MULTIMOORA for the EU member states updated with fuzzy number theory./Neraiskiuju skaiciu teorija papildytas multimoora metodas Europos sajungos valstybiu nariu issivystymo vertinimui.

Brauers, Willem K.M. ; Balezentis, Alvydas ; Balezentis, Tomas 等

1. Introduction

Multi-Objective Optimization (MOO) methods deal with problems of compromise selection of the best solutions from the set of available alternatives A = {[A.sub.1]; [A.sub.2];...; [A.sub.j];...; [A.sub.n]} according to objectives C = {[C.sub.1]; [C.sub.2];...; [C.sub.i];... [C.sub.m]}. Usually neither of the alternatives satisfies all the objectives therefore satisfactory decision is made instead of optimal one. Roy (1996) presented the following pattern of MOO problems: 1) [alpha] choosing problem--choosing the best alternative from A; 2) [beta] sorting problem--classifying alternatives of A into relatively homogenous groups; 3) [gamma] ranking problem--ranking alternatives of A from best to worst; 4) 8 describing problem--describing alternatives of A in terms of their peculiarities and features. Hence, during last few decades there were many Multi- Objective methods developed. Usually MOO techniques are classified into multiple objective decision making (MODM) and multiple attribute decision making (MADM). While MODM deals with continuous optimization problems and virtually infinite set of alternatives, MADM methods are aimed at discrete optimization and finite set of pre-defined alternatives. In this article term MOO will refer to MADM. The MOO methodology and methods were overviewed by Guitouni and Martel (1998) and Zavadskas et al. (2008b). Kaplinski (2009) presented an overview of advances in MOO science.

The MOO procedure usually consists of three basic stages: 1) identification of alternatives; 2) selection of objectives or indicators; 3) the choice of the problem with the appropriate MOO method (Roy 2005). Whereas the first stage is quite unequivocal the remaining two could raise some questions. Objectives can encompass non-subjective as well as subjective attributes (Liang, Wang 1991; Heragu 1997; Chou et al. 2008). Non-subjective indicators (attributes) are quantitative, e.g., investment costs. Subjective indicators are qualitative such as stakeholders' opinions. Therefore, decision making often relies on complex as well as on vague issues. Zadeh, the Founder of fuzzy logic (1965), proposed employing the fuzzy set theory as a modeling tool for complex systems that are hard to define exactly in crisp numbers. Fuzzy logic hence allows coping with vague, imprecise and ambiguous input and knowledge (Kahraman 2008; Kahraman and Kaya 2010). Linguistic variables expressed in fuzzy numbers were introduced by Zadeh (1975a, 1975b, 1975c) and applied in many studies (Liang 1999; Chen 2000; Chou et al. 2008; Torlak et al. 2011). Grey numbers were also applied in the decision making branch (Zavadskas et al. 2008a, 2008c; Lin et al. 2008; Zavadskas et al. 2010a; Peldschus et al. 2010) when creating MOO methods suitable for fuzzy inputs (1).

The question of extending the existing MOO methods to the fuzzy environment is of high importance. The Analytic Hierarchy Process (AHP) was initially proposed by Saaty (1980) and extended into fuzzy environment (van Laarhoven, Pedrycz 1983; Leung, Cao 2000). The simple additive weight (SAW) method (MacCrimmon 1968) was updated with fuzzy numbers theory and integrated with other decision making techniques (Chou et al. 2008). Technique for the Order Preference by Similarity to Ideal Solution (TOPSIS) was introduced by Hwang and Yoon (1981) and updated with fuzzy number theory (Chen 2000; Liu 2009a; Zavadskas and Antucheviciene 2006). The Method of Complex Proportional Assessment (COPRAS) (Zavadskas et al. 1994) was improved by applying fuzzy number technique (Zavadskas, Antucheviciene 2007). Zavadskas and Turskis introduced another method ARAS (2010), extended with grey and triangular fuzzy number (Turskis and Zavadskas 2010a, 2010b). Liang and Ding (2003) developed fuzzy MOO method based on a-cut concept. Peldschus and Zavadskas (2005) applied fuzzy game theory in multiple objective evaluation. Hence, updating MOO methods with fuzzy number theory is important.

Brauers and Zavadskas (2006) introduced Multi-Objective Optimization by Ratio Analysis (MOORA) on basis of previous research by Brauers (2004). In 2010 these authors developed this method further which became MULTIMOORA (MOORA plus the full multiplicative form). Numerous examples of application of these methods are present (Brauers et al. 2007, 2008, 2010; Brauers and Ginevicius 2009, 2010; Brauers and Zavadskas 2009a, 2009b; Balezentis and Balezentis 2010; Balezentis et al. 2010; Chakraborty 2010). However MULTIMOORA has not been updated with fuzzy numbers theory yet. This article deals with the issue of updating MULTIMOORA method with triangular fuzzy number theory and applying the fuzzy MULTIMOORA in international comparison of the European Union Member States.

The article is therefore organized in the following way. Section 2 deals with fuzzy set theory. The following Section 3 focuses on MULTIMOORA method. The proposed fuzzy MULTIMOORA method is described in Section 4. Section 5 undertakes a numerical example where the European Union (EU) Member States are compared on a basis of structural indicators and the new method. The data covers the period of 2000-2008. Section 6 makes a distinction between cardinal and ordinal scales in MULTIMOORA. Section 7 brings the application of the Multi-Objective Optimization on the European Union Member States based on MULTIMOORA.

2. The fuzzy set theory and triangular fuzzy numbers

Fuzzy sets and fuzzy logic are powerful mathematical tools for modeling uncertain systems. A fuzzy set is an extension of a crisp set. Crisp sets only allow full membership or non-membership, while fuzzy sets allow partial membership. The theoretical fundaments of fuzzy set theory are overviewed by Chen (2000).

In a universe of discourse X, a fuzzy subset [??] of X is defined with a membership function [[mu].sub.[??]](x) which maps each element x [member of] X to a real number in the interval [0; 1]. The function value of [[mu].sub.[??]](x) resembles the grade of membership of x in [??]. The higher the value of [[mu].sub.[??]](x), the higher the degree of membership of x in [??] (Keufmann and Gupta 1991). Noteworthy, in this study any variable with tilde will denote a fuzzy number.

A fuzzy number [??] is described as a subset of real number whose membership function [[mu].sub.[??]](x) is a continuous mapping from the real line R to a closed interval [0; 1], which has the following characteristics: 1) [[mu].sub.[??]](x) = 0, for all x [member of](-[infinity];a] [union] [c; [infinity]); 2) [[mu].sub.[??]](x) is strictly increasing in [a; b] and strictly decreasing in [d; c]; 3) [[mu].sub.[??]](x) = 1, for all x [member of] [b; d], where a, b, d, and c are real numbers, and -[infinity] < a [less than or equal to] b [less than or equal to] d [less than or equal to] c < [infinity]. When b = d a fuzzy number [??] is called a triangular fuzzy number (Fig. 1) represented by a triplet (a, b, c).

Triangular fuzzy numbers will therefore be used in this study to characterize the alternatives. The membership function [[mu].sub.[??]](x) is thus defined as:

[FIGURE 1 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

In addition, the parameters a, b, and c in (1) can be considered as indicating respectively the smallest possible value, the most promising value, and the largest possible value that describe a fuzzy event (Torlak et al. 2011).

Let [??] and [??] be two positive fuzzy numbers (Liang, Ding 2003). Hence, the main algebraic operations of any two positive fuzzy numbers [??] = (a, b, c) and B = (d, e, f) can be defined in the following way (Zavadskas, Antucheviciene 2007):

1. Addition [direct sum] :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

2. Subtraction [??]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

3. Multiplication [cross product]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

4. Division [??]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

The vertex method will be applied to measure the distance between two fuzzy numbers. Let [??] = (a, b, c) and B = (d, e, f) be two triangular fuzzy numbers. Then, the vertex method can be applied to measure the distance between these two fuzzy numbers:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Fuzzy numbers can be applied in two ways when forming the response matrix of alternatives on objectives. First, fuzzy numbers can represent the values of linguistic variables (Zadeh 1975a, 1975b, 1975c) when deciding either on the importance of criteria or performing qualitative evaluation of alternatives. For the latter purpose Chen (2000) describes the following fuzzy numbers identifying values of linguistic variables from scale Very poor to Very good: Very poor - (0, 0, 1); Poor - (0, 1, 3); Medium poor - (1, 3, 5); Fair - (3, 5, 7); Medium good - (5, 7, 9); Good - (7, 9, 10); Very good - (9, 10, 10). Second, the fuzzy numbers can represent monetary (quantitative) terms. It can be done either through direct input of certain fuzzy numbers into the response matrix or by aggregation of raw data (e. g. time series). For example, if there are costs "approximately equal to $200" estimated, the sum can be represented by triangular fuzzy number (190, 200, 210). Moreover, the fuzzy numbers can embody expected rate of growth. For example, if there is level of unemployment of 5 per cent with expected growth of 10 per cent, a triangular fuzzy number (5, 5.5, 6.1) can summarize these characteristics. As for time series data, a fuzzy number can represent the dynamics of certain indicator during past t periods:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where [a.sub.p] represents the value of certain indicator during period p (p = 1, 2,..., t).

The results of comparison of alternatives based on fuzzy numbers are also expressed in fuzzy numbers. The fuzzy numbers therefore need to be converted into crisp ones in order to identify the most promising alternative. There are four defuzzification methods commonly employed: (i) the centered method (or centre of area--COA); (ii) the Mean-of-maximum (MOM); (iii) the [alpha]-cut method; and (iv) the signed distance method (Zhao and Govind 1991; Yao and Wu 2000). In this study the COA method will be applied to obtain the Best Non-fuzzy Performance (BNP) value:

[BNP.sub.[??]] = (c - a) + (b - a)/3 + a, (8)

where a, b and c are respectively the lower, modal, and upper values of fuzzy number [??] = (a, b, c) (2) (Triantaphyllou 2000; Zavadskas and Antucheviciene 2006). Moreover, the robustness as well as precision of multi-criteria optimization can be improved by applying either intuitionist fuzzy numbers (Zhang, Liu 2010) or two-tuple linguistic representation (Liu 2009b).

3. The MULTIMOORA method

As already said earlier, Multi-Objective Optimization by Ratio Analysis (MOORA) method was introduced by Brauers and Zavadskas (2006) on the basis of previous research (Brauers 2004). Brauers, Zavadskas (2010) and Brauers, Ginevicius (2010) extended the method and in this way it became more robust as MULTIMOORA (MOORA plus the full multiplicative form). These methods have been applied in numerous studies (Brauers et al. 2007, 2010; Brauers, Ginevicius 2009; Brauers, Zavadskas 2009a, 2009b; Brauers, Ginevicius 2010; Balezentis et al. 2010) focused on regional studies, international comparisons and investment management.

MOORA method begins with matrix X where its elements [x.sub.ij] denote ith alternative of jth objective (i = 1, 2,..., m and j = 1, 2,..., n). MOORA method consists of two parts: the ratio system and the reference point approach. MacCrimmon (1968) defines two stages of weighting, namely normalization and voting on significance of objectives. The issue of weighting is discussed by Brauers, Zavadskas (2010); Zavadskas et al. (2010b), while the problem of normalization is analyzed by Brauers (2007) and Turskis et al. (2009). The MULTIMOORA method includes internal normalization and treats originally all the objectives equally important. In principle all stakeholders interested in the issue only could give more importance to an objective. Therefore they could either multiply the dimensionless number representing the response on an objective with a significance coefficient or they could decide beforehand to split an objective into different sub-objectives (Brauers, Ginevicius 2009).

The Ratio System of MOORA. Ratio system defines data normalization by comparing alternative of an objective to all values of the objective:

[x*.sub.ij] = [x.sub.ij]/[square root of [m.summation over (i=1)] [x.sup.2.sub.ij]

where [x*.sub.ij] denotes ith alternative of jth objective (in this case jth structural indicator of ith state). Usually these numbers belong to the interval [-1; 1]. These indicators are added (if desirable value of indicator is maxima) or subtracted (if desirable value is minima) and summary index of state is derived in this way:

[y.sup.*.sub.i] = [g.summation over (j=1)] [x.sup.*.sub.ij] - [g.summation over (j=g+1)] [x.sup.*.sub.ij], (10)

where g = 1,..., n denotes number of objectives to be maximized. Then every ratio is given the rank: the higher the index, the higher the rank.

The Reference Point of MOORA. Reference point approach is based on the Ratio System. The Maximal Objective Reference Point (vector) is found according to ratios found in formula (9). The jth coordinate of the reference point can be described as [r.sub.j] = max [x.sup.*.sub.ij] in case of maximization. Every coordinate of this vector represents maxima or minima of certain objective (indicator). Then every element of normalized responses matrix is recalculated and final rank is given according to deviation from the reference point and the Min-Max Metric of Tchebycheff:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

The Full Multiplicative Form and MULTIMOORA. Brauers and Zavadskas (2010) proposed MOORA to be updated by the Full Multiplicative Form method embodying maximization as well as minimization of purely multiplicative utility function. Overall utility of the ith alternative can be expressed as dimensionless number:

[U.sup.'.sub.i] = [A.sub.i]/[B.sub.i]. (12)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the product of objectives of the ith alternative to be maximized with g = 1,..., n being the number of objectives (structural indicators) to be maximized and where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the product of objectives of the ith alternative to be minimized with n - g being the number of objectives (indicators) to be minimized. Thus MULTIMOORA summarizes MOORA (i.e. Ratio System and Reference point) and the Full Multiplicative Form. Ameliorated Nominal Group and Delphi techniques can also be used to reduce remaining subjectivity (Brauers and Zavadskas 2010).

