European Union member states preparing for Europe 2020. An application of the MULTIMOORA method.
Brauers, Willem Karel M. ; Balezentis, Alvydas ; Balezentis, Tomas 等
1. Introduction
The Lisbon Strategy (European Council 2000) was adopted in 2000 and
focused on turning the European Union (EU) into "the most
competitive and dynamic knowledge-based economy in the world" by
2010.
Today it is obvious that many goals of the Lisbon Strategy for 2010
were not met. Therefore the new strategy Europe 2020 was initiated in
2010 (European Commission 2010). The new strategy aims to the creation
of a growing and sustainable European economy. Furthermore, greater
policy coordination between national governments and the EU is an
additional focal point of the Strategy Europe 2020.
Twenty two objectives, 10 originating from statistics and 12 from
statistics and forecasts, important for the future, will characterize
the 27 EU-Countries economies as a preparation for 2020. Nevertheless
these data concern only the economic guidelines of Strategy Europe 2020
and not the climate and energy targets.
As all these data are expressed in different units the exercise
needs an adequate Decision Support System for optimization. For the
researcher in multi-objective decision support systems the choice
between many methods is not very easy. Indeed numerous theories were
developed since the forerunners: Condorcet (the Condorcet Paradox,
against binary comparisons, 1785, LVIII), Gossen (law of decreasing
marginal utility, 1853) Minkowski (Reference Point 1896, 1911) and
Pareto (Pareto Optimum and Indifference Curves analysis 1906, 1927) and
pioneers like Kendall (ordinal scales, since 1948), Miller and Starr
(Multiplicative Form 1969), Roy et al. (ELECTRE, since 1966), Hwang and
Yoon (TOPSIS 1981), Brans et al. (PROMETHEE since 1984), Saaty (AHP,
since 1988) and Opricovic, Tzeng (VIKOR 2004; Fouladgar et al. 2012).
Already in 1983 at least 96 methods for Multi-Objective Optimization
existed (Despontin et al. 1983). Since then numerous other methods
appeared. Therefore, we only cited the probably most used methods for
Multi-Objective Optimization.
Twenty two objectives from 27 countries result in 594 data, which
means that methods of partial aggregation can not be used. Scharlig
(1985, 1996) gave the name of partial aggregation to: the Electre Group
(Electre I, Electre Iv, Electre Is, Electre TRI, Electre II, Electre III
and Electre IV) and Promethee.
The Analytic Hierarchy Process (AHP) of Saaty (1988) uses weights.
The use of weights was introduced by Churchman and Ackoff (1954) and
Churchman et al. (1957). This Additive Weighting Procedure (MacCrimmon,
1968: 29-33) was called SAW, Simple Additive Weighting Method, by Hwang
and Yoon (1981: 99). Weights defined in that way for using in
Multi-Objective Optimization have the disadvantage to possess a double
content namely normalization on the one side and giving importance on
the other. In addition, for the European example, 22 weights have to be
defined or 44 normalization and importance exercises.
Also the methods of partial aggregation use weights. Even more, all
these methods are expert oriented with qualitative statements as a
basis. As conclusion one may say that for this application preference
has to be given to a method of total aggregation and a method not using
weights.
The article is organized in the following way. Section 2 deals with
the objectives characterizing the economies of the EU Countries for the
present and for the future. Section 3 focuses on further considerations
on methods for Multi-Objective Optimization. The MOORA method is
explained in section 4 and MULTIMOORA in section 5, both methods
satisfying the conditions of total aggregation and of not using weights.
Section 6 describes the numerical example where the European Union
States are compared on basis of structural indicators and of the new
method. Finally conclusions are drawn: which of these countries are the
best prepared for 2020 from the economic point of view? A Dominance
Theory, summarizing the three obtained ordinal numbers per country,
ranks the 27 countries for that purpose.
2. Choice of objectives characterizing the economies of the EU
countries in the present and in the future
The Choice of Objectives characterizing the Economies of a Country
for the Present and for the Future is restricted to the available data.
Afonso et al. (2011) mention that four variables have short-run impact
on ratings namely the level of GDP per capita, real GDP growth, the
public debt level and the government balance. Government effectiveness,
the level of external debt and external reserves are important long-run
determinants.
The national economies to be studied concern the Economies of the
European Union Member States. We selected twenty-two objectives, ten
originating from statistics and twelve from statistics and forecasts
important for the future, in order to characterize the 27 EU-Countries
economies as shown in the following Table 1.
