mh2IFSM_mh3IFSM_ae2IFSM_ae3IFSM_ah2IFSM_ah3
IFSM_me21.0000.958 **0.927 **0.920 **0.971 **0.949 **0.927 **0.872 **
IFSM_me3 1.0000.898 **0.922 **0.939 **0.952 **0.898 **0.884 **
IFSM_mh2 1.0000.929 **0.945 **0.917 **1.000 **0.870 **
IFSM_mh3 1.0000.926 **0.936 **0.929 **0.911 **
IFSM_ae2 1.0000.958 **0.945 **0.887 **
IFSM_ae3 1.0000.917 **0.921 **
IFSM_ah2 1.0000.870 **
IFSM_ah3 1.000
IFSM differ with distance measure parameters (2 or 3)IFSM differ with distance measures (Hamming or Euclidean) IFSM differ with pattern (based on (1,0) or based on max, min values)IFSM differ with all elements: distance measures parameters, distance measure function, and pattern

** p = 0.01.

The choice between the Euclidean and Hamming distance and the method for determining the coordinates of the pattern objects does not have a large impact on the ranking positions of the cities. High values of the Spearman coefficient suggest slight changes in the ranking position of the cities. If the Hamming distance for two parameters is used in IFSM, the choice of the method for determining the coordinates of the pattern objects is irrelevant. In both cases, the value of the Spearman coefficient was equal to one, which means the same ranking of the cities in terms of satisfaction with public administration services. The lowest similarity of rankings (Spearman coefficient value equal to 0.870) was observed for the IFSM with the Hamming distance with two and three parameters, respectively, for the coordinates of the pattern objects determined based on the values (max, min) and (1,0). The Information Transfer Measures for the IFSMes are presented in Table 10.
Table 10

The Information Transfer Measures for IFSMes.

CriteriaIFSM_me2IFSM_me3IFSM_mh2IFSM_mh3IFSM_ae2IFSM_ae3IFSM_ah2IFSM_ah3
C10.879 **0.862 **0.906 **0.881 **0.891 **0.716 **0.906 **0.870 **
C20.762 **0.764 **0.781 **0.786 **0.781 **0.593 **0.781 **0.803 **
C30.668 **0.650 **0.716 **0.691 **0.679 **0.503 **0.716 **0.649 **
C40.739 **0.739 **0.764 **0.766 **0.733 **0.556 **0.764 **0.750 **
C50.858 **0.860 **0.816 **0.816 **0.845 **0.647 **0.816 **0.795 **

** p = 0.01.

The criteria are well represented by the IFSM. The largest information transfer occurred for C1 criterion (regardless of the IFSM variant). The smallest information transfer was recorded for C3 criterion. All the criteria are the least represented by the IFSM_ae3 variant. The same values of Spearman coefficients were observed for the two variants: IFSM_mh2 and IFSM_ah2. This result is not surprising since equal rankings were obtained using these variants of the methods (see Table 10).

4.3. Comparative Analysis and Implications

Hellwig’s method uses only the concept of a positive pattern object (named as the pattern of development), while the well known TOPSIS method [34] used the concept of pattern and anti-pattern object (ideal and anty-ideal solution, respectively). The TOPSIS with many modifications in the fuzzy and intuitionistic fuzzy environment has been proposed and applied in real-life problems [19,35,36]. Similarly, a description of variants of the IFTes with respect to pattern objects and distance measures used in comparative analysis is presented in Table 11 (for details see [19]).
Table 11

A classification of variants of the IFTes with respect to pattern objects, and distance measure.

MethodsPattern ObjectsThe Distance MeasureNumber of Parameters in the Distance Measure
IFT_me2based on max and min valuesEuclidean distancedistance based on two parameters
IFT_me3based on max and min valuesEuclidean distancedistance based on three parameters
IFT_mh2based on max and min valuesHamming distancedistance based on two parameters
IFT_mh3based on max and min valuesHamming distancedistance based on three parameters
IFT_ae2based on (1,0) and (0,1) valuesEuclidean distancedistance based on two parameters
IFT_ae3based on (1,0) and (0,1) valuesEuclidean distancedistance based on three parameters
IFT_ah2based on (1,0) and (0,1) valuesHamming distancedistance based on two parameters
IFT_ah3based on (1,0) and (0,1) valuesHamming distancedistance based on three parameters
The coordinates of an intuitionistic fuzzy anti-pattern object based on (0,1) values or max and min in the IFT are presented in Table 12. The coordinates of an intuitionistic fuzzy anti-pattern object based on max and min values are presented in Table 13.
Table 12

The coordinates of an intuitionistic fuzzy anti-pattern object based on (0,1) values (used in IFT method).