4. The fuzzy MULTIMOORA method

The fuzzy MULTIMOORA begins with response matrix [??] with [[??].sub.ij] = ([x.sub.ij1], [x.sub.ij2], [x.sub.ij3]) being the ith alternative of the jth objective (i = 1, 2,..., m and j = 1, 2,..., n).

4.1. The fuzzy Ratio System

The Ratio System defines normalization of the fuzzy numbers [[??].sub.ij] resulting in matrix of dimensionless numbers. The normalization is performed by comparing appropriate values of fuzzy numbers:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

The normalization is followed by computation of summarizing ratios [[??].sub.ij] for each ith alternative. The normalized ratios are added or subtracted according to formulas (2) or (3) respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

where g = 1, 2,..., n stands for number of indicators to be maximized. Then each ratio [[??].sub.ij] = ([y.sup.*.sub.i1], [y.sup.*.sub.i2], [y.sup.*.sub.i3]) is de-fuzzified by applying Eq. 8:

[BNP.sub.i] = ([y.sup.*.sub.i3] - [y.sup.*.sub.i1]) + ([y.sup.*.sub.i2] - [y.sup.*.sub.i1])/3 + [y.sup.*.sub.i1], (15)

where [BNP.sub.i] denotes the best non-fuzzy performance value of the ith alternative. Consequently, the alternatives with higher BNP values are attributed with higher ranks.

4.2. The fuzzy Reference Point

The fuzzy Reference Point approach is based on the fuzzy Ratio System. The Maximal Objective Reference Point (vector) [??] is found according to ratios found in formula (13). The jth coordinate of the reference point resembles the fuzzy maxima or minima of jth criterion [[??].sup.+.sub.j], where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Then every element of normalized responses matrix is recalculated and final rank is given according to deviation from the reference point (Eq. 6) and the Min-Max Metric of Tchebycheff:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

4.3. The fuzzy Full Multiplicative Form

Overall utility of the ith alternative can be expressed as dimensionless number by employing Eq. 5:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the product of objectives of the ith alternative to be maximized with g = 1,..., n being the number of objectives (structural indicators) to be maximized and where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the product of objectives of the ith alternative to be minimized with n - g being the number of objectives (indicators) to be minimized. Formula (4) is applied when computing these variables. Since overall utility [[??].sup.'.sub.i] is fuzzy number, Eq. 8 has to be used to rank the alternatives. The higher the BNP, the higher the rank of certain alternative.

Thus fuzzy MULTIMOORA summarizes fuzzy MOORA (i. e. fuzzy Ratio System and fuzzy Reference Point) and the fuzzy Full Multiplicative Form.

5. A comparison of the European Union Member States according to fuzzy MULTIMOORA

The fuzzy MULTIMOORA was applied when comparing EU Member States. Empirical analysis of EU Member States' efforts in seeking Lisbon goals began with definition of system of structural indicators (Table 1). The system consists of 12 indicators from the shortlist of structural indicators. Directions of optimization were also attributed to each of the indicator. For example, rising level of unemployment has negative economic and social consequences (Martinkus et al. 2009; Korpysa 2010) therefore it should be minimized.

The indicators are measured in different dimensions. The volume index of GDP per capita in Purchasing Power Standards (PPS) is expressed in relation to the European Union (EU-27) average set to equal 100. If the index of a country is higher than 100, this country's level of GDP per head is higher than the EU average and vice versa. Labor productivity per person employed is measured as GDP in PPS per person employed relative to EU-27 average (EU-27 = 100). The employment rate is calculated by dividing the number of persons aged 15 to 64 in employment by the total population of the same age group. The employment rate of older workers is calculated by dividing the number of persons aged 55 to 64 in employment by the total population of the same age group. The indicator Youth education attainment level is defined as the percentage of young people aged 20-24 years having attained at least upper secondary education attainment level. Gross domestic expenditure on R&D is expressed as a percentage of GDP. Business investment is defined as total gross fixed capital formation expressed as a percentage of GDP, for the private sector. Comparative price levels are the ratio between Purchasing power parities and market exchange rate for each country shown in relation to the EU average (EU-27=100). The share of persons with an equivalised disposable income below the risk-of-poverty threshold, which is set at 60% of the national median equivalised disposable income (after social transfers) is resembled by At-risk-of-poverty rate indicator. Long-term unemployment rate is number of persons that have been unemployed for more than 12 months expressed as the percentage of total labor force. Greenhouse gas emissions indicator presents annual total emissions (CO2 equivalents) in relation to "Kyoto base year". In general the base year is 1990 for the non-fluorinated gases and 1995 for the fluorinated gases. Gross inland consumption of energy divided by GDP (kilogram of oil equivalent per 1000 Euro) results in the Energy intensity of the economy indicator. However, the application of MULTIMOORA method enables to summarize all these indicators expressed in different dimensions.

Data covering these indicators and the period 2000-2008 were obtained from EUROSTAT Structural Indicators database and are available from the authors upon request. Due to limited data availability three time points were chosen for analysis, namely years 2000, 2004 and 2008. The data therefore cover 27 Member States, 3 years and 12 structural indicators, 972 observations in total.

The initial data (Annex A, Table Al) were aggregated by employing Eq. 8. Minimal values, geometric means and maximum values (denoted as min, average and max respectively in Table A2, Annex A) were obtained for each indicator thus creating the fuzzy response matrix [??] (Table A2) containing 324 fuzzy numbers. The data were internally normalized by applying Eq. 13: each response [x.sub.ijk], k = 1,2,3, was divided by respective ratio presented in the last row of Table A2 (Annex A). Hence the fuzzy normalized response matrix [[??].sup.*] was formed (Table A3, Annex A).

Aggregation of normalized fuzzy ratios was performed according to Eq. 14. In this way the summarizing fuzzy ratios [[??].sup.*.sub.i] = ([y.sup.*.sub.i1], [y.sup.*.sub.i2], [y.sup.*.sub.i3]) were obtained and de-fuzzified by applying Eq. 15:

[BNP.sub.i] = ([y.sup.*.sub.i3] - [y.sup.*.sub.i1] + ([y.sup.*.sub.i2] - [y.sup.*.sub.i1]/3 + [y.sup.*.sub.i1]. (19)

BNP expressed in crisp numbers enabled to attribute each EU Member State with appropriate rank (Table A4, Annex A).

The fuzzy Reference Point relies on ratios retrieved by fuzzy Ratio System. Table A5a (Annex A) presents the coordinates of fuzzy vector [??], which were obtained by applying Eq. 16. Afterwards, the countries were ranked according Eq. 17 (Table A5b, Annex A). Since the distances were expressed in crisp numbers, no de-fuzziness was necessary.

Finally, the fuzzy Full Multiplicative Form was applied according to Eq. 18. Computation of fuzzy products [[??].sub.i] and [[??].sub.i] was a prerequisite for further calculations (Eq. 4, 5). Since [[??].sup.'.sub.i] is also a fuzzy number, Eq. 8 was applied to transform it into a crisp one (Annex B, Table B1). MULTIMOORA should summarize ranks from the Ratio System, Reference Point, and the Full Multiplicative Form.

6. Cardinal and ordinal scales in MULTIMOORA

Does there not exist a problem when MULTIMOORA has to totalize ranks from the Ratio System, Reference Point and the Full Multiplicative Form? Indeed adding up of ranks, ranks mean an ordinal scale (1st, 2nd, 3rd etc.) signifies a return to a cardinal operation (1 + 2 + 3 +...). Is this allowed?

The answer is "no" following the Noble prize Winner Arrow:

6.1. The impossibility theorem of arrow

"Obviously, a cardinal utility implies an ordinal preference but not vice versa" (Arrow 1974).

6.2. The rank correlation method

The method of correlation of ranks consists of totalizing ranks. Rank correlation was introduced first by psychologists such as Spearman (1904, 1906 and 1910) and later taken over by the statistician Kendall in 1948. He argues (Kendall 1948): "we shall often operate with these numbers as if they were the cardinals of ordinary arithmetic, adding them, subtracting them and even multiplying them", but he never gives a proof of this statement. In his later work this statement is dropped (Kendall and Gibbons 1990).

In ordinal ranking 3 is farther away from 1 than 2 from 1, but Kendal (1948) goes too far (Table 2).

For Kendal B is far away from A as it has 7 ranks before and A only 4, whereas it is not true cardinally.

In addition a supplemental notion, the statistical term of Correlation, is introduced. Suppose the statistical universe is just represented by two experts, for us it could be two methods. If they both rank in a same order different items to reach a certain goal, it is said that the correlation is perfect. However, perfect correlation is a rather exceptional situation. The problem is then posited: how in other situations correlation is measured. Therefore, the following Spearman's coefficient is used (Kendall 1948: 8):

[rho] = 1 - 6[summation] [D.sup.2]/N([N.sup.2] - 1), (20)

where d stands for the difference between paired ranks, and n for the number of items ranked.

According to this formula, perfect correlation yields the coefficient of one. An acceptable correlation reaches the coefficient of one as much as possible. No correlation at all yields a coefficient of zero. If the series are exactly in reverse order, there will be a negative correlation of minus one, as shown in the following example (Table 3).

This table shows that the sum of ranks in the case of an ordinal scale has no sense. Correlation leads to: [rho] = 1-6x112 / (7(49 - 1)) = -1. However, as addition of ranks is not allowed also a subtraction, the difference D, is not permitted.

Most people will better understand the ordinal problem by the way of a qualitative scale, e. g.:

1st very good;

2nd moderate;

3rd very bad.

But equally one could say:

1st very good;

2nd good;

3rd more or less good;

4th moderate;

5th more or less low;

6th low;

7th very low.

How is the first 2nd comparable with the second 2nd?, etc.

6.3. Arbitrary methods to go from an ordinal scale to a cardinal scale

1. Arithmetical Progression: 1, 2, 3, 4, 5,...

The ordinal scale 5 gets 1 cardinal point with all variations possible e.g. an additional point 1, etc.

The ordinal scale 4 gets 2 cardinal points etc.

The best one in the ordinal scale gets the most cardinal points in an arithmetical progression.

2. A Geometric Progression: 1, 2, 4, 8, 16,.

3. The Fundamental Scale of Saaty (1987): 1, 3, 5, 7, 9.

4. The Normal Scale of Lootsma (1987):

[e.sup.o] = 1; [e.sup.1] = 2.7; [e.sup.2] = 7.4; [e.sup.3] = 20.1...

5. The Stretched Scale of Lootsma (1987):

[e.sup.o] = 1;

[e.sup.2] = 7.4;

[e.sup.4] = 54.6;

[e.sup.6] = 403.4...

6. The Point of View of the Psychologists (Miller 1956):

Ordinal Scales: 1, 2, 3, 4, 5, 6, 7.

After 7 an individual would no more know the cardinal significance compared to the previous 7 ones.

In fact infinite variations are possible. All stress an acceleration or a dis-acceleration process but are not aware of a possible trend break. The full multiplicative method with its huge numbers illustrates the best this trend break as shown in next Table 4.

With the usual Arithmetical Progression: 1, 2, 3, 4, 5,... the distance from the rank 4 to 5 would be the same as from 3 to 4 which is certainly not the case here. In addition all the other progressions fail to discover a trend break too.

Summarizing all these statements the following axioms are proposed.

6.4. Axioms on Ordinal and Cardinal Scales

1. A deduction of an Ordinal Scale, a ranking, from cardinal data is always possible.

2. An Ordinal Scale can never produce a series of cardinal numbers.

3. An Ordinal Scale of a certain kind, a ranking, can be translated in an ordinal scale of another kind.

In application of axiom 3 we shall translate the rankings of three methods of MULTIMOORA into an other ordinal scale based on Dominance, being Dominated, Transitivity and Equability.

6.5. Dominance, being Dominated, Transitiveness and Equability

The three methods of MULTIMOORA are assumed to have the same importance. Stakeholders, or their representatives like experts, may give a different importance in an ordinal ranking but this is not the case with the three methods of MULTIMOORA. These three methods represent all existing methods in multi-objective optimization with dimensionless measures and consequently all the three have the same important significance.

Dominance (3)

Absolute Dominance means that an alternative, solution or project dominates in ranking all other alternatives, solutions or projects which are all being dominated. This absolute dominance shows as rankings for MULTIMOORA: (1-1-1).

General Dominance in two of the three methods is of the form with a < b < c <d:

(d-a-a) is generally dominating (c-b-b);

(a-d-a) is generally dominating (b-c-b);

(a-a-d) is generally dominating (b-b-c);

and further transitiveness plays fully.

Transitiveness. If a dominates b and b dominates c than also a will dominate c. Overall Dominance of one alternative on another: (a-a-a) overall dominating (b-b-b), see Table 5.