No. 1, 17 and 21 Government Budget Deficit
The IMF correlates the Government Budget Deficit to GDP for a given
year and not to GNP. Let us remember that the Gross Domestic Product
(GDP) is a territorial concept, the Value Added realized on the
territory of a country during a certain year. On the contrary the Gross
National Product (GNP) is a personal concept related to the citizens and
the permanent residents of a country (1). In fact the Government Budget
Deficit is a part of the yearly Government Budget and rather has to be
related to that budget and not to GDP, otherwise some countries are
punished and others are rewarded. Indeed at the time of the EU 15 the
government budget of Sweden compared to 56.6% of its GDP (Belgian
Federal Department of Finance, Documentatieblad 2008, 46), which means
that for instance a government deficit of 3% of GDP corresponds to a
deficit of 5.3% of its government budget. At the other side Ireland with
34% government expenses compared to its GDP (Belgian Federal Department
of Finance, op.cit. 46) may run a deficit of 8.8% of its government
budget. One could conclude that a country with high government expenses
compared to GDP is punished in that way.
Interesting to know is that the government budget deficit is linked
to the government debt. Indeed the deficit and the increase in the
interest payments on the government debt determine an increase in the
government debt. In this way the government debt is composed of primary
expenditures and of interest payments. Consequently, the impact of the
government on the government debt runs through the budget deficit and
the obtained rates of interest on the government debt.
No. 2, 18 and 22 Government Debt
The International Monetary Fund (IMF) once again relates Government
Debt to Gross Domestic Product. Government Debt is distinct from the
debt of the households, which depends on household expenditures and
savings. In Europe the savings ratio of the households is the highest in
Spain, Germany and Belgium, translated in Belgium of a participation
ratio of the Belgians in their own government budget of around 46%. The
International Monetary Fund does not separate domestic from external
financing for the government debt, though an important issue. Related to
GDP Japan has one of the highest government debts of the world, probably
around 200% of GDP, but mainly financed by the own population. As a
consequence the interest rate on the government debt is extremely low,
sometimes 1% on 10-year government bonds. On the contrary Greece with a
government debt of 142.8% in 2010; estimated at 157.7% in 2011 and at
166.1% in 2012, had to pay an interest rate of 13.2% in 2010 increased
to 16.69% on June 11, 2011 (European Central Bank 2011). The reason is
that around 71% of the Greek government debt is in the hands of
foreigners, mainly foreign banks.
Once the foreign part of government debt is very high it is
difficult to bring it down. Previously in Europe devaluation of the
national currencies was helpful, which is impossible with the Euro,
unless as a function of a basket of the main other currencies in the
world, comparable to the system of drawing rights of the International
Monetary Fund. Otherwise, one has to foresee international transfer
payments or reduction or complete remission of the government debt in
the hands of foreigners. Nevertheless, one has to question the
possibility to foresee a fiscal outlook until 2035 as has been done by
Gagnon (2011). Any way we stopped our forecasts until the end of 2012.
No. 3 Current Account Deficit in the Balance of Payments (BoP)
The Current Account Balance is the sum of net exports of goods,
services, net income, and net current transfers. Once again the IMF
correlates the Current Account Deficit in the Balance of Payments to GDP
and not as a part of the total of the Current Account of the Balance of
Payments for a certain period (2). Nevertheless it may happen that in
typical export oriented countries like the Benelux countries the total
of their Current Account of the Balance of Payments is larger than their
GDP.
No. 4 GDP per Capita in Purchasing Power Parity (PPP)
Purchasing Power Parity means that the GDP per country is not
compared on basis of exchange rates but on basis of the consumer basket
with its prices of a country compared with the consumer basket with its
prices, the purchasing power, of a standard country, namely the United
States. In principle the GDP per Capita in a Purchasing Power Parity
(PPP) would measure the real purchasing power of the consumer in a
country.
No. 5, 16 and 20 GDP Growth Rate
GDP Growth Rate represents the per cent of GDP growth compared to
previous year in constant prices.
No. 6 Inflation
Inflation is expressed in averages for the year, not end-of-period
data. A consumer price index (CPI) measures changes in the prices of
goods and services that households consume. Such changes affect the real
purchasing power of consumer income. As all the prices of different
goods and services do not change at the same rate, a price index can
only reflect their average movement. A price index is assigned a value
compared to some reference period. Price indices can be used to measure
differences in price levels between different countries at the same
point in time. For Euro Countries, consumer prices are calculated based
on harmonized prices.