ParameterC1C2C3C4C5
ν 11111
μ 00000
π 00000
Table 13

The coordinates of an intuitionistic fuzzy anti-pattern object based on max and min values (used in IFT method).

ParameterC1C2C3C4C5
ν 0.8360.7100.7650.4500.772
μ 0.1280.2710.2210.5010.088
π 0.0360.0190.0140.0480.140
Descriptive statistics and box plots for the values of IFT coefficients are presented in Table 14 and Figure 3.
Table 14

Descriptive statistics for IFT values.

Descriptive StatisticsSatisfaction with Administration
IFT_me2IFT_me3IFT_mh2IFT_mh3IFT_ae2IFT_ae3IFT_ah2IFT_ah3
Min0.0580.0700.0380.0500.3050.5090.2890.513
Max0.9200.8950.9480.9110.7850.8140.7930.827
Range0.8620.8240.9100.8610.4800.3050.5040.314
Average0.5700.5630.5790.5650.5840.6680.5880.690
Standard deviation0.1730.1660.1860.1730.0970.0640.1030.066
Figure 3

Box plots for IFT values.

Determining the coordinates of the pattern object based on the values (max, min) resulted in the average IFT values presenting a very similar level with a highly corresponding variability range. In this case, choosing the distance measure and taking into account the degree of uncertainty in its measurement are of no great importance. Establishing the coordinates of the pattern object based on the intuitionistic values (1,0) significantly reduced the variability range of the IFT values. Moreover, for this type of pattern object, the IFT has become more sensitive to the number of parameters included in measuring the distance between cities and the pattern object. It is evident that the average IFT values increased after taking into account the degree of uncertainty for both the Euclidean and Hamming distances. The variability range of IFT values, in this case, is also the smallest among all the analyzed IFT variants. The Spearman coefficients between IFT measures are presented in Table 15.
Table 15

Spearman coefficients between IFTes.

CoefficientIFT_me2IFT_me3IFT_mh2IFT_mh3IFT_ae2IFT_ae3IFT_ah2IFT_ah3
IFT_me21.0000.996 **0.994 **0.998 **0.986 **0.996 **0.972 **0.999 **
IFT_me3 1.0000.995 **0.994 **0.997 **0.987 **0.995 **0.972 **
IFT_mh2 1.0000.997 **0.999 **0.983 **1.000 **0.971 **
IFT_mh3 1.0000.995 **0.978 **0.997 **0.911 **
IFT_ae2 1.0000.987 **0.999 **0.975 **
IFT_ae3 1.0000.983 **0.995 **
IFT_ah2 1.0000.971 **
IFT_ah3 1.000
IFT measures differ with distance measure parameters (2 or 3)IFT measures differ with distance measures (Hamming or Euclidean) IFT measures differ with pattern (based on (1,0) or based on max, min values)IFT measures differ with all elements: distance measures parameters, distance measure function, and pattern

** p = 0.01.

The total consistency of the city rankings using the IFT (Spearman coefficient value equal to 1) was obtained for the Hamming distance with two parameters and for the coordinates of pattern objects determined based on the values (max, min) and (1,0), respectively. However, the lowest still very high consistency of rankings was found in two cases. In the first one, the IFT values were calculated by defining the pattern object coordinates based on the value (1,0) and using the Hamming distance for 2 and 3 parameters, respectively. The second case was very similar and the difference was only in the method used for determining the pattern object coordinates. The Information Transfer Measures for the IFTes are presented in Table 16.
Table 16

The Information Transfer Measures for the IFTes.