Equability

Absolute Equability has the form: (e-e-e) for 2 alternatives.

Partial Equability of 2 on 3 exists e. g. (5-e-7) and (6-e-3).

A distinction can be made if a classification shows equability but one of the two alternatives belongs to a higher classified group.

Circular Reasoning

Despite all distinctions in classification some contradictions remain possible in a kind of Circular Reasoning. In such a case the same ranking is given.

7. Application on the Multi-Objective Optimization of the European Union Member

States based on MULTIMOORA

All Member States were assigned either of three roles in the European world-system. Best performing states with ranks from 1 to 9 were considered as Core states (Group 1), those possessing ranks 10-18--as Semi-Peripheral states (Group 2), and those with ranks 19-27--as Peripheral States (Group 3). It should be noted that all European states are unequivocally semi-peripheral at least in the total world-system, thus the given classification is only valid in the context of the European world-system (for the global world-system see for instance: Clark 2010).

Beside the general characteristics given above additional remarks have to be made for application on the European situation:

--We have to repeat again that with ranking by dominance the application remains in the ordinal sphere.

--We have to repeat again that the three methods have the same importance.

--Due to limited data availability and to limit the number of calculations only the years 2000, 2004 and 2008 were selected. In that way the response matrix was already composed of 972 elements.

--Also the choice of the years 2000, 2004 and 2008 has an historical meaning. In 2000 the European Union was only composed of 15 countries, the so-called EU-15: the original countries (1957) BENELUX (Belgium, Netherlands, Luxemburg), France, Germany and Italy; UK, Ireland and Denmark (1973); Greece (1981); Spain and Portugal (1986).

On May 1, 2004 the EU extended with 10 members: Poland, Lithuania, Latvia, Estonia, Slovenia, Slovakia, Czech Republic, Hungary, Cyprus and Malta. Consequently these countries were not member in 2000, a half time in 2004 and full time in 2008. Nevertheless their data are also assembled for 2000 and 2004.

On January 1, 2007 Romania and Bulgaria joined the Union meaning that they were not present in 2000 and 2004. Nevertheless their data are also used for 2000 and 2004.

--No Equability in ranking was found between the EU members.

--No Absolute Dominance was present in the three methods.

--General Dominance: Sweden with (1-5-7) dominates Luxemburg (2-2-19) and further all the others by transitiveness.

Table 6 and Annex D show the final results for the European Member States on basis of Dominance.

The application of a theory of Dominance to solve the ordinal problem was successful. If the transition from cardinal to ordinal is possible but from ordinal to cardinal not then the solution has to be found in the transition from one ordinal system to another one. Let us hope that in this way the old discussion between cardinal and ordinal is solved once for all.

Given the recession of 2009 a trend break occurred which was certainly fatal for Ireland, Greece, Portugal and even perhaps for the UK. Standard&Poor's gives a credit rating of BB+ to Greece, which means classifying its government bonds as "junk" paper. Before March 2009 Ireland had the highest rating of AAA but since then it went down over AA+, AA, AA-, A+ to A. Portugal has even A -. Of course this is only a single indicator. Bur the rating offices take into account many criteria (4). Probably Ireland, Portugal and Greece will have to substitute Group 2 (Semi Periphery) by Group 3 (Periphery). One can even wonder if UK can stay in Group 1. Consequently similar research on the year 2009 would be very useful.

8. Conclusion

Fuzzy logic handles vague problems in various areas. Fuzzy numbers can represent either quantitative or qualitative variables. The quantitative fuzzy variables can embody crisp numbers, aggregates of historical data (i.e. time series) or forecasts. The qualitative fuzzy variables may be applied when dealing with ordinal scales. The MULTIMOORA method was therefore updated with fuzzy number theory. Vertex method was used when measuring the distances between fuzzy numbers. Centre of area method was applied for defuzzification.

The MULTIMOORA method consists of three parts, namely Ratio System, Reference Point and Full Multiplicative Form. Accordingly, each of them was modified and thus updated with triangular fuzzy number theory. The fuzzy Ratio System defines internal normalization, aggregation of criteria into single ratios and defuzzification. The fuzzy Reference Point approach relies on definition of the Maximal Objective Reference Point as well as measurement of distances between certain coordinates of the Reference Point and every alternative according to vertex method. The fuzzy Full Multiplicative Form embodies maximization of a purely multiplicative utility function and defuzzification. The fuzzy MULTIMOORA summarizes these three approaches under the form of three sets of ranking, which means: of an ordinal order. At that moment the problem is set: what to do with these three sets of rankings. With small responses matrices no problems did arrive. The solution was mostly easy to see. For large matrices it is much more complicated.

At that occasion three Axioms on Ordinal and Cardinal Scales are proposed:

1. A deduction of an Ordinal Scale, a ranking, from cardinal data is always possible.

2. An Ordinal Scale can never produce a series of cardinal numbers.

3. An Ordinal Scale of a certain kind, a ranking, can be translated in an ordinal scale of another kind.

In application of axiom 3 the rankings of the three methods of MULTIMOORA were translated into an other ordinal scale based on Dominance, being Dominated, Transitivity and Equability.

The three methods of MULTIMOORA are assumed to have the same importance. These three methods represent all existing methods with dimensionless measures in multi-objective optimization and all the three have an important significance.

Fuzzy MULTIMOORA ranked the EU Member States in three groups based on the cited domination principles and according to their performance in reaching the goals of the Lisbon Strategy 2000-2008. As table 6 suggests, the best performing countries (Group 1) are Sweden, Luxemburg Finland, Austria, the Netherlands Denmark Belgium, UK and Germany. Group 2 consists of, France, Ireland, Spain, Italy, Slovenia, Portugal, Czech Republic, Greece, and Estonia. Group 3 encompasses the less performing states, namely Cyprus, Hungary, Poland, Lithuania, Malta, Latvia, Romania, Slovakia and Bulgaria. The three groups are called successively: Core, Semi-Periphery and Periphery in comparison with what is done on world level.

Given the recession of 2009 a trend break occurred which was certainly fatal for Ireland, Greece, Portugal and even perhaps for the UK. Consequently new research on 2010 would be very useful. Nevertheless no link has to be made with the period from before 2010. The changes are too profound.

doi: 10.3846/20294913.2011.580566

Annex A. The fuzzy MOORA method

Table A1. The initial data for the research of EU Member
States performance in seeking Lisbon goals (2000-2008)

           1. GDP per capita in PPS         2. Labor productivity
                                            per person employed

         2000       2004       2008       2000       2004       2008

BE        126        121        115      136.6      131.7      125.4
BG         28         34         41       30.4       33.7       37.2
CZ         68         75         80       61.8         68       71.9
DK        132        126        120      110.5      108.6        101
DE        118        116        116        108      108.1      106.9
EE         45         57         67       46.9       57.4       63.8
IE        131        142        135      127.4      135.2      130.1
EL         84         94         94       93.6      101.1      102.1
ES         97        101        103      103.7        102      103.6
FR        115        110        108        125      120.6      121.2
IT        117        107        102        126      112.1      109.4
CY         89         90         96         85       82.8       87.2
LV         37         46         57       40.2       45.7         52
LT         39         50         62       42.7       53.3         62
LU        244        253        276      175.9      169.6      175.7
HU         55         63         64       57.7       67.4       71.2
MT         84         77         76       96.7         90       86.9
NL        134        129        134      114.4      112.2      114.4
AT        131        127        124      120.6      117.5        114
PL         48         51         56       55.2       61.5         62
PT         81         77         78       71.5       69.3       73.5
RO         26         34         42       23.6       34.4       50.2
SI         80         86         91       76.2         82       84.3
SK         50         57         72         58       65.4       79.2
FI        117        116        117      114.8      112.9      111.8
SE        128        126        122      114.3      114.9      112.3
UK        119        124        116      110.7      113.8      109.7

     3. Employment rate by            4. Employment rate of
     gender; Total                    older workers by gender;
                                      Total

         2000       2004       2008       2000       2004       2008

BE       60.5       60.3       62.4       26.3         30       34.5
BG       50.4       54.2         64       20.8       32.5         46
CZ         65       64.2       66.6       36.3       42.7       47.6
DK       76.3       75.7       78.1       55.7       60.3         57
DE       65.6         65       70.7       37.6       41.8       53.8
EE       60.4         63       69.8       46.3       52.4       62.4
IE       65.2       66.3       67.6       45.3       49.5       53.7
EL       56.5       59.4       61.9         39       39.4       42.8
ES       56.3       61.1       64.3         37       41.3       45.6
FR       62.1       63.8       64.9       29.9       37.8       38.2
IT       53.7       57.6       58.7       27.7       30.5       34.4
CY       65.7       68.9       70.9       49.4       49.9       54.8
LV       57.5       62.3       68.6         36       47.9       59.4
LT       59.1       61.2       64.3       40.4       47.1       53.1
LU       62.7       62.5       63.4       26.7       30.4       34.1
HU       56.3       56.8       56.7       22.2       31.1       31.4
MT       54.2         54       55.3       28.5       31.5       29.2
NL       72.9       73.1       77.2       38.2       45.2         53
AT       68.5       67.8       72.1       28.8       28.8         41
PL         55       51.7       59.2       28.4       26.2       31.6
PT       68.4       67.8       68.2       50.7       50.3       50.8
RO         63       57.7         59       49.5       36.9       43.1
SI       62.8       65.3       68.6       22.7         29       32.8
SK       56.8         57       62.3       21.3       26.8       39.2
FI       67.2       67.6       71.1       41.6       50.9       56.5
SE         73       72.1       74.3       64.9       69.1       70.1
UK       71.2       71.7       71.5       50.7       56.2         58

     5. Youth education               6. Gross domestic expenditure
     attainment level                  on R&D (GERD;

         2000       2004       2008       2000       2004       2008

BE       81.7       81.8       82.2       1.97       1.86       1.92
BG       75.2       76.1       83.7       0.52        0.5       0.49
CZ       91.2       91.4       91.6       1.21       1.25       1.47
DK         72       76.2         71       2.24       2.48       2.72
DE       74.7       72.8       74.1       2.45       2.49       2.63
EE         79       80.3       82.2        0.6       0.85       1.29
IE       82.6       85.3       87.7       1.12       1.23       1.43
EL       79.2         83       82.1       0.59       0.55       0.58
ES         66       61.2         60       0.91       1.06       1.35
FR       81.6       81.8       83.4       2.15       2.15       2.02
IT       69.4       73.4       76.5       1.05        1.1       1.18
CY         79       77.6       85.1       0.24       0.37       0.46
LV       76.5       79.5         80       0.44       0.42       0.61
LT       78.9         85       89.1       0.59       0.75        0.8
LU       77.5       72.5       72.8       1.65       1.63       1.62
HU       83.5       83.5       83.6       0.79       0.87          1
MT       40.9         51         53       0.26       0.53       0.54
NL       71.9         75       76.2       1.82       1.81       1.63
AT       85.1       85.8       84.5       1.94       2.26       2.67
PL       88.8       90.9       91.3       0.64       0.56       0.61
PT       43.2       49.6       54.3       0.76       0.77       1.51
RO       76.1       75.3       78.3       0.37       0.39       0.58
SI         88       90.5       90.2       1.39        1.4       1.66
SK       94.8       91.7       92.3       0.65       0.51       0.47
FI       87.7       84.5       86.2       3.35       3.45       3.73
SE       85.2         86       85.6       3.61       3.62       3.75
UK       76.7         77       78.2       1.81       1.68       1.88

     7. Business investment           8. Comparative price levels

         2000       2004       2008       2000       2004       2008

BE       19.1       18.2         21        102      106.7      111.1
BG       12.1       17.6       27.7       38.7         42       50.2
CZ       24.4         21         19       48.1       55.4       72.8
DK       18.5       17.4         19      130.2      139.5      141.2
DE       19.7       16.1       17.5      106.5      104.7      103.8
EE         22       27.1         24       57.2         63         78
IE       19.6       20.9       16.6      114.8      125.9      127.6
EL       17.9       18.7       16.6       84.8       87.6         94
ES       22.7       24.7         25         85         91       95.4
FR       16.4       16.2       18.5      105.8      109.9      110.8
IT         18       18.1       18.5       97.5      104.9      105.6
CY       11.9       12.1       20.4         88       91.2       90.5
LV       22.9       24.4       24.5       58.8       56.1       72.6
LT       16.4       18.8       20.2       52.6       53.5       64.7
LU         17       17.3       16.1      101.5        103      119.1
HU       20.2         19         18       49.2         62       68.1
MT       13.3       10.2         11       73.2       73.2       78.8
NL       18.8       15.6       16.9        100      106.1        104
AT       22.5       20.8         21      101.8      103.3      105.1
PL       21.4       14.7       17.5       57.9       53.2       69.1
PT       24.1       20.3         20         83       87.4         87
RO       15.4       18.7       26.4       42.5       43.3       60.9
SI       22.4       21.5       24.6       72.8       75.5       82.3
SK       23.8       22.2         23       44.4       54.9       70.2
FI       17.6       16.5         19      120.8      123.8      124.3
SE       15.2       14.1       16.8      127.6      121.4      114.5
UK       15.9       14.9       14.4      119.9      108.5      100.1