No. 7 Government Bond Yields
Yields of Government Bonds concern Government bond yields with
maturity close to 10 years.
No. 8 Employment Rate
The Employment Rate is the rate of people employed as a percentage
of the age group 15-64.
No. 9 Unemployment Rate
The Unemployment rate is the rate of people unemployed as a
percentage of the professional population (people disposed to work in
the age group 15-64).
No. 10 Tertiary Educational Attainment
Tertiary Educational Attainment is the rate of people who finished
any form of higher education as a percentage of the age group 30-34.
No. 11 Median Age
Median Age is the median of the ages of the entire population.
No. 12 Proportion of population aged 0-14
Population aged 0-14 is compared to total population.
No. 13 Proportion of population aged 15-64
Population aged 15-64 is compared to total population.
No. 14 Proportion of population aged 65 and over
Population aged 65 and over is compared to total population.
No. 15 and 19 GDP per capita (EU-15 = 100)
GDP per capita is a percentage compared to the GDP per capita with
EU-15 = 100. In the year 2000 the European Union was composed of the
original 15 member states. In 2004 the 10 new countries joined and
finally in 2007 Romania and Bulgaria. In 2013 Croatia will join.
The data for the 22 objectives are given in the following Table 2.
3. Choice of a method for multi-objective optimization
Based on previous analysis, methods of partial aggregation and
methods based on weights are excluded for large decision matrices.
How is it possible to exclude Weights? Therefore it is necessary to
read the decision matrix, in the European example composed of 594
elements, not in the horizontal but in the vertical way. By making
averages per column of the decision matrix dimensionless measures are
obtained.
Reference Point Methods like TOPSIS (Hwang, Yoon 1981) and VIKOR
(Opricovic, Tzeng 2004) do not use weights but rather dimensionless
measures, but they are overtaken by MOORA which is composed of two
different dimensionless based methods, each controlling each other.
Chakraborty (2011) for decision making in manufacturing compared
MOORA to TOPSIS and VIKOR concerning: computational time, simplicity,
mathematical calculations and stability. For all these characteristics
MOORA was superior to TOPSIS and VIKOR.
Later (Brauers, Zavadskas 2010) MULTIMOORA overtook MOORA by
assembling three instead of two methods and, in fact, by assembling all
possible methods with dimensionless measures.
The Ratio System is the first part of MOORA. In MOORA the decision
matrix is read not in the horizontal but in the vertical way. The
obtained ratios are used as inputs in the Reference Point Method where
they are compared with the coordinates of a Reference Point.
In order to make the series of dimensionless measures methods
complete a Full Multiplicative Form is added, whereby its factors loose
their identity and ipso facto become dimensionless.
4. Multi-objective optimization by ratio analysis (MOORA)
4.1. The two parts of MOORA
The method starts with a matrix of responses of different
alternatives on different objectives:
([x.sub.ij])
with: [x.sub.ij] as the response of alternative j on objective i
i = 1, 2, ..., n as the objectives. j = 1, 2, ..., m as the
alternatives.
MOORA goes for a ratio system in which the response of an
alternative on an objective is compared to a denominator, which is
representative for all alternatives concerning that objective.
If the sum of each alternative per objective is chosen for this
denominator, the traditional formula of averages is obtained:
[x.sup.o.sub.ij] = [x.sub.ij]/[m.summation over (j=1)][x.sub.ij]
(1)
This formula may lead to unexpected results. Indeed, for instance
in the case of productivity growth some sectors, regions or countries
may show a decrease instead of an increase in productivity i.e. a
negative number. In this way the sum in the denominator could become
negative and ipso facto all ratios become negative. Even worse the
denominator can become zero and division by zero means a senseless
operation.
Brauers, Zavadskas (2006) still tested eight other formulas. They
all had main disadvantages with exception of the formula where for the
denominator the square root of the sum of squares of each alternative
per objective was chosen:
[x.sup.*.sub.ij] = [x.sub.ij]/[square root of [m.summation over
(j=1)][x.sup.2.sub.ij]] (2)
with: [x.sub.ij] = response of alternative j on objective i.
j = 1, 2, ..., m; m the number of alternatives.
i = 1, 2, ... n; n the number of objectives.