CriteriaIFT_me2IFT_me3IFT_mh2IFT_mh3IFT_ae2IFT_ae3IFT_ah2IFT_ah3
C10.896 **0.891 **0.906 **0.896 **0.904 **0.875 **0.906 **0.870 **
C20.763 **0.764 **0.781 **0.769 **0.779 **0.800 **0.781 **0.803 **
C30.702 **0.705 **0.716 **0.726 **0.702 **0.654 **0.716 **0.649 **
C40.742 **0.741 **0.764 **0.766 **0.756 **0.740 **0.764 **0.750 **
C50.848 **0.846 **0.816 **0.818 **0.828 **0.823 **0.816 **0.795 **

** p = 0.01.

All the criteria are very well represented by the IFT. It is not possible to identify the IFT variant following which the information transfer for all the criteria is the highest or the lowest. As with the IFSM, an identical information transfer for each criterion was observed for IFT_mh2 and IFT_ah2. Regardless of the IFT variant, C3 was the least represented, whereas C1 received the strongest representation of all the criteria applied. Spearman coefficients between the IFSM and the IFT measures are presented in Table 17.
Table 17

Spearman coefficients between the IFSMes and the IFTes.

CoefficientIFT_me2IFT_me3IFT_mh2IFT_mh3IFT_ae2IFT_ae3IFT_ah2IFT_ah3
IFSM_me20.995 **0.995 **0.988 **0.985 **0.993 **0.988 **0.988 **0.972 **
IFSM_me30.990 **0.991 **0.981 **0.979 **0.986 **0.991 **0.981 **0.977 **
IFSM_mh20.996 **0.995 **1.000 **0.997 **0.999 **0.983 **1.000 **0.971 **
IFSM_mh30.989 **0.991 **0.990 **0.991 **0.991 **0.993 **0.990 **0.986 **
IFSM_ae20.997 **0.997 **0.993 **0.989 **0.997 **0.991 **0.993 **0.977 **
IFSM_ae30.992 **0.993 **0.987 **0.983 **0.992 **0.997 **0.987 **0.988 **
IFSM_ah20.996 **0.995 **1.000 **0.997 **0.999 **0.983 **1.000 **0.971 **
IFSM_ah30.972 **0.972 **0.971 **0.965 **0.975 **0.995 **0.971 **1.000 **

** p = 0.01.

The compared measures, regardless of the distance used and the coordinates of the pattern object, rank cities in a very similar way in terms of satisfaction with public administration services. The lowest value of the Spearman coefficient was 0.971, which means very high consistency of all the obtained rankings. In the case of using the pattern object, the coordinates of which were determined based on intuitionistic values (1,0), and the Hamming distance, it did not matter whether the IFSM or the IFT measure was used in ranking the cities (Spearman coefficient value was 1). In this case, it was also irrelevant to include the uncertainty parameter in the calculation of the distance between the cities and the pattern city. Identical results of the city rankings using the IFSM, and the IFT were also recorded for the combination of the Hamming measure with two parameters and the coordinates of the pattern objects determined based on max and min values. Taking into account the uncertainty parameter in this case resulted in slight changes in the ranking position of some cities.