     9. At-risk-of-poverty rate       10. Long-term unemployment rate
     after social transfers

         2000       2004       2008       2000       2004       2008

BE         13       14.3       14.7        3.7        4.1        3.3
BG         14         15       21.4        9.4        7.2        2.9
CZ          8       10.4          9        4.2        4.2        2.2
DK         10       10.9       11.8        0.9        1.2        0.5
DE         10       12.2       15.2        3.8        5.5        3.8
EE         18       20.2       19.5        6.3          5        1.7
IE         20       20.7       15.4        1.6        1.6        1.7
EL         20       19.9       20.1        6.2        5.6        3.6
ES         18       19.9       19.6        4.6        3.4          2
FR         16       13.5       13.4        3.5        3.8        2.9
IT         18       19.1       18.7        6.3          4        3.1
CY         15         15       16.2        1.2        1.2        0.5
LV         16       19.2       25.6        7.9        4.6        1.9
LT         17       20.7         20          8        5.8        1.2
LU         12       12.7       13.4        0.5          1        1.6
HU         11       13.5       12.4        3.1        2.7        3.6
MT         15       13.7       14.6        4.5        3.4        2.5
NL         11       10.7       10.5        0.8        1.6          1
AT         12       12.8       12.4          1        1.4        0.9
PL         16       20.5       16.9        7.4       10.3        2.4
PT         21       20.4       18.5        1.7          3        3.7
RO         17         18       23.4        3.8        4.8        2.4
SI         11       12.2       12.3        4.1        3.2        1.9
SK       13.3       13.3       10.9       10.3       11.8        6.6
FI         11         11       13.6        2.8        2.1        1.2
SE          8       11.3       12.1        1.4        1.5        0.8
UK         19         18       18.8        1.4          1        1.4

     11. Greenhouse gas emissions     12. Energy intensity
                                      of the economy

         2000       2004       2008       2000       2004       2008

BE      100.9      101.3       92.9      243.7      229.3      199.8
BG         59       60.6       62.6     1362.4     1139.3      944.2
CZ       75.6       74.8       72.5      659.1      660.2      525.3
DK       99.1       98.7       92.6      112.5      111.9      103.1
DE       83.2       81.2       77.8      166.0      166.1      151.1
EE       44.5       49.3       49.6      812.7      687.5      570.5
IE      123.6      122.8        123      137.0      123.0      106.5
EL      120.9      125.7      122.8      204.6      186.8      170.0
ES      133.6      147.5      142.3      196.2      198.1      176.4
FR       98.9       98.1       93.6      179.1      179.4      166.7
IT      106.3        111      104.7      146.6      150.5      142.6
CY      172.8      176.4      193.9      237.1      215.5      213.4
LV       38.1       41.1       44.4      441.0      387.0      308.7
LT         39       44.2       48.9      571.2      547.4      417.5
LU       75.5      100.7       95.2      165.3      185.6      154.6
HU       79.2       81.2       75.1      487.5      435.3      401.4
MT      126.9      140.6      144.2      191.3      217.4      194.9
NL      101.2      102.9       97.6      184.8      191.6      171.6
AT      102.7      116.3      110.8      140.3      151.7      138.1
PL       86.1       85.3       87.3      488.7      442.1      383.5
PT      137.1      142.8      132.2      197.5      201.3      181.5
RO       56.3       64.2       60.3      913.4      768.3      614.6
SI      101.9      107.7      115.2      299.2      289.6      257.5
SK       66.6       68.7       66.1      796.4      729.1      519.7
FI       98.2        114       99.7      246.3      257.4      217.8
SE       95.1       97.2       88.3      177.4      177.5      152.1
UK       87.2       85.4       81.4      144.5      131.0      113.7

Table A2. Fuzzy response matrix [??]

                                   1

j                  min      average          max

BE               115.00       120.58       126.00
BG                28.00        33.92        41.00
CZ                68.00        74.17        80.00
DK               120.00       125.90       132.00
DE               116.00       116.66       118.00
EE                45.00        55.60        67.00
IE               131.00       135.92       142.00
EL                84.00        90.54        94.00
ES                97.00       100.30       103.00
FR               108.00       110.96       115.00
IT               102.00       108.49       117.00
CY                89.00        91.62        96.00
LV                37.00        45.95        57.00
LT                39.00        49.45        62.00
LU               244.00       257.32       276.00
HU                55.00        60.53        64.00
MT                76.00        78.92        84.00
NL               129.00       132.31       134.00
AT               124.00       127.30       131.00
PL                48.00        51.56        56.00
PT                77.00        78.65        81.00
RO                26.00        33.36        42.00
SI                80.00        85.55        91.00
SK                50.00        58.98        72.00
FI               116.00       116.67       117.00
SE               122.00       125.31       128.00
UK               116.00       119.62       124.00
Sum             274,978      301,706      335,446
                    524          549          579

                                   2

j                  min      average          max

BE               125.40       131.15       136.60
BG                30.40        33.65        37.20
CZ                61.80        67.10        71.90
DK               101.00       106.62       110.50
DE               106.90       107.67       108.10
EE                46.90        55.59        63.80
IE               127.40       130.86       135.20
EL                93.60        98.86       102.10
ES               102.00       103.10       103.70
FR               120.60       122.25       125.00
IT               109.40       115.61       126.00
CY                82.80        84.98        87.20
LV                40.20        45.71        52.00
LT                42.70        52.06        62.00
LU               169.60       173.71       175.90
HU                57.70        65.18        71.20
MT                86.90        91.11        96.70
NL               112.20       113.66       114.40
AT               114.00       117.34       120.60
PL                55.20        59.48        62.00
PT                69.30        71.41        73.50
RO                23.60        34.41        50.20
SI                76.20        80.76        84.30
SK                58.00        66.97        79.20
FI               111.80       113.16       114.80
SE               112.30       113.83       114.90
UK               109.70       111.39       113.80
Sum             236,452      255,155      275,759
                    486          505          525

                                   3

j                  min      average          max

BE                60.30        61.06        62.40
BG                50.40        55.92        64.00
CZ                64.20        65.26        66.60
DK                75.70        76.69        78.10
DE                65.00        67.05        70.70
EE                60.40        64.28        69.80
IE                65.20        66.36        67.60
EL                56.50        59.23        61.90
ES                56.30        60.48        64.30
FR                62.10        63.59        64.90
IT                53.70        56.63        58.70
CY                65.70        68.47        70.90
LV                57.50        62.64        68.60
LT                59.10        61.50        64.30
LU                62.50        62.87        63.40
HU                56.30        56.60        56.80
MT                54.00        54.50        55.30
NL                72.90        74.37        77.20
AT                67.80        69.44        72.10
PL                51.70        55.22        59.20
PT                67.80        68.13        68.40
RO                57.70        59.86        63.00
SI                62.80        65.52        68.60
SK                56.80        58.65        62.30
FI                67.20        68.61        71.10
SE                72.10        73.13        74.30
UK                71.20        71.47        71.70
Sum             104,826      111,483      120,374
                    324          334          347

                                   4

j                  min      average          max

BE                26.30        30.08        34.50
BG                20.80        31.45        46.00
CZ                36.30        41.94        47.60
DK                55.70        57.63        60.30
DE                37.60        43.89        53.80
EE                46.30        53.30        62.40
IE                45.30        49.38        53.70
EL                39.00        40.36        42.80
ES                37.00        41.15        45.60
FR                29.90        35.08        38.20
IT                27.70        30.75        34.40
CY                49.40        51.31        54.80
LV                36.00        46.79        59.40
LT                40.40        46.58        53.10
LU                26.70        30.25        34.10
HU                22.20        27.88        31.40
MT                28.50        29.71        31.50
NL                38.20        45.06        53.00
AT                28.80        32.40        41.00
PL                26.20        28.65        31.60
PT                50.30        50.60        50.80
RO                36.90        42.86        49.50
SI                22.70        27.85        32.80
SK                21.30        28.18        39.20
FI                41.60        49.27        56.50
SE                64.90        68.00        70.10
UK                50.70        54.88        58.00
Sum              39,441       49,176       62,528
                    199          222          250

                                   5

j                  min      average          max

BE                81.70        81.90        82.20
BG                75.20        78.24        83.70
CZ                91.20        91.40        91.60
DK                71.00        73.03        76.20
DE                72.80        73.86        74.70
EE                79.00        80.49        82.20
IE                82.60        85.17        87.70
EL                79.20        81.42        83.00
ES                60.00        62.35        66.00
FR                81.60        82.26        83.40
IT                69.40        73.04        76.50
CY                77.60        80.50        85.10
LV                76.50        78.65        80.00
LT                78.90        84.23        89.10
LU                72.50        74.23        77.50
HU                83.50        83.53        83.60
MT                40.90        47.99        53.00
NL                71.90        74.34        76.20
AT                84.50        85.13        85.80
PL                88.80        90.33        91.30
PT                43.20        48.82        54.30
RO                75.30        76.56        78.30
SI                88.00        89.56        90.50
SK                91.70        92.92        94.80
FI                84.50        86.12        87.70
SE                85.20        85.60        86.00
UK                76.70        77.30        78.20
Sum             161,567      169,429      178,389
                    402          412          422

                                   6

j                  min      average          max

BE                 1.86         1.92         1.97
BG                 0.49         0.50         0.52
CZ                 1.21         1.31         1.47
DK                 2.24         2.47         2.72
DE                 2.45         2.52         2.63
EE                 0.60         0.87         1.29
IE                 1.12         1.25         1.43
EL                 0.55         0.57         0.59
ES                 0.91         1.09         1.35
FR                 2.02         2.11         2.15
IT                 1.05         1.11         1.18
CY                 0.24         0.34         0.46
LV                 0.42         0.48         0.61
LT                 0.59         0.71         0.80
LU                 1.62         1.63         1.65
HU                 0.79         0.88         1.00
MT                 0.26         0.42         0.54
NL                 1.63         1.75         1.82
AT                 1.94         2.27         2.67
PL                 0.56         0.60         0.64
PT                 0.76         0.96         1.51
RO                 0.37         0.44         0.58
SI                 1.39         1.48         1.66
SK                 0.47         0.54         0.65
FI                 3.35         3.51         3.73
SE                 3.61         3.66         3.75
UK                 1.68         1.79         1.88
Sum                  65           73           86
                      8            9            9

                                   7

j                  min      average          max

BE                18.20        19.40        21.00
BG                12.10        18.07        27.70
CZ                19.00        21.35        24.40
DK                17.40        18.29        19.00
DE                16.10        17.71        19.70
EE                22.00        24.28        27.10
IE                16.60        18.95        20.90
EL                16.60        17.71        18.70
ES                22.70        24.11        25.00
FR                16.20        17.00        18.50
IT                18.00        18.20        18.50
CY                11.90        14.32        20.40
LV                22.90        23.92        24.50
LT                16.40        18.40        20.20
LU                16.10        16.79        17.30
HU                18.00        19.05        20.20
MT                10.20        11.43        13.30
NL                15.60        17.05        18.80
AT                20.80        21.42        22.50
PL                14.70        17.66        21.40
PT                20.00        21.39        24.10
RO                15.40        19.66        26.40
SI                21.50        22.80        24.60
SK                22.20        22.99        23.80
FI                16.50        17.67        19.00
SE                14.10        15.33        16.80
UK                14.40        15.05        15.90
Sum               8,322        9,889       12,355
                     91           99          111

                                   8

j                  min      average          max

BE               102.00       106.54       111.10
BG                38.70        43.37        50.20
CZ                48.10        57.89        72.80
DK               130.20       136.88       141.20
DE               103.80       104.99       106.50
EE                57.20        65.51        78.00
IE               114.80       122.63       127.60
EL                84.80        88.72        94.00
ES                85.00        90.37        95.40
FR               105.80       108.81       110.80
IT                97.50       102.60       105.60
CY                88.00        89.89        91.20
LV                56.10        62.10        72.60
LT                52.60        56.68        64.70
LU               101.50       107.58       119.10
HU                49.20        59.22        68.10
MT                73.20        75.02        78.80
NL               100.00       103.34       106.10
AT               101.80       103.39       105.10
PL                53.20        59.71        69.10
PT                83.00        85.78        87.40
RO                42.50        48.21        60.90
SI                72.80        76.76        82.30
SK                44.40        55.52        70.20
FI               120.80       122.96       124.30
SE               114.50       121.05       127.60
UK               100.10       109.20       119.90
Sum             202,131      225,220      254,215
                    450          475          504

                                   9

j                  min      average          max

BE                13.00        13.98        14.70
BG                14.00        16.50        21.40
CZ                 8.00         9.08        10.40
DK                10.00        10.88        11.80
DE                10.00        12.29        15.20
EE                18.00        19.21        20.20
IE                15.40        18.54        20.70
EL                19.90        20.00        20.10
ES                18.00        19.15        19.90
FR                13.40        14.25        16.00
IT                18.00        18.59        19.10
CY                15.00        15.39        16.20
LV                16.00        19.89        25.60
LT                17.00        19.16        20.70
LU                12.00        12.69        13.40
HU                11.00        12.26        13.50
MT                13.70        14.42        15.00
NL                10.50        10.73        11.00
AT                12.00        12.40        12.80
PL                16.00        17.70        20.50
PT                18.50        19.94        21.00
RO                17.00        19.27        23.40
SI                11.00        11.82        12.30
SK                10.90        12.45        13.30
FI                11.00        11.81        13.60
SE                 8.00        10.30        12.10
UK                18.00        18.59        19.00
Sum               5,527        6,604        8,057
                     74           81           90