[x.sup.*.sub.ij] = a dimensionless number representing the response
of alternative j on objective i.
Dimensionless Numbers, having no specific unit of measurement, are
obtained for instance by deduction, multiplication or division. The
normalized responses of the alternatives on the objectives belong to the
interval [0; 1]. However, sometimes the interval could be [-1; 1].
Indeed, for instance, in the case of productivity growth some sectors,
regions or countries may show a decrease instead of an increase in
productivity i.e. a negative dimensionless number.
For optimization these responses are added in case of maximization
and subtracted in case of minimization:
[y.sup.*.sub.j] = [i=g.summation over (i=1)][x.sup.*.sub.ij] -
[i=n.summation over (i=g+1)][x.sup.*.sub.ij], (3)
with: i = 1, 2, ..., g as the objectives to be maximized.
i = g + 1, g + 2, ..., n as the objectives to be minimized.
[y.sup.*.sub.j] as the assessment of alternative j with respect to
all objectives.
An ordinal ranking in a descending order of the [y.sup.*.sub.j]
shows the final preference.
For the second part of MOORA the Reference Point Theory is chosen
with the Min-Max Metric of Tchebycheff as given by the following formula
(Karlin, Studden 1966: 280):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
with [absolute value of [r.sub.i] - [x.sup.*.sub.ij]] the absolute
value if [x.sup.*.sub.ij] is larger than [r.sub.i] for instance by
minimization.
This reference point theory starts from the already normalized
ratios as defined in the MOORA method, namely formula (1). Preference is
given to a reference point possessing as co-ordinates the dominating
co-ordinates per attribute of the candidate alternatives and which is
designated as the Maximal Objective Reference Point. This approach is
called realistic and non-subjective as the co-ordinates, which are
selected for the reference point, are realized in one of the candidate
alternatives. The alternatives A (10; 100), B (100; 20) and C (50; 50)
will result in the Maximal Objective Reference Point [R.sub.m] (100;
100).
Given the composition of equation (4) the results are ranked in an
ascending order.
4.2. The Importance given to an objective by the attribution method
in MOORA
It may look that one objective can not be much more important than
another one as all their ratios are smaller than one (see formula 2)
Nevertheless it may turn out to be necessary to stress that some
objectives are more important than others. In order to give more
importance to an objective its ratios could be multiplied with a
Significance Coefficient.
In the Ratio System in order to give more importance to an
objective its response on an alternative under the form of a
dimensionless number could be multiplied with a Significance
Coefficient:
[[??].sup.*.sub.j] = [i=g.summation over
(i=1)][s.sub.i][x.sup.*.sub.ij] - [i=n.summation over
(i=g+1)][s.sub.i][x.sup.*.sub.ij], (5)
with: i = 1, 2,., g as the objectives to be maximized.
i = g + 1, g + 2,..., n as the objectives to be minimized.
[s.sub.i] = the significance coefficient of objective i.
[[??].sup.*.sub.j] = the total assessment with significance
coefficients of alternative j with respect to all objectives.
For the Reference Point Approach the place of the significance
coefficient would be:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The Attribution of Sub-Objectives represents another solution. Take
the example of the purchase of fighter planes (Brauers 2002). For
economics, the objectives concerning the fighter planes are threefold:
price, employment and balance of payments, but also there is military
effectiveness. In order to give more importance to military defense,
effectiveness is broken down in, for instance, the maximum speed, the
power of the engines and the maximum range of the plane. Anyway, the
Attribution Method is more refined than a significance coefficient
method could be as the attribution method succeeds in characterizing an
objective better. For instance, for employment two sub-objectives
replace a significance coefficient of two and in this way characterize
the direct and indirect side of employment separately.