5. Conclusions

The paper proposes the IFSM as a method for measuring complex phenomena based on survey data. Most frequently, this type of data takes the form of ordinal data. In this case the measurement results at the level of individual respondents are not required. The suggested method allows measuring complex phenomena from aggregated ordinal data offered by public statistics. The proposed approach adopts the transformation of aggregated ordinal data into intuitionistic fuzzy sets. The IFSM construction, as with other synthetic measures, requires the researcher to make subjective decisions regarding, e.g., the choice of the distance measure and how to determine the coordinates of the pattern object. Therefore, this paper provides a comparative analysis addressing the two most popular distances for the intuitionistic fuzzy sets, the Euclidean distance and the Hamming distance. Both two and three parameters of the intuitionistic fuzzy sets were taken into account in the distance calculation. The construction of a pattern object based on the intuitionistic values was also proposed and compared with the classical approach, where the coordinates of the pattern object are determined based on the maximum and minimum criterion assessments observed in the research sample. In addition, the findings collected using the IFSM were compared with the IFT since both methods are very similar in their construction and use the idea of pattern (reference) objects. The empirical example presented in the paper as well as the comparative analyses carried out for different variants of the IFSM method allowed formulating the following conclusions: in each of the eight analyzed variants of the synthetic measure construction, the mean values of IFT for the cities were higher than in the case of IFSM. Furthermore, in each of these cases the variability range of IFT values was lower than that of IFSM. This is primarily true when the coordinates of the pattern objects were established based on the intuitionistic values (1,0); in the case of the pattern object coordinates determined based on the values (max, min), very similar changes in the ranges of their value variability were observed for the IFSM and IFT, depending on the selected distance measure and the number of parameters included in it; determining the coordinates of the pattern objects based on the value (1,0) caused that the values of IFSM and IFT changed in an opposite way as a result of the applied distance and taking into account the degree of uncertainty. The increase in the value of the IFT measure for the cities occurred along with the decrease in the value of IFSM and vice versa. It should be noted, however, that the increase in IFT values took place at a reduced variability range. In the case of IFSM such a large reduction in variability was not observed. Therefore, the application of the IFSM in the variant with the pattern object, the coordinates of which are determined based on the intuitionistic values (1,0), allowed for differentiating cities to a greater extent in terms of the complex phenomenon, i.e., satisfaction with public administration services; for the analyzed data set, the ranking of cities determined on the basis of both IFSM and IFT values turned out to be a little sensitive to the choice of the distance measure and the method for determining the coordinates of pattern objects. The values of the correlation coefficients for the obtained rankings were very high, reaching the value of 1 in some cases. Slightly greater consistency of the rankings was obtained for the IFT, which suggests a somewhat higher sensitivity of the IFSM to the choice of the distance measure and the method for determining the coordinates of the pattern object. In the case of both methods, the highest ranking consistency was recorded using the Hamming distance for two parameters and the coordinates of pattern objects established based on the values (max, min) and (1,0), respectively. Therefore, including the third parameter in measuring the distance, taking the form of the degree of uncertainty, changes the position of cities in the rankings, although in the presented example these changes were small and referred to some cities only. The least consistent rankings for both measures were also observed for the Hamming distance, however, for a different number of parameters combined with: It should be highlighted that, despite the high consistency of the obtained rankings, the values of measures for cities were diversified, which suggests a different level of residents’ satisfaction with public administration services. It is of particular importance in the context of monitoring the analyzed phenomenon over time because the same position of a city in the ranking does not imply that the level of the phenomenon is not going to increase over time; pattern objects, the coordinates of which were determined based on the values (1,0); pattern objects, the coordinates of which were determined based on the values (1,0) and (max, min). it is difficult to identify, from among the IFSM and the IFT, a better method in terms of representing the particular criteria. All criteria are very well represented by both the IFSM and the IFT. In either case, the largest transfer of information was recorded for C1 and the smallest for C3. A certain limitation of the proposed method for transforming ordinal data is that there is no possibility to differentiate categories on the side of “positive” and “negative” responses. This may have an impact on the synthetic measure values and the ranking positions of the assessed objects. One of the directions for further research on the IFSM will be presenting some proposals in this area. The influence of data distribution on the results of object ranking using the IFSM will also be analyzed.
Table A6

Values of function and rank of criteria.