                                  10

j                  min      average          max

BE                 3.30         3.69         4.10
BG                 2.90         5.81         9.40
CZ                 2.20         3.39         4.20
DK                 0.50         0.81         1.20
DE                 3.80         4.30         5.50
EE                 1.70         3.77         6.30
IE                 1.60         1.63         1.70
EL                 3.60         5.00         6.20
ES                 2.00         3.15         4.60
FR                 2.90         3.38         3.80
IT                 3.10         4.27         6.30
CY                 0.50         0.90         1.20
LV                 1.90         4.10         7.90
LT                 1.20         3.82         8.00
LU                 0.50         0.93         1.60
HU                 2.70         3.11         3.60
MT                 2.50         3.37         4.50
NL                 0.80         1.09         1.60
AT                 0.90         1.08         1.40
PL                 2.40         5.68        10.30
PT                 1.70         2.66         3.70
RO                 2.40         3.52         4.80
SI                 1.90         2.92         4.10
SK                 6.60         9.29        11.80
FI                 1.20         1.92         2.80
SE                 0.80         1.19         1.50
UK                 1.00         1.25         1.40
Sum                 164          369          790
                     13           19           28

                                  11

j                  min      average          max

BE                92.90        98.29       101.30
BG                59.00        60.72        62.60
CZ                72.50        74.29        75.60
DK                92.60        96.75        99.10
DE                77.80        80.70        83.20
EE                44.50        47.74        49.60
IE               122.80       123.13       123.60
EL               120.90       123.12       125.70
ES               133.60       141.02       147.50
FR                93.60        96.84        98.90
IT               104.70       107.30       111.00
CY               172.80       180.80       193.90
LV                38.10        41.12        44.40
LT                39.00        43.85        48.90
LU                75.50        89.79       100.70
HU                75.10        78.46        81.20
MT               126.90       137.03       144.20
NL                97.60       100.54       102.90
AT               102.70       109.79       116.30
PL                85.30        86.23        87.30
PT               132.20       137.30       142.80
RO                56.30        60.18        64.20
SI               101.90       108.13       115.20
SK                66.10        67.12        68.70
FI                98.20       103.73       114.00
SE                88.30        93.45        97.20
UK                81.40        84.63        87.20
Sum             248,335      272,337      297,810
                    498          522          546

                                  12

j                  min      average          max

BE               199.82       223.51       243.68
BG               944.16      1135.85      1362.36
CZ               525.30       611.44       660.22
DK               103.13       109.07       112.47
DE               151.12       160.92       166.12
EE               570.51       683.12       812.71
IE               106.52       121.52       137.00
EL               169.95       186.56       204.57
ES               176.44       189.97       198.07
FR               166.74       174.97       179.36
IT               142.59       146.54       150.53
CY               213.39       221.72       237.06
LV               308.74       374.91       441.00
LT               417.54       507.30       571.22
LU               154.61       168.04       185.63
HU               401.35       439.99       487.54
MT               191.27       200.85       217.38
NL               171.58       182.46       191.56
AT               138.06       143.24       151.71
PL               383.54       435.97       488.67
PT               181.53       193.22       201.25
RO               614.57       755.52       913.36
SI               257.54       281.52       299.15
SK               519.68       670.74       796.44
FI               217.79       239.91       257.39
SE               152.08       168.55       177.45
UK               113.66       129.10       144.54
Sum           3,248,647    4,549,209    6,118,770
                  1,802        2,133        2,474

Table A3. Normalized fuzzy response matrix X
(objectives divided by their square roots)

                                      1

              [X.sub.ij1]      [X.sub.ij2]      [X.sub.ij3]

BE               0.219            0.220            0.218
BG               0.053            0.062            0.071
CZ               0.130            0.135            0.138
DK               0.229            0.229            0.228
DE               0.221            0.212            0.204
EE               0.086            0.101            0.116
IE               0.250            0.247            0.245
EL               0.160            0.165            0.162
ES               0.185            0.183            0.178
FR               0.206            0.202            0.199
IT               0.195            0.198            0.202
CY               0.170            0.167            0.166
LV               0.071            0.084            0.098
LT               0.074            0.090            0.107
LU               0.465            0.468            0.477
HU               0.105            0.110            0.111
MT               0.145            0.144            0.145
NL               0.246            0.241            0.231
AT               0.236            0.232            0.226
PL               0.092            0.094            0.097
PT               0.147            0.143            0.140
RO               0.050            0.061            0.073
SI               0.153            0.156            0.157
SK               0.095            0.107            0.124
FI               0.221            0.212            0.202
SE               0.233            0.228            0.221
UK               0.221            0.218            0.214

                                      2

              [X.sub.ij1]      [X.sub.ij2]      [X.sub.ij3]

BE               0.258            0.260            0.260
BG               0.063            0.067            0.071
CZ               0.127            0.133            0.137
DK               0.208            0.211            0.210
DE               0.220            0.213            0.206
EE               0.096            0.110            0.121
IE               0.262            0.259            0.257
EL               0.192            0.196            0.194
ES               0.210            0.204            0.197
FR               0.248            0.242            0.238
IT               0.225            0.229            0.240
CY               0.170            0.168            0.166
LV               0.083            0.090            0.099
LT               0.088            0.103            0.118
LU               0.349            0.344            0.335
HU               0.119            0.129            0.136
MT               0.179            0.180            0.184
NL               0.231            0.225            0.218
AT               0.234            0.232            0.230
PL               0.114            0.118            0.118
PT               0.143            0.141            0.140
RO               0.049            0.068            0.096
SI               0.157            0.160            0.161
SK               0.119            0.133            0.151
FI               0.230            0.224            0.219
SE               0.231            0.225            0.219
UK               0.226            0.221            0.217

                                      3

             [X.sub.ij1]      [X.sub.ij2]      [X.sub.ij3]

BE               0.186            0.183            0.180
BG               0.156            0.167            0.184
CZ               0.198            0.195            0.192
DK               0.234            0.230            0.225
DE               0.201            0.201            0.204
EE               0.187            0.193            0.201
IE               0.201            0.199            0.195
EL               0.175            0.177            0.178
ES               0.174            0.181            0.185
FR               0.192            0.190            0.187
IT               0.166            0.170            0.169
CY               0.203            0.205            0.204
LV               0.178            0.188            0.198
LT               0.183            0.184            0.185
LU               0.193            0.188            0.183
HU               0.174            0.170            0.164
MT               0.167            0.163            0.159
NL               0.225            0.223            0.223
AT               0.209            0.208            0.208
PL               0.160            0.165            0.171
PT               0.209            0.204            0.197
RO               0.178            0.179            0.182
SI               0.194            0.196            0.198
SK               0.175            0.176            0.180
FI               0.208            0.205            0.205
SE               0.223            0.219            0.214
UK               0.220            0.214            0.207

                                      4

              [X.sub.ij1]      [X.sub.ij2]      [X.sub.ij3]

BE               0.132            0.136            0.138
BG               0.105            0.142            0.184
CZ               0.183            0.189            0.190
DK               0.280            0.260            0.241
DE               0.189            0.198            0.215
EE               0.233            0.240            0.250
IE               0.228            0.223            0.215
EL               0.196            0.182            0.171
ES               0.186            0.186            0.182
FR               0.151            0.158            0.153
IT               0.139            0.139            0.138
CY               0.249            0.231            0.219
LV               0.181            0.211            0.238
LT               0.203            0.210            0.212
LU               0.134            0.136            0.136
HU               0.112            0.126            0.126
MT               0.144            0.134            0.126
NL               0.192            0.203            0.212
AT               0.145            0.146            0.164
PL               0.132            0.129            0.126
PT               0.253            0.228            0.203
RO               0.186            0.193            0.198
SI               0.114            0.126            0.131
SK               0.107            0.127            0.157
FI               0.209            0.222            0.226
SE               0.327            0.307            0.280
UK               0.255            0.247            0.232

                                      5

              [X.sub.ij1]      [X.sub.ij2]      [X.sub.ij3]

BE               0.203            0.199            0.195
BG               0.187            0.190            0.198
CZ               0.227            0.222            0.217
DK               0.177            0.177            0.180
DE               0.181            0.179            0.177
EE               0.197            0.196            0.195
IE               0.205            0.207            0.208
EL               0.197            0.198            0.197
ES               0.149            0.151            0.156
FR               0.203            0.200            0.197
IT               0.173            0.177            0.181
CY               0.193            0.196            0.201
LV               0.190            0.191            0.189
LT               0.196            0.205            0.211
LU               0.180            0.180            0.183
HU               0.208            0.203            0.198
MT               0.102            0.117            0.125
NL               0.179            0.181            0.180
AT               0.210            0.207            0.203
PL               0.221            0.219            0.216
PT               0.107            0.119            0.129
RO               0.187            0.186            0.185
SI               0.219            0.218            0.214
SK               0.228            0.226            0.224
FI               0.210            0.209            0.208
SE               0.212            0.208            0.204
UK               0.191            0.188            0.185

                                      6

              [X.sub.ij1]      [X.sub.ij2]      [X.sub.ij3]

BE               0.231            0.224            0.213
BG               0.061            0.059            0.056
CZ               0.150            0.153            0.159
DK               0.279            0.289            0.293
DE               0.305            0.295            0.284
EE               0.075            0.102            0.139
IE               0.139            0.147            0.154
EL               0.068            0.067            0.064
ES               0.113            0.128            0.146
FR               0.251            0.246            0.232
IT               0.131            0.130            0.127
CY               0.030            0.040            0.050
LV               0.052            0.056            0.066
LT               0.073            0.083            0.086
LU               0.201            0.191            0.178
HU               0.098            0.103            0.108
MT               0.032            0.049            0.058
NL               0.203            0.205            0.196
AT               0.241            0.265            0.288
PL               0.070            0.070            0.069
PT               0.094            0.112            0.163
RO               0.046            0.051            0.063
SI               0.173            0.173            0.179
SK               0.058            0.063            0.070
FI               0.417            0.410            0.402
SE               0.449            0.428            0.405
UK               0.209            0.209            0.203

                                      7

              [X.sub.ij1]      [X.sub.ij2]      [X.sub.ij3]

BE               0.200            0.195            0.189
BG               0.133            0.182            0.249
CZ               0.208            0.215            0.220
DK               0.191            0.184            0.171
DE               0.176            0.178            0.177
EE               0.241            0.244            0.244
IE               0.182            0.191            0.188
EL               0.182            0.178            0.168
ES               0.249            0.242            0.225
FR               0.178            0.171            0.166
IT               0.197            0.183            0.166
CY               0.130            0.144            0.184
LV               0.251            0.241            0.220
LT               0.180            0.185            0.182
LU               0.176            0.169            0.156
HU               0.197            0.192            0.182
MT               0.112            0.115            0.120
NL               0.171            0.171            0.169
AT               0.228            0.215            0.202
PL               0.161            0.178            0.193
PT               0.219            0.215            0.217
RO               0.169            0.198            0.238
SI               0.236            0.229            0.221
SK               0.243            0.231            0.214
FI               0.181            0.178            0.171
SE               0.155            0.154            0.151
UK               0.158            0.151            0.143

                                      8

              [X.sub.ij1]      [X.sub.ij2]      [X.sub.ij3]

BE               0.227            0.224            0.220
BG               0.086            0.091            0.100
CZ               0.107            0.122            0.144
DK               0.290            0.288            0.280
DE               0.231            0.221            0.211
EE               0.127            0.138            0.155
IE               0.255            0.258            0.253
EL               0.189            0.187            0.186
ES               0.189            0.190            0.189
FR               0.235            0.229            0.220
IT               0.217            0.216            0.209
CY               0.196            0.189            0.181
LV               0.125            0.131            0.144
LT               0.117            0.119            0.128
LU               0.226            0.227            0.236
HU               0.109            0.125            0.135
MT               0.163            0.158            0.156
NL               0.222            0.218            0.210
AT               0.226            0.218            0.208
PL               0.118            0.126            0.137
PT               0.185            0.181            0.173
RO               0.095            0.102            0.121
SI               0.162            0.162            0.163
SK               0.099            0.117            0.139
FI               0.269            0.259            0.247
SE               0.255            0.255            0.253
UK               0.223            0.230            0.238

                                      9

              [X.sub.ij1]      [X.sub.ij2]      [X.sub.ij3]

BE               0.175            0.172            0.164
BG               0.188            0.203            0.238
CZ               0.108            0.112            0.116
DK               0.135            0.134            0.131
DE               0.135            0.151            0.169
EE               0.242            0.236            0.225
IE               0.207            0.228            0.231
EL               0.268            0.246            0.224
ES               0.242            0.236            0.222
FR               0.180            0.175            0.178
IT               0.242            0.229            0.213
CY               0.202            0.189            0.180
LV               0.215            0.245            0.285
LT               0.229            0.236            0.231
LU               0.161            0.156            0.149
HU               0.148            0.151            0.150
MT               0.184            0.177            0.167
NL               0.141            0.132            0.123
AT               0.161            0.153            0.143
PL               0.215            0.218            0.228
PT               0.249            0.245            0.234
RO               0.229            0.237            0.261
SI               0.148            0.145            0.137
SK               0.147            0.153            0.148
FI               0.148            0.145            0.152
SE               0.108            0.127            0.135
UK               0.242            0.229            0.212