4.3. The importance given to an objective by the attribution method
in the application for the economies of the EU-countries
After the given military example the importance given to an
objective in the application for the Economies of the EU-countries is
not done by the introduction of significance coefficients but by
sub-objectives. However, the sub-objectives are already present but are
depending on a super objective:
The Super Objective of Economic Importance represented by 10
objectives:
1. Current account deficit of the Balance of Payments (2010)
2. GDP per capita in Purchasing Power Parity (2010)
3. GDP growth rate (2010)
4. Inflation (2010)
5. Employment rate (2010)
6. Unemployment rate (2010)
7. GDP per capita index number with basis EU-15 = 100 (2011)
8. GDP growth rate (2011)
9. GDP per capita index number with basis EU-15 = 100 (2012)
10. GDP growth rate (2011)
The Super Objective of Public Finance represented by 7 objectives:
1. Government Budget Deficit (2010)
2. Government Debt (2010)
3. Government bond yields (2010)
4. Government Budget Deficit (2011)
5. Government Debt (2011)
6. Government Budget Deficit (2012)
7. Government Debt (2012)
The Super Objective of the Population Pyramid represented by 5
objectives:
1. Tertiary education (2010)
2. Median Age (2010)
3. Proportion of population aged 0-14 (2010)
4. Proportion of population aged 15-64 (2010)
5. Proportion of population aged 65 or more (2010)
5. MULTIMOORA
As MOORA is based on dimensionless measures why not go further by
adding the remaining form which uses dimensionless measures namely the
Multiplicative Form? We even prefer to speak of the
"Full-Multiplicative Form". Otherwise it could refer to a
combination with linearity.
The following n-power form for multi-objectives is called from now
on a Full-Multiplicative Form:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
with: j = 1, 2, ..., m; m the number of alternatives.
i = 1, 2, ..., n; n being the number of objectives.
[x.sub.ij] = response of alternative j on objective i.
[U.sub.j] = overall utility of alternative j.
The overall utilities ([U.sub.j]), obtained by multiplication of
different units of measurement, become dimensionless. The outcome of
this presentation is nonlinear, which presents an advantage, as the
utility function of human behavior toward several objectives has to be
considered as nonlinear.
Rule
In the full-multiplicative form the relation between the utilities
U does not change if more importance is given to an objective by
multiplying it by a factor. Indeed, at that moment all alternatives are
multiplied with that factor.
Consequence 1
In the full-multiplicative form the introduction of weights is
meaningless. Indeed weights are here, in fact, multiplying coefficients.
Consequence 2
In the full-multiplicative form an attribute of the size 10, 102,
103, 106, 109 etc. can be replaced by the unit size without changing the
relationship between the utilities of the alternatives.
This consequence is extremely important for attributes expressed in
monetary units. Instead of expressing an attribute in tens, hundreds,
thousands, millions, billions, for instance, of dollars, the use of one
digit in the integer part is sufficient.
Is it possible to give more importance to an objective? Allocating
an exponent to an objective signifies stressing the importance of this
objective.
How is it possible to combine a minimization problem with the
maximization of the other objectives? Therefore, the objectives to be
minimized are denominators in the formula:
[U'.sub.j] = [A.sub.j]/[B.sub.j], (7)
with: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
j = 1, 2, ..., m; m the number of alternatives;
i as the number of objectives;
g as the number of objectives to be maximized;
with: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
n-g = the number of objectives to be minimized;
with: [U'.sub.j] the utility of alternative j with objectives
to be maximized and objectives to be minimized.
In the Full Multiplicative Form a problem may arise for zero and
negative values making the results senseless. Therefore the index number
100 replaces the zero number. At that moment for instance 96.6
substitutes the negative value of minus 3.4. Consequently, 103.4
represents the positive value of 3.4.
Summarizing, one can conclude that the MULTIMOORA method is
composed of three parts: the Ratio System, the Reference Point Method
and the Full Multiplicative Form. The three methods were already
discussed but separately by Brauers (2004), whereas the name of
MULTIMOORA was given by Brauers and Zavadskas in 2010.
6. MULTIMOORA as applied on the multi-objective optimization of the
economies of the EU Member countries
The initial data were translated in dimensionless ratios according
to Eqs. 2 and 3, i.e. the Ratio System of MOORA. Subsequently Eq. 4 used
the ratios obtained in Eq. 2 to calculate the distances to the Reference
Point of MOORA. Finally, the Full Multiplicative Form used the initial
data to rank the Member States according to Eq. 6 and 7. Following Table
3 presents the results of multi-objective optimization.
7. The theory of dominance
How to make a synthesis between the results of the three
approaches: Ratio System, Reference Point Method, which uses the ratios
obtained in the ratio system as coordinates, and the Full Multiplicative
Form?
The use of the Rank Correlation Method is misleading, see Appendix
A. Instead Brauers et al. developed a Theory of Dominance (Brauers,
Zavadskas 2011).