CitySc1Sc2Sc3Sc4Sc5Rank C1Rank C2Rank C3Rank C4Rank C5
Aalborg0.5050.3150.3230.7450.6223212521
Amsterdam0.1590.2040.1110.6780.1663843591029
Ankara0.1530.3070.1370.4830.0693923544139
Antalya0.2940.5670.2950.6830.18119231925
Antwerpen0.2970.5250.2480.1680.204186398023
Athina−0.245−0.175−0.4600.174−0.4117476807973
Barcelona−0.0950.275−0.1740.3660.0476728756441
Belfast0.1930.3290.2580.5340.0483520363240
Beograd−0.297−0.258−0.1530.316−0.6617678747082
Berlin−0.253−0.2850.3090.284−0.0787581297350
Białystok0.3980.2640.3870.6370.141732181534
Bologna0.0760.0490.0130.5520.0995154672836
Bordeaux0.1940.2290.2760.5070.2193439333619
Braga0.0290.3570.1610.427−0.2535918525462
Bratislava0.177−0.0030.3920.420−0.2673664165763
Bruxelles0.2640.5770.2750.4520.214251344522
Bucharest0.0000.0020.3160.272−0.5206461277477
Budapest0.0640.2310.1470.546−0.0695438532949
Burgas0.0790.1760.0040.661−0.2295047681257
Cardiff0.3340.3820.3420.6150.377121522198
Cluj-Napoca0.3490.2850.4470.451−0.449112664774
Diyarbakir−0.1960.0000.2190.181−0.0797062427851
Dortmund0.045−0.0920.0290.406−0.0555869666047
Dublin0.3050.4100.3730.6450.1441710191333
Essen0.197−0.1660.2510.4450.1473275384932
Gdańsk0.3160.2690.4030.6260.0231530141643
Genève0.5580.5460.2110.4520.00024434645
Glasgow0.0950.2400.1620.4970.1614836503930
Graz0.3120.4560.4200.7230.366168949
Groningen0.3630.4090.1360.7470.43210115515
Hamburg0.3260.0260.2790.5180.1671356323528
Helsinki0.018−0.1070.3130.5050.2576170283717
Irakleio−0.327−0.124−0.4500.463−0.3137772794366
Istanbul0.0450.2990.0890.407−0.2405725625960
København0.2900.2570.4100.7330.62220331132
Košice0.2230.2210.3900.559−0.2532940172761
Kraków0.282−0.0010.1710.5850.0712263482238
Lefkosia0.0700.5660.1100.601−0.231533612059
Leipzig0.3750.2670.1840.4770.175831464227
Liège0.2600.5260.2530.3790.084275376237
Lille0.0860.3050.1650.3370.1774924496826
Lisboa−0.2340.016−0.0410.385−0.3357258706170
Ljubljana0.1960.2740.4680.525−0.328332953368
London0.1220.2450.2380.6200.2604435401816
Luxembourg0.4190.4020.5550.6910.3026132713
Madrid0.1220.1930.0620.3680.0134345646344
Málaga0.1000.2490.1310.432−0.0104734575246
Malmö0.173−0.0270.3390.5740.419376623256
Manchester0.3680.4340.3700.6390.14999211431
Marseille0.0710.152−0.0760.208−0.2095248727755
Miskolc0.2780.3100.2040.623−0.1682322441754
Munich0.1470.1150.3710.5750.2974050202414
Naples−0.473−0.272−0.4410.090−0.3578180788271
Oslo−0.162−0.1270.1620.4180.2696973515815
Ostrava0.1150.0440.4070.687−0.286465512865
Oulu0.2010.0130.1890.4480.3183159454810
Oviedo0.0050.067−0.1030.295−0.0926253737252
Palermo−0.708−0.408−0.5440.059−0.5148382838376
Paris0.1210.2180.1310.4530.1404542564435
Piatra Neamt0.2680.2030.3990.346−0.3822444156772
Podgorica−0.0780.0970.2950.316−0.5246651307178
Praha0.019−0.0240.4980.568−0.230606532658
Rennes0.2480.3430.3320.5450.460281924304
Reykjavík−0.243−0.086−0.0520.432−0.2177368715156
Riga−0.0290.013−0.4600.366−0.4756560816575
Rome−0.672−0.439−0.4660.120−0.6168283828180
Rostock0.4500.1410.4440.5190.21544973421
Rotterdam0.1280.2770.3180.5370.1964227263124
Skopje−0.3600.0940.1280.256−0.6568052587681
Sofia−0.201−0.1500.0620.361−0.3297174656669
Stockholm0.0580.0220.4070.5000.3135657133811
Strasbourg0.3220.3660.2350.4240.2381416415518
Tallinn0.2840.2340.4840.666−0.281213741164
Tirana0.0030.192−0.0360.486−0.5366346694079
Turin−0.343−0.251−0.3150.258−0.1307977777553
Tyneside conurbation0.2600.4060.2620.4430.2192612355020
Valletta0.2230.4000.4180.5810.3033014102312
Verona−0.119−0.1170.1740.420−0.3156871475667
Vilnius0.0590.2190.1110.428−0.0655541605348
Warszawa0.145−0.0310.0850.5890.0324167632142
Wien0.4450.3580.4410.7000.389517867
Zagreb−0.341−0.261−0.2110.336−0.6617879766983
Zurich0.6100.517 0.583 0.7180.50117153
Table A7

Values of IFTOPSIS coefficients for cities.