                                     10

              [X.sub.ij1]      [X.sub.ij2]      [X.sub.ij3]

BE               0.257            0.192            0.146
BG               0.226            0.302            0.334
CZ               0.172            0.176            0.149
DK               0.039            0.042            0.043
DE               0.296            0.224            0.196
EE               0.133            0.196            0.224
IE               0.125            0.085            0.060
EL               0.281            0.260            0.221
ES               0.156            0.164            0.164
FR               0.226            0.176            0.135
IT               0.242            0.222            0.224
CY               0.039            0.047            0.043
LV               0.148            0.214            0.281
LT               0.094            0.199            0.285
LU               0.039            0.048            0.057
HU               0.211            0.162            0.128
MT               0.195            0.175            0.160
NL               0.062            0.057            0.057
AT               0.070            0.056            0.050
PL               0.187            0.295            0.366
PT               0.133            0.139            0.132
RO               0.187            0.183            0.171
SI               0.148            0.152            0.146
SK               0.515            0.484            0.420
FI               0.094            0.100            0.100
SE               0.062            0.062            0.053
UK               0.078            0.065            0.050

                                     11

             [X.sub.ij1]      [X.sub.ij2]      [X.sub.ij3]

BE               0.186            0.188            0.186
BG               0.118            0.116            0.115
CZ               0.145            0.142            0.139
DK               0.186            0.185            0.182
DE               0.156            0.155            0.152
EE               0.089            0.091            0.091
IE               0.246            0.236            0.226
EL               0.243            0.236            0.230
ES               0.268            0.270            0.270
FR               0.188            0.186            0.181
IT               0.210            0.206            0.203
CY               0.347            0.346            0.355
LV               0.076            0.079            0.081
LT               0.078            0.084            0.090
LU               0.152            0.172            0.185
HU               0.151            0.150            0.149
MT               0.255            0.263            0.264
NL               0.196            0.193            0.189
AT               0.206            0.210            0.213
PL               0.171            0.165            0.160
PT               0.265            0.263            0.262
RO               0.113            0.115            0.118
SI               0.204            0.207            0.211
SK               0.133            0.129            0.126
FI               0.197            0.199            0.209
SE               0.177            0.179            0.178
UK               0.163            0.162            0.160

                                     12

              [X.sub.ij1]      [X.sub.ij2]      [X.sub.ij3]

BE               0.111            0.105            0.099
BG               0.524            0.533            0.551
CZ               0.291            0.287            0.267
DK               0.057            0.051            0.045
DE               0.084            0.075            0.067
EE               0.317            0.320            0.329
IE               0.059            0.057            0.055
EL               0.094            0.087            0.083
ES               0.098            0.089            0.080
FR               0.093            0.082            0.073
IT               0.079            0.069            0.061
CY               0.118            0.104            0.096
LV               0.171            0.176            0.178
LT               0.232            0.238            0.231
LU               0.086            0.079            0.075
HU               0.223            0.206            0.197
MT               0.106            0.094            0.088
NL               0.095            0.086            0.077
AT               0.077            0.067            0.061
PL               0.213            0.204            0.198
PT               0.101            0.091            0.081
RO               0.341            0.354            0.369
SI               0.143            0.132            0.121
SK               0.288            0.314            0.322
FI               0.121            0.112            0.104
SE               0.084            0.079            0.072
UK               0.063            0.061            0.058

Table A4. The final results of the fuzzy Ratio
System (RS) of MOORA

                     [y.sup.*.sub.i]

         [y.sup.*    [y.sup.*    [y.sup.*       [BNP.        Rank
States    .sub.i1]    .sub.i2]    .sub.i3]      sub.i]        (RS)

BE           0.616       0.534       0.435       0.528          11
BG          -0.581      -0.378      -0.129      -0.363          27
CZ           0.408       0.403       0.429       0.413          13
DK           0.915       0.879       0.843       0.879           3
DE           0.697       0.650       0.565       0.638           8
EE           0.091       0.203       0.358       0.217          19
IE           0.642       0.607       0.569       0.606           9
EL           0.227       0.146       0.061       0.145          22
ES           0.341       0.326       0.317       0.328          14
FR           0.641       0.562       0.450       0.551          10
IT           0.315       0.283       0.234       0.277          16
CY           0.290       0.275       0.288       0.285          15
LV           0.036       0.217       0.372       0.208          20
LT           0.033       0.184       0.353       0.190          21
LU           0.998       0.995       0.984       0.992           2
HU           0.253       0.238       0.182       0.224          18
MT           0.044       0.034       0.015       0.031          23
NL           0.791       0.764       0.712       0.756           6
AT           0.829       0.802       0.781       0.804           5
PL          -0.141      -0.035       0.085      -0.030          24
PT           0.291       0.244       0.256       0.264          17
RO          -0.175      -0.055       0.069      -0.054          25
SI           0.467       0.459       0.456       0.460          12
SK          -0.128      -0.134      -0.061      -0.108          26
FI           0.865       0.846       0.804       0.838           4
SE           1.137       1.067       1.007       1.071           1
UK           0.762       0.701       0.631       0.698           7

Table A5. The fuzzy Reference Point (RP) of MOORA
A5a - Maximal Objective Reference Point:

                         1

j       [r.sub.j1]   [r.sub.j2]   [r.sub.j3]

r         0.465        0.468        0.477

                         2

j       [r.sub.j1]   [r.sub.j2]   [r.sub.j3]

r         0.349        0.344        0.335

                         3

j       [r.sub.j1]   [r.sub.j2]   [r.sub.j3]

r         0.234        0.230        0.225

                         4

j       [r.sub.j1]   [r.sub.j2]   [r.sub.j3]

r         0.327        0.307        0.280

                         5

j       [r.sub.j1]   [r.sub.j2]   [r.sub.j3]

r         0.228        0.226        0.224

                         6

j       [r.sub.j1]   [r.sub.j2]   [r.sub.j3]

r         0.449        0.428        0.405

                         7

j       [r.sub.j1]   [r.sub.j2]   [r.sub.j3]

r         0.251        0.244        0.249

                         8

j       [r.sub.j1]   [r.sub.j2]   [r.sub.j3]

r         0.086        0.091        0.100

                         9

j       [r.sub.j1]   [r.sub.j2]   [r.sub.j3]

r         0.108        0.112        0.116

                         10

j       [r.sub.j1]   [r.sub.j2]   [r.sub.j3]

r         0.039        0.042        0.043

                         11

j       [r.sub.j1]   [r.sub.j2]   [r.sub.j3]

r         0.076        0.079        0.081

                         12

j       [r.sub.j1]   [r.sub.j2]   [r.sub.j3]

r         0.057        0.051        0.045

Annex B. The fuzzy Full Multiplicative Form and fuzzy MULTIMOORA

Table B1. Te fuzzy Full Multiplicative Form (MF)

State   [A.sub.i1]   [A.sub.i2]   [A.sub.i3]   [B.sub.i1]   [B.sub.i2]

BE       6.33E+10     8.84E+10     1.26E+11     81229192     1.21E+08
BG       3.98E+08     1.43E+09     5.41E+09     87525501     2.87E+08
CZ       2.05E+10     3.47E+10     5.99E+10     32240603     80841425
DK       1.41E+11     1.96E+11     2.71E+11      6216945     12792050
DE        8.7E+10     1.22E+11     1.88E+11     46374847     72006311
EE       6.15E+09      1.8E+10     5.35E+10     44436590     1.55E+08
IE       7.57E+10     1.18E+11     1.83E+11     37000885     55552085
EL       1.25E+10     1.77E+10     2.33E+10     1.25E+08     2.04E+08
ES       2.55E+10     4.22E+10     6.98E+10     72131495     1.46E+08
FR       6.46E+10     8.91E+10     1.18E+11     64165873     88772369
IT       2.18E+10     3.22E+10     4.97E+10     81222166     1.28E+08
CY        5.3E+09     1.09E+10      2.6E+10     24336703     49704816
LV       2.27E+09      5.6E+09     1.44E+10     20061080     78102282
LT       3.04E+09     8.08E+09     1.89E+10     17473448     92256740
LU       1.31E+11     1.73E+11     2.32E+11      7108890     19116414
HU       4.71E+09     8.74E+09     1.37E+10     44043797     77979617
MT        1.1E+09     2.69E+09     5.39E+09     60852740        1E+08
NL       7.37E+10     1.12E+11     1.64E+11     14066815     22088363
AT       9.41E+10     1.39E+11     2.41E+11     15588698     21769174
PL       2.62E+09     4.66E+09     8.12E+09     66834784     2.26E+08
PT       1.19E+10     1.94E+10     4.09E+10     62643874     1.21E+08
RO       5.61E+08     1.94E+09     7.88E+09     59996905     1.49E+08
SI       2.29E+10      3.8E+10     6.38E+10     39929745     80673819
SK       3.36E+09      7.5E+09     2.04E+10      1.1E+08     2.89E+08
FI       1.69E+11     2.38E+11     3.35E+11     34102820     69290315
SE       2.78E+11     3.41E+11     4.15E+11      9840525     23355097
UK       8.52E+10     1.09E+11     1.37E+11     16670117     27764214

State   [B.sub.i3]   [U.sub.i1]   [U.sub.i2]   [U.sub.i3]

BE       1.65E+08     382.6751     733.2964     1.02E+19
BG       8.61E+08     0.461969     4.977532     4.74E+17
CZ       1.59E+08     129.3744     429.2268     1.93E+18
DK       22284777     6346.014     15315.89     1.68E+18
DE       1.23E+08     707.2274     1693.321     8.71E+18
EE          4E+08     15.38156     116.3191     2.38E+18
IE       76034200     995.5973     2122.429     6.76E+18
EL       3.01E+08     41.58832     86.78519     2.91E+18
ES       2.55E+08     100.1222     289.2566     5.03E+18
FR       1.19E+08     540.4122     1003.993     7.59E+18
IT       2.12E+08     102.5437     250.9532     4.04E+18
CY       81494291     65.04372      218.485     6.32E+17
LV       2.87E+08     7.879871     71.63826      2.9E+17
LT       2.99E+08     10.14279     87.61634      3.3E+17
LU       47732500      2735.72     9052.587     1.65E+18
HU       1.31E+08     35.94482     112.1021     6.04E+17
MT       1.67E+08     6.612322     26.76281     3.28E+17
NL       36808511      2002.01     5065.027      2.3E+18
AT       33230329     2832.278     6391.682     3.75E+18
PL       6.22E+08      4.21494     20.67204     5.43E+17
PT       1.95E+08     61.22833      160.627     2.56E+18
RO       4.01E+08     1.397506     13.02349     4.73E+17
SI       1.43E+08      159.785     471.6043     2.55E+18
SK       6.03E+08      5.56885     25.96284     2.24E+18
FI       1.39E+08     1219.223     3437.218     1.14E+19
SE       39945657     6960.107     14579.86     4.08E+18
UK       40198084     2120.383     3916.003     2.29E+18

                      Rank
State   [BNP.sub.1]   (MF)

BE        3.41E+18      2
BG        1.58E+17     23
CZ        6.44E+17     17
DK        5.61E+17     18
DE         2.9E+18      3
EE        7.92E+17     13
IE        2.25E+18      5
EL        9.69E+17     10
ES        1.68E+18      6
FR        2.53E+18      4
IT        1.35E+18      8
CY        2.11E+17     20
LV        9.66E+16     27
LT         1.1E+17     25
LU         5.5E+17     19
HU        2.01E+17     21
MT        1.09E+17     26
NL        7.67E+17     14
AT        1.25E+18      9
PL        1.81E+17     22
PT        8.54E+17     11
RO        1.58E+17     24
SI        8.49E+17     12
SK        7.47E+17     16
FI        3.81E+18      1
SE        1.36E+18      7
UK        7.62E+17     15

Annex C. Summary table for the three Methods of Fuzzy MULTIMOORA

Table C1. Final ranks of a fuzzy MULTIMOORA for EU member
states (2000-2004-2008)

                            Ranks

                                                             Final
                                           The               rank
                  The        The          Fuzzy               by
                 Fuzzy      Fuzzy          Full               Sum
                 Ratio    Reference   Multiplicative         MULTI
State            System     Point          Form        Sum   MOORA

Austria            5          3             9          17      3
Belgium            11         6             2          19      5
Bulgaria           27        27             23         77     27
Cyprus             15        24             20         59     20
Czech Republic     13        16             17         46     16
Denmark            3          4             18         25     10
Estonia            19        19             13         51     18
Finland            4          9             1          14      2
France             10        10             4          24      8
Germany            8          8             3          19      4
Greece             22        17             10         49     17
Hungary            18        18             21         57     19
Ireland            9         11             5          25      9
Italy              16        12             8          36     13
Latvia             20        23             27         70     24
Lithuania          21        21             25         67     22
Luxembourg         2          2             19         23      7
Malta              23        22             26         71     25
Netherlands        6          1             14         21      6
Poland             24        20             22         66     21
Portugal           17        15             11         43     15
Romania            25        25             24         74     26
Slovakia           26        26             16         68     23
Slovenia           12        14             12         38     14
Spain              14        13             6          33     12
Sweden             1          5             7          13      1
United Kingdom     7          7             15         29     11

                  The                Rank        Group
                 Fuzzy    Group   Correction   Correction
                 Ratio     by         by           by
State            System    Sum    Dominance    Dominance

Austria            5        1         4            --
Belgium            11       1         7            --
Bulgaria           27       3         --           --
Cyprus             15       3         19           --
Czech Republic     13       2         --           --
Denmark            3        2         6            1
Estonia            19       2         --           --
Finland            4        1         3            --
France             10       1         10           --
Germany            8        1         9            --
Greece             22       2         --           --
Hungary            18       3         20           --
Ireland            9        1         11           2
Italy              16       2         --           --
Latvia             20       3         --           --
Lithuania          21       3         --           --
Luxembourg         2        1         2            --
Malta              23       3         23           --
Netherlands        6        1         5            --
Poland             24       3         --           --
Portugal           17       2         --           --
Romania            25       3         25           --
Slovakia           26       3         26           --
Slovenia           12       2         --           --
Spain              14       2         --           --
Sweden             1        1         --           --
United Kingdom     7        2         8            1

Annex D. Theory of Dominance, Domination and Transitivity

1. Principles

1. Staying in the ordinal sphere with ranking by dominance.

2. The three methods have the same importance.

3. Overall dominance is ranked on the first place. Will seldom occur.

4. Three groups are considered: Core (in principle first 9), Semi-Periphery (next 9), Periphery (last 9). If countries are ex-aequo but a country is Semi-Periphery or Periphery in one of the methods then it is inferior to the other country.