CityIFT_me2IFT_me3IFT_mh2IFT_mh3IFT_ae2IFT_ae3IFT_ah2IFT_ah3
Aalborg0.8350.8240.8730.8500.7370.7770.7510.797
Amsterdam0.6380.6280.6580.6430.6230.6880.6320.711
Ankara0.6210.6160.6270.6030.6120.7000.6150.731
Antalya0.7470.7340.7840.7430.6930.7550.7020.786
Antwerpen0.6690.6560.6800.6540.6380.6990.6440.719
Athina0.2400.2450.2180.2290.3960.5500.3880.566
Barcelona0.4930.4900.4950.4800.5400.6510.5420.684
Belfast0.6540.6470.6660.6520.6290.6910.6360.709
Beograd0.2470.2490.2300.2350.4050.5530.3950.570
Berlin0.4350.4370.4150.4300.4980.5940.4980.612
Białystok0.7250.7040.7490.7050.6740.7320.6830.758
Bologna0.5560.5520.5620.5380.5750.6710.5790.701
Bordeaux0.6720.6570.6770.6490.6380.7070.6430.729
Braga0.5350.5300.5500.5320.5670.6600.5720.689
Bratislava0.5430.5390.5490.5500.5660.6470.5720.671
Bruxelles0.7180.7020.7410.7010.6730.7370.6780.764
Bucharest0.4440.4430.4320.4320.5060.6100.5070.634
Budapest0.5690.5600.5850.5790.5850.6520.5920.671
Burgas0.5200.5160.5440.5240.5630.6570.5690.690
Cardiff0.7790.7610.7900.7610.6970.7420.7050.758
Cluj-Napoca0.5790.5710.6150.5940.5940.6620.6080.691
Diyarbakir0.4570.4560.4420.4370.5120.6340.5120.663
Dortmund0.4830.4780.4800.4710.5320.6320.5330.655
Dublin0.7380.7300.7580.7390.6790.7330.6880.753
Essen0.5750.5610.5770.5610.5810.6510.5870.673
Gdańsk0.6880.6710.7150.6790.6540.7180.6640.745
Genève0.6990.6900.7390.7190.6650.7220.6770.747
Glasgow0.6250.6200.6280.6200.6110.6800.6160.697
Graz0.8070.7900.8310.7900.7200.7710.7280.795
Groningen0.7520.7260.7960.7500.6920.7290.7090.750
Hamburg0.6470.6290.6570.6320.6240.6870.6320.709
Helsinki0.5970.5990.5980.6060.5930.6710.5990.696
Irakleio0.2880.2930.2840.2960.4330.5740.4250.598
Istanbul0.5180.5160.5280.5120.5570.6580.5600.688
København0.8030.7950.8370.8190.7180.7650.7310.788
Košice0.5960.5860.6250.6030.6040.6760.6140.704
Kraków0.6060.5930.6200.6000.6030.6750.6110.700
Lefkosia0.5810.5780.6210.6110.6000.6860.6110.725
Leipzig0.6680.6380.6870.6560.6380.6860.6480.705
Liège0.6710.6550.6910.6620.6430.7020.6500.724
Lille0.6110.5970.6130.5890.6030.6790.6070.701
Lisboa0.3750.3740.3820.3830.4810.6000.4790.624
Ljubljana0.5920.5890.6240.6130.6020.6730.6130.700
London0.6750.6670.6880.6750.6410.7050.6490.726
Luxembourg0.8270.8170.8470.8160.7310.7850.7370.811
Madrid0.5590.5530.5570.5380.5740.6650.5760.689
Málaga0.5790.5760.5820.5750.5870.6690.5900.690
Malmö0.6760.6740.6870.6840.6390.7030.6480.725
Manchester0.7570.7560.7740.7710.6880.7440.6960.765
Marseille0.4580.4540.4460.4460.5140.6180.5150.639
Miskolc0.6130.6010.6450.6220.6120.6640.6250.685
Munich0.6760.6570.6910.6730.6410.6970.6500.717
Naples0.1750.1760.1570.1570.3620.5370.3550.549
Oslo0.5270.5300.5210.5240.5510.6240.5560.640
Ostrava0.5620.5590.5940.5880.5850.6620.5970.692
Oulu0.6360.6350.6310.6260.6140.6930.6170.716
Oviedo0.4580.4560.4500.4440.5170.6280.5170.651
Palermo0.0710.0790.0380.0500.3050.5090.2890.513
Paris0.6080.5970.6120.5850.6030.6850.6060.710
Piatra Neamt0.5560.5490.5700.5500.5750.6490.5830.675
Podgorica0.4420.4400.4390.4320.5100.6180.5110.646
Praha0.5480.5440.5700.5650.5730.6410.5830.664
Rennes0.7570.7370.7680.7250.6880.7480.6930.772
Reykjavík0.3880.3980.3900.4140.4850.5970.4830.618
Riga0.3340.3380.3140.3320.4470.5770.4410.598
Rome0.0580.0700.0450.0620.3110.5110.2930.515
Rostock0.7190.6960.7390.7070.6670.7160.6770.736
Rotterdam0.6660.6440.6830.6580.6380.6970.6460.719
Skopje0.3410.3400.3220.3170.4520.5850.4460.612
Sofia0.3780.3850.3730.3900.4760.5830.4740.600
Stockholm0.6450.6360.6540.6360.6210.6780.6300.695
Strasbourg0.7000.6830.7060.6720.6550.7200.6590.742
Tallinn0.6240.6200.6700.6520.6220.6790.6390.705
Tirana0.4320.4320.4390.4350.5100.6280.5110.664
Turin0.2940.2960.2790.2800.4260.5700.4220.588
Tyneside conurbation0.6990.6850.7070.6840.6530.7060.6590.721
Valletta0.7420.7110.7670.7280.6820.7200.6930.736
Verona0.4260.4250.4270.4220.5040.6180.5040.645
Vilnius0.5510.5490.5550.5570.5720.6540.5750.673
Warszawa0.5560.5460.5680.5470.5760.6600.5820.687
Wien0.8290.8160.8410.8090.7270.7770.7330.798
Zagreb0.2340.2460.2140.2300.3990.5430.3860.557
Zurich0.9200.8950.9480.9110.7850.8140.7930.827
Table A8