2. Ranking

Overall dominance in the three methods is not present.

I. Core
1. General dominance in two of the three methods:
Sweden (1-5-7)

--Dominates Luxemburg            1) Ratio System;
(2-2-19) in:                        Dominated in Reference
                                    Point.

                                 2) Multiplicative Form.

--Dominates Austria              1) Ratio System;
(5-3-9) in:                         Dominated in Reference
                                    Point.

                                 2) Multiplicative Form.

--Dominates Finland              1) Ratio System; Dominated
(4-9-1) in:                         in Multiplicative
                                    Form.

                                 2) Reference Point.

--Dominates all the others in 2 methods

2. Dominance in two of the three
methods: Luxemburg (2-2-19)

3. Dominance in two of the three methods:
Finland (4-9-1) dominated by Luxemburg.

Dominates Austria in 2 methods.

4. Austria (5-3-9)               2 x dominated by Finland.

5. Netherlands (6-1-14)          2 x dominated by Austria.

6. Denmark (3-4-18)              2 x dominated by the Netherlands.

7. Belgium (11-6-2)              2 x dominated by Denmark.

8. UK (7-7-15)                   2 x dominated by Belgium.

9. Germany (8-8-3)               2 x dominated by UK.

II. Semi-Periphery

10. France (10-10-4)             overall dominated by Germany.

11. Ireland (9-11-5)             2 x dominated by France.

12. Spain (14-13-6)              overall dominated by Ireland.

13. Italy (16-12-8)              2 x dominated by Spain.

14. Slovenia (12-14-12)          2 x dominated by Italy.

15. Portugal (17-15-11)          2 x dominated by Slovenia.

16. Czech (13-16-17)             2 x dominated by Portugal.

17. Greece (22-17-10)            2 x dominated by Czech Republic.

18. Estonia (19-19-13)           2 x dominated by Greece.

III. Periphery

19. Cyprus (15-24-20)            2 x dominated by Estonia.

20. Hungary (18-18-21)           2 x dominated by Cyprus.

21. Poland (24-20-22)            2 x dominated by Hungary.

22. Lithuania (21-21-25)         2 x dominated by Poland.

23. Malta (23-22-26)             overall dominated by Lithuania.

24. Latvia (20-23-27)            2 x dominated by Malta.

25. Romania (25-25-24)           2 x dominated by Latvia.

26. Slovakia (26-26-16)          2 x dominated by Romania.

27. Bulgaria (27-27-23)          overall dominated by Slovakia.

References

Arrow, K. J. 1974. General Economic Equilibrium: Purpose, Analytic Techniques, Collective Choice, American Economic Review 64: 253-272.

Balezentis, A.; Balezentis, T.; Valkauskas, R. 2010. Evaluating Situation of Lithuania in the European Union: Structural Indicators and MULTIMOORA Method, Technological and Economic Development of Economy 16(4): 578-602. doi:10.3846/tede.2010.36

Balezentis, A.; Balezentis, T. 2010. Europos Sajungos valstybiu nariu kaimo darnaus vystymo vertinimas [Evaluation of sustainable rural development in the European Union Member States], Management Theory & Studies for Rural Business & Infrastructure Development 23(4): 16-24. (In Lithuanian)

Brauers, W. K. 2004. Optimization Methods for a Stakeholder Society, a Revolution in Economic Thinking by Multi-Objective Optimization. Boston: Kluwer Academic Publishers.

Brauers, W. K. 2007. What is meant by normalization in decision making? International Journal of Management and Decision Making 8(5/6): 445-460. doi:10.1504/IJMDM.2007.013411

Brauers, W. K. M.; Ginevicius, R. 2009. Robustness in Regional Development Studies. The Case of Lithuania, Journal of Business Economics and Management 10(2): 121-140. doi:10.3846/1611-1699.2009.10.121-140

Brauers, W. K. M.; Ginevicius, R. 2010. The economy of the Belgian regions tested with MULTIMOORA, Journal of Business Economics and Management 11(2): 173-209. doi:10.3846/jbem.2010.09

Brauers, W. K. M.; Ginevicius, R.; Podvezko, V. 2010. Regional development in Lithuania considering multiple objectives by the MOORA method, Technological and Economic Development of Economy 16(4): 613-640. doi:10.3846/tede.2010.38

Brauers, W. K. M.; Ginevicius, R.; Zavadskas E. K.; Antucheviciene J. 2007. The European Union in a transition economy, Transformation in Business & Economics 6(2): 21-37.

Brauers, W. K. M.; Zavadskas, E. K. 2006. The MOORA method and its application to privatization in a transition economy, Control and Cybernetics 35(2): 445-469.

Brauers, W. K.; Zavadskas, E. K. 2009a. Robustness of the multi-objective MOORA method with a test for the facilities sector, Technological and Economic Development of Economy 15(2): 352-375. doi:10.3846/1392-8619.2009.15.352-375

Brauers, W. K. M.; Zavadskas, E. K. 2009b. Multi-objective optimization with discrete alternatives on the basis of ratio analysis, Intelektine ekonomika [Intellectual Economics] 2(6): 30-41.

Brauers, W. K. M.; Zavadskas, E. K. 2010. Project management by MULTIMOORA as an instrument for transition economies, Technological and Economic Development of Economy 16(1): 5-24. doi:10.3846/ tede.2010.01

Brauers, W. K. M.; Zavadskas, E. K. 2011. MULTIMOORA Optimization Used to Decide on a Bank Loan to Buy Property, Technological and Economic Development of Economy 17(1): 174-188. doi:10.3846 /13928619.2011.560632

Brauers, W. K. M.; Zavadskas, E. K.; Turskis, Z.; Vilutiene, T. 2008. Multi-objective contractor's ranking by applying the MOORA method, Journal of Business Economics and Management 9(4): 245-255. doi:10.3846/1611-1699.2008.9.245-255

Chakraborty, S. 2010. Applications of the MOORA method for decision making in manufacturing environment, The International Journal of Advanced Manufacturing Technology. 54 (9-12): 1155-1166. In Press. doi:10.1007/s00170-010-2972-0

Chen, C. T. 2000. Extensions of the TOPSIS for group decision-making under fuzzy environment, Fuzzy Sets and Systems 114: 1-9. doi:10.1016/S0165-0114(97)00377-1

Chou, S. Y.; Chang, Y. H.; Shen, C. Y. 2008. A fuzzy simple additive weighting system under group decision-making for facility location selection with objective/subjective attributes, European Journal of Operational Research 189: 132-145. doi:10.1016/j.ejor.2007.05.006

Clark, R. 2010. World-System Mobility and Economic Growth 1981-2000, Social Forces 88(3): 1123-1151.

Guitouni, A.; Martel, J. M. 1998. Tentative guidelines to help choosing an appropriate MCDA method, European Journal of Operational Research 109: 501-521. doi:10.1016/S0377-2217(98)00073-3

Heragu, S. 1997. Facilities Design. Boston: PWS Publishing.

Hwang, C. L.; Yoon, K. 1981. Multiple Attribute Decision Making Methods and Applications. Berlin: Springer Verlag.

Kahraman, C. 2008. Multi-Criteria Decision Making Methods and Fuzzy Sets. In: Kahraman, C. (ed.) Fuzzy Multi-Criteria Decision Making. Springer. doi:10.1007/978-0-387-76813-7_1

Kahraman, C.; Kaya, 1. 2010. Investment analyses using fuzzy probability concept, Technological and Economic Development of Economy 16(1): 43-57. doi:10.3846/tede.2010.03

Kaplinski, O. 2009. Sapere Aude: Professor Edmundas Kazimieras Zavadskas, Inzinerine Ekonomika Engineering Economics (5): 113-119.

Kendall, M. G. 1948. Rank Correlation Methods. London: Griffin.

Kendall, M. G.; Gibbons, J. D. 1990. Rank Correlation Methods. London: Edward Arnold.

Keufmann, A.; Gupta, M. M. 1991. Introduction to Fuzzy Arithmetic: Theory and Application. New York: Van Nostrand Reinhold.

Korpysa, J. (2010). Unemployment as a Main Determinant of Entrepreneurship, Transformations in Business & Economics 9(1): 109-123.

van Laarhoven, P. J. M.; Pedrycz, W., 1983. A fuzzy extension of Saaty's priority theory, Fuzzy Sets and Systems 11: 229-241.

Leung, L. C.; Cao, D. 2000. On consistency and ranking of alternatives in fuzzy AHP, European Journal of Operational Research 124(1): 102-113. doi:10.1016/S0377-2217(99)00118-6

Liang, G. S. 1999. Fuzzy MCDM based on ideal and anti-ideal concepts, European Journal of Operational Research 112: 682-691. doi:10.1016/S0377-2217(97)00410-4

Liang, G. S.; Ding, J. F. 2003. Fuzzy MCDM based on the concept of a-cut, Journal of Multi-Criteria Decision Analysis 12(6): 299-310. doi:10.1002/mcda.366

Liang, G. S.; Wang, M. J. J. 1991. A fuzzy multi-criteria decision-making method for facility site selection, International Journal of Production Research 29(11): 2313-2330. doi:10.1080/00207549108948085

Lin, Y. H.; Lee, P. C; Chang, T. P.; Ting, H. I. 2008. Multi-attribute group decision making model under the condition of uncertain information, Automation in Construction 17(6): 792-797. doi:10.1016/j. autcon.2008.02.011

Liu, P. 2009a. Multi-attribute decision-making method research based on interval vague set and TOPSIS method, Technological and Economic Development of Economy 15(3): 453-463. doi:10.3846/1392 8619.2009.15.453-463

Liu, P. 2009b. A novel method for hybrid multiple attribute decision making, Knowledge-Based Systems 22(5): 388-391. doi:10.1016/j.knosys.2009.02.001

Lootsma, F. A. 1987. Numerical Scaling of Human Judgement in Pairwise-comparison Methods for Fuzzy Multi-criteria Decision Analysis, in: NATO Advanced Study Institute, Val d'Isere (F), July 1987, Thu 39.

MacCrimmon, K. R. 1968. Decision making among multiple attribute alternatives: A survey and consolidated approach. RAND Memorandum, RM-4823-ARPA. The RAND Corporation, Santa Monica, Calif.

Martinkus, B.; Stoskus, S.; Berzinskiene, D. 2009. Changes of Employment through the Segmentation of Labour Market in the Baltic States, Inzinerine Ekonomika--Engineering Economics (3): 41-48.

Mukaidono, M. 2001. Fuzzy Logic for Beginners. River Edge (NJ): World Scientific Publishing Co.

Miller, G. A. 1956. The magical number seven plus or minus two: some limits on our capacity for processing information, Psychological Review 63: 81-97. doi:10.1037/h0043158

Peldschus, F.; Zavadskas, E. K. 2005. Fuzzy matrix games multi-criteria model for decision-making in engineering, Informatica 16(1): 107-120.

Peldschus, F.; Zavadskas, E. K.; Turskis, Z.; Tamosaitiene, J. 2010. Sustainable Assessment of Construction Site by Applying Game Theory, Inzinerine Ekonomika--Engineering Economics 21(3): 223-237.

Roy, B. 1996. Multicriteria methodology for decision aiding. Dordrecht: Kluwer.

Roy, B. 2005. Paradigms and challenges. In: Figueira, J.; Greco, S.; Ehrgott, M. (eds.) Multiple Criteria Decision Analysis--State of the Art Survey. Springer.