Rank of IFTOPSIS coefficients for cities.

CityIFT_me2IFT_me3IFT_mh2IFT_mh3IFT_ae2IFT_ae3IFT_ah2IFT_ah3
Aalborg22222424
Amsterdam3233313131313131
Ankara3636374135253720
Antalya11109108797
Antwerpen2625272728262727
Athina7980798080797979
Barcelona6161616161566154
Belfast2927302829303033
Beograd7878787878787878
Berlin6868706970727072
Białystok1414141614131412
Bologna5251535551445338
Bordeaux2423283026202821
Braga5757565756505649
Bratislava5656575257595759
Bruxelles1615151715111510
Bucharest6666686868696869
Budapest4848474748554758
Burgas5959585958535847
Cardiff7788710811
Cluj-Napoca4646424344494246
Diyarbakir6564656565616562
Dortmund6262626262626263
Dublin1311131113121313
Essen4747495049574957
Gdańsk2021182019181816
Genève1917171417151715
Glasgow3435363637363642
Graz56665565
Groningen101279914714
Hamburg3032323330323234
Helsinki4138453945434543
Irakleio7777767676767676
Istanbul6060596059525951
København65536656
Košice4243384038403837
Kraków4041414240414140
Lefkosia4444403843334024
Leipzig2729252625342535
Liège2526222421242225
Lille3840434439384339
Lisboa7373727372707270
Ljubljana4342393742423941
London2322232122222322
Luxembourg43343232
Madrid5050545653465450
Málaga4545484846454848
Malmö2120241924232423
Manchester88107119109
Marseille6465646364676468
Miskolc3737343536473453
Munich2224212223272129
Naples8181818181818181
Oslo5858605860656067
Ostrava4949464547484645
Oulu3331353434293530
Oviedo6363636463636364
Palermo8282838383838383
Paris3939444641354432
Piatra Neamt5353505352585055
Podgorica6767676767686765
Praha5555514954605161
Rennes991113108118
Reykjavík7171717171717171
Riga7575757475757575
Rome8383828282828282
Rostock1516161516191619
Rotterdam2828262527282628
Skopje7474747574737473
Sofia7272737273747374
Stockholm3130333233393344
Strasbourg1719202318172017
Tallinn3534292932372936
Tirana6969666666646660
Turin7676777777777777
Tyneside conurbation1818191820211926
Valletta1213121212161218
Verona7070697069666966
Vilnius5452555155545556
Warszawa5154525450515252
Wien34454343
Zagreb8079807979808080
Zurich11111111
  1 in total

1.  Intuitionistic Fuzzy TOPSIS as a Method for Assessing Socioeconomic Phenomena on the Basis of Survey Data.

Authors:  Ewa Roszkowska; Marta Kusterka-Jefmańska; Bartłomiej Jefmański
Journal:  Entropy (Basel)       Date:  2021-05-01       Impact factor: 2.524
  1 in total
  1 in total

1.  On Some Extension of Intuitionistic Fuzzy Synthetic Measures for Two Reference Points and Entropy Weights.

Authors:  Ewa Roszkowska; Bartłomiej Jefmański; Marta Kusterka-Jefmańska
Journal:  Entropy (Basel)       Date:  2022-08-05       Impact factor: 2.738
  1 in total

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