Saaty, T. L. 1980. The Analytic Hierarchy Process. New York: McGraw-Hill.

Saaty, T. L. 1987. What is the Analytic Hierarchy Process? In: Mathematical Models for Decision Support, Nato Advanced Study Institute, Val d'Isere (F), July 26, 1987, Mon 82.

Spearman, C. 1904. The proof and measurement of association between two things, American Journal of Psychology 15: 72-101. doi:10.2307/1412159

Spearman, C. 1906. A footrule for measuring correlation, British Journal of Psychology 2: 89-108.

Spearman, C. 1910. Correlation calculated from faulty data, British Journal of Psychology 3: 271-295.

Triantaphyllou, E. 2000. Multi-Criteria Decision Making Methods: a Comparative Study. Dordrecht: Kluwer.

Torlak, G.; Sevkli, M.; Sanal, M.; Zaim, S. 2011. Analyzing business competition by using fuzzy TOPSIS method: An example of Turkish domestic airline industry, Expert Systems with Applications 38(4): 3396-9406. doi:10.1016/j.eswa.2010.08.125

Turskis, Z.; Zavadskas, E. K. 2010a. A Novel Method for Multiple Criteria Analysis: Grey Additive Ratio Assessment (ARAS-G) Method, Informatica 21(4): 597-610.

Turskis, Z.; Zavadskas, E. K. 2010b. A New Fuzzy Additive Ratio Assessment Method (ARAS-F). Case Study: the Analysis of Fuzzy Multiple Criteria in Order to Select the Logistic Center Location, Transport 25(4): 423-432. doi:10.3846/transport.2010.52

Turskis, Z.; Zavadskas, E. K.; Peldschus, F. 2009. Multi-criteria Optimization System for Decision Making in Construction Design and Management, Inzinerine Ekonomika--Engineering Economics (1): 7-17.

Yao, J. S.; Wu, K. 2000. Ranking fuzzy numbers based on decomposition principle and signed distance, Fuzzy Sets and Systems 116: 275-288. doi:10.1016/S0165-0114(98)00122-5

Vertrouwen in Ierland slinkt met de dag. 2010, De Tijd, November 25.

Zadeh, L. A. 1975a. The concept of a linguistic variable and its application to approximate reasoning--I, Information Sciences 8(3): 199-249. doi:10.1016/0020-0255(75)90036-5

Zadeh, L. A. 1975b. The concept of a linguistic variable and its application to approximate reasoning--II, Information Sciences 8(4): 301-357. doi:10.1016/0020-0255(75)90046-8

Zadeh, L. A. 1975c. The concept of a linguistic variable and its application to approximate reasoning--III, Information Sciences 9(1): 43-80. doi:10.1016/0020-0255(75)90017-1

Zadeh, L. A. 1965. Fuzzy sets, Information and Control 8(1): 338-353. doi:10.1016/S0019-9958(65)90241-X

Zavadskas, E. K.; Antucheviciene. 2007. Multiple criteria evaluation of rural building's regeneration alternatives, Building and Environment 42(1): 436-451. doi:10.1016/j.buildenv.2005.08.001

Zavadskas, E. K.; Antucheviciene, J. 2006. Development of an indicator model and ranking of sustainable revitalization alternatives of derelict property: A Lithuanian case study, Sustainable Development 14(5): 287-299. doi:10.1002/sd.285

Zavadskas, E. K., Kaklauskas, A., Sarka, V. 1994. The new method of multicriteria complex proportional assessment of projects. Technological and Economic Development of Economy 1(3): 131-139.

Zavadskas, E. K.; Kaklauskas, A.; Turskis, Z.; Tamosaitiene, J. 2008a. Selection of the effective dwelling house walls by applying attributes values determined at intervals, Journal of Civil Engineering and Management 14(2): 85-93. doi:10.3846/1392-3730.2008.14.3

Zavadskas, E. K.; Liias, R.; Turskis, Z. 2008b. Multi-attribute decision-making methods for assessment of quality in bridges and road construction: State-of-the-art surveys, Baltic Journal of Road and Bridge Engineering 3(3): 152-160. doi:10.3846/1822-427X.2008.3.152-160

Zavadskas, E. K.; Turskis, Z. 2010. A new additive ratio assessment (ARAS) method in multi-criteria decision making, Technological and Economic Development of Economy 16(2): 159-172. doi:10.3846/ tede.2010.10

Zavadskas, E. K.; Turskis, Z.; Tamosaitiene, J.; Marina, V. 2008c. Multicriteria selection of project managers by applying grey criteria, Technological and Economic Development of Economy 14(4): 462-477. doi:10.3846/1392-8619.2008.14.462-477

Zavadskas, E. K.; Vilutiene, T.; Turskis, Z.; Tamosaitiene, J. 2010a. Contractor selection for construction works by applying SAW-G and TOPSIS GREY techniques, Journal of Business Economics and Management 11(1): 34-55. doi:10.3846/jbem.2010.03

Zavadskas, E. K.; Turskis, Z.; Ustinovichius, L.; Shevchenko, G. 2010b. Attributes Weights Determining Peculiarities in Multiple Attribute Decision Making Methods, Inzinerine Ekonomika--Engineering Economics 21(1): 32-43.

Zhang, X.; Liu, P. 2010. Method for aggregating triangular fuzzy intuitionistic fuzzy information and its application to decision making, Technological and Economic Development of Economy 16(2): 280-290. doi:10.3846/tede.2010.18

Zhao, R.; Govind, R. 1991. Algebraic characteristics of extended fuzzy numbers, Information Science 54(1): 103-130. doi:10.1016/0020-0255(91)90047-X

Zopounidis, C.; Pardalos, P. M.; Baourakis, G. 2001. Fuzzy Sets in Management, Economics and Marketing. River Edge (NJ): World Scientific Publishing Co. doi:10.1142/9789812810892

Willem K. M. Brauers (1), Alvydas Balezentis (2), Tomas Balezentis (3)

(1) Vilnius Gediminas Technical University, Sauletekio al. 10, LT-10223 Vilnius, Lithuania

(2) Mykolas Romeris University, Valakupiu g. 5, LT-10101 Vilnius, Lithuania

(3) Vilnius University, Sauletekio al. 9, LT-10222 Vilnius, Lithuania

E-mails: (1) willem.brauers@ua.ac.be; (2) a.balezentis@gmail.com (corresponding author); (3) t.balezentis@gmail.com

Received 3 November 2010; accepted 16 March 2011

(1) Mukaidono (2001) presents an interesting introduction to fuzzy logic. Zopounidis et al. (2001) with "Fuzzy sets in Management, Economics and Marketing" are perhaps nearer to the topic of this article.

(2) Mode is the measurement with the maximum frequency if there is one. As there is only a lower limit and an upper limit the average of both is taken.

(3) Brauers and Zavadskas (2011) developed the theory of Dominance for the first time in January 2011.

(4) Vertrouwen in Ierland slinkt met de dag, De Tijd, November 25, 2010. These figures are considered as confidential, but the newspaper takes the responsibility of publication.

Willem K. M. BRAUERS was graduated as: Ph.D. in economics (Un. of Leuven), Master of Arts (in economics) of Columbia Un. (New York), Master in Economics, in Management and Financial Sciences, in Political and Diplomatic Sciences and Bachelor in Philosophy all of the Un. of Leuven). He is professor ordinarius at the Faculty of Applied Economics of the University of Antwerp, Honorary Professor at the University of Leuven, the Belgian War College, the School of Military Administrators and the Antwerp Business School. He was a research fellow in several American institutions like Rand Corporation, the Institute for the Future, the Futures Group and extraordinary advisor to the Center for Economic Studies of the University of Leuven. He was consultant in the public sector, such as the Belgian Department of National Defense, the Department of Industry in Thailand, the project for the construction of a new port in Algeria (the port of Arzew) and in the private sector such as the international seaport of Antwerp and in electrical works. He was Chairman of the Board of Directors of SORCA Ltd.Brussels, Management Consultants for Developing Countries, linked to the worldwide group of ARCADIS and Chairman of the Board of Directors of MARESCO Ltd. Antwerp, Marketing Consultants. At the moment he is General Manager of CONSULTING, Systems Engineering Consultants. Brauers is member of many international scientific organizations. His specialization covers: Optimizing Techniques with Different Objectives, Forecasting Techniques, Input-Output Techniques and Public Sector Economics such as for National Defense and for Regional Sub-optimization. His scientific publications consist of seventeen books and several hundreds of articles and reports.

Alvydas BALEZENTIS. Ph. D. (HP) in management and administration, is Professor at the Department of Strategic Management in Mykolas Romeris University. While working at the Parliament of the Republic of Lithuania, Ministry of Agriculture, and Institute of Agrarian Economics he contributed to creation and fostering of the Lithuanian rural development policy at various levels. His scientific interests cover areas of innovatics, strategic management, sustainable development and rural development.

Tomas BALEZENTIS is student of economics (economic analysis) at the Faculty of Economics in Vilnius University. His working experience includes traineeship at the European Parliament and working at the Training Centre of the Ministry of Finance. His scientific interests: quantitative methods in social sciences, multi-criteria decision making, European integration processes.

Table 1. System of structural indicators used in
analysis of EU Member States' development during
2000-2008

     Structural indicators                     Desirable
                                                values
I. General economic background

1    GDP per capita in PPS (EU-27 = 100)          Max
2    Labor productivity per person employed       Max

II. Employment

3    Employment rate                              Max
4    Employment rate of older workers             Max

III. Innovation and research

5    Youth education attainment level             Max
6    Gross domestic expenditure on R&D            Max

IV. Economic reform

7    Business investment                          Max
8    Comparative price levels                     Min

V. Social cohesion

9    At-risk-of-poverty rate                      Min
10   Long-term unemployment rate                  Min

VI. Environment

11   Greenhouse gas emissions                     Min
12   Energy intensity of the economy              Min

Table 2. Ordinal versus cardinal:
comparing the price of one commodity

    Ordinal   Cardinal

       1
       2
       3
       4
A      5       6.03$
       6       6.02$
       7       6.01$
B      8         6$

Table 3. Negative rank order correlations

Items   Expert 1   Expert 2      D          D2

1          1          7          -6         36
2          2          6          -4         16
3          3          5          -2          4
4          4          4           0          0
5          5          3           2          4
6          6          2           4         16
7          7          1           6         36
E                                          112

Table 4. Ranking of Scenarios for the Belgian
Regions by the Full-Multiplicative
Method at the Year 1996

1    Scenario IX     Optimal           203,267
                     Economic Policy
                     in Wallonia and
                     Brussels

2    Scenario X      Optimal           196,306
                     Economic Policy
                     in Wallonia and
                     Brussels even
                     agreeing on the
                     Partition of
                     the National
                     Public Debt

3    Scenario VII    Flanders asks     164,515
                     for the
                     Partition of
                     the National
                     Public Debt

4    Scenario VIII   No Solidarity     158,881
                     at all

5    Scenario II     Unfavorable       90
                     Growth Rate for
                     Flanders

6    Scenario IV     an Unfavorable    87
                     Growth Rate for
                     Flanders and at
                     that moment
                     asks also for
                     the Partition
                     of the National
                     Public Debt

7    Scenario III    Partition of      54
                     the National
                     Public Debt

8    Scenario I      the Average       51
                     Belgian

9    Scenario V      Average Belgian   49
                     but as
                     compensation
                     Flanders asks
                     for the
                     Partition of
                     the National
                     Public Debt

10   Scenario O      Status Quo        43

11   Scenario VI     Flanders asks     42
                     for the
                     Partition of
                     the National
                     Public Debt

Source: Brauers, Ginevicius 2010.

Table 5. European Member States overall
dominating other European Memeber

Overall dominating        Overall being
                            dominated

Germany (8-8-3)         France (10-10-4)
Ireland (9-11-5)         Spain (14-13-6)
Lithuania (21-21-25)    Malta (23-22-26)
Slovakia (26-26-16)    Bulgaria (27-27-23)

Table 6. MOO Ranking on basis of 12 Structural
Indicators for the 27 Member States of the EU

Ranking   Member States with
          MULTIMOORA Rankings

          Core (Group 1)

1         Sweden (1-5-7)
2         Luxemburg (2-2-19)
3         Finland (4-9-1)
4         Austria (5-3-9)
5         Netherlands (6-1-14)
6         Denmark (3-4-18)
7         Belgium (11-6-2)
8         UK (7-7-15)
9         Germany (8-8-3)

          Semi-Periphery (Group 2)

10        France (10-10-4)
11        Ireland (9-11-5)
12        Spain (14-13-6)
13        Italy (16-12-8)
14        Slovenia (12-14-12)
15        Portugal (17-15-11)
16        Czech (13-16-17)
17        Greece (22-17-10)
18        Estonia (19-19-13)

          Periphery (Group 3)

19        Cyprus(15-24-20)
20        Hungary (18-18-21)
21        Poland (24-20-22)
22        Lithuania (21-21-25)
23        Malta (23-22-26)
24        Latvia (20-23-27)
25        Romania (25-25-24)
26        Slovakia (26-26-16)
27        Bulgaria (27-27-23)

For details see: Annex C.