7.1. Axioms on ordinal and cardinal scales
1. A deduction of an Ordinal Scale, a ranking, from cardinal data
is always possible (Arrow 1974).
2. An Ordinal Scale can never produce a series of cardinal numbers
(Arrow).
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 ordinal scales of
the three methods of MULTIMOORA in another one based on Dominance, being
Dominated, Transitivity and Equability.
7.2. Dominance, being dominated, transitiveness and equability
Stakeholders or their representatives may give a different
importance to objectives in a multi-objective problem, but this is not
the case with the three methods of MULTIMOORA. These three methods
represent all possible methods with dimensionless measures in
multi-objective optimization and one can not argue that one method is
better than or is of more important than the other ones.
Dominance
Absolute Dominance means that an alternative, solution or project
is dominating 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 with a P b P c P d (P
preferred to) is for instance of the form:
(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 on transitiveness plays fully.
Transitiveness
If a dominates b and b dominates c than a will also dominate c.
Overall Dominance of one alternative on another
For instance (a-a-a) is overall dominating (b-b-b) which is being
overly dominated by (a-a-a).
Equability
Absolute Equability has the form: for instance (e-e-e) for 2
alternatives.
Partial Equability of 2 on 3 exists e. g. (5-e-7) and (6-e-3).
Circular Reasoning
Despite all distinctions in classification some contradictions
remain possible in a kind of Circular Reasoning.
We can cite the case of:
Object A (11-20-14) dominates generally Object B (14-16-15).
Object B (14-16-15) dominates generally Object C (15-19-12)
but Object C (15-19-12) dominates generally Object A (11-20-14).
In such a case the same ranking is given to the three objects.
The same rules apply for the three methods of MULTIMOORA with no
significance coefficients proposed as the three methods are considered
to have the same importance.
In the following Table 4 the final classification of MULTIMOORA is
presented.
If there is General Dominance in two of the three methods the
domination is indicated with a large bold figure.
7.3. Some special remarks
Overall Dominance of one Alternative on another is noted in the
following cases:
-- The Netherlands on Denmark
-- Poland on Cyprus
-- United Kingdom on Italy and Romania
-- Italy on Romania
-- Portugal on Greece.
The group of ten which only joined the EU in 2004 is doing quite
well and especially these countries: Estonia (which even joined the EURO
Group), Slovakia, Slovenia, Lithuania, and the Czech Republic.
The so called PIIGS are indeed the last classified together with
Romania, but also the United Kingdom is not very well classified.
8. Conclusion
We prefer to estimate the economic worth of the European Union
Member States by Multi-Objective Optimization towards 2020. Therefore 22
objectives with an actual and future outlook were selected to
characterize each EU Member State.
Next problem was the choice of an effective method of
Multi-Objective Optimization. This method has to use complete and not
partial aggregation, as an overall view of the countries is needed, and
has to avoid the use of weights, being dual on normalization and
importance. Therefore methods based on dimensionless measures are
preferred. Responding to all these conditions MULTIMOORA was finally
chosen. In addition MULTIMOORA is composed of three approaches, each
controlling each other. In this way all possible methods based on
dimensionless measures are included.
Having the results of the three approaches, Ratio Analysis System,
Reference Point Approach and Full Multiplicative Form, the problem
remains how to come to a final and unique solution. For that purpose the
correlation of ranks is senseless. A Theory of Dominance is rather
preferred.
The final results classify Sweden, first followed by Luxemburg,
Finland, the Netherlands and Denmark. Some of the ten countries, which
joined the EU in 2004, are doing quite well, leaded by Estonia, which
even joined the EMU. As expected the PIIGS countries are classified at
the bottom, but joined by an unforeseen United Kingdom. In this way we
have an idea how the European countries are advancing on economic terms
to the European Strategy for 2020.
On basis of the outcomes it would perhaps be possible, eventually
with more available data, to come to ratings comparable to the Credit
Rating Agencies ratings. Further investigation in this sense could be
useful.
APPENDIX A
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: 1): "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: 1) goes too far (Table A1).
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 different items in a same order 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 is
correlation measured in other situations. Therefore, the following
Spearman's coefficient is used (Kendall 1948: 8):
[rho] = 1 - [6[summation][D.sup.2]/N([N.sup.2] - 1)] (A1)
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 A2).
This table shows that the sum of ranks in the case of an ordinal
scale has no sense. Correlation leads to:
[rho] = 1 - [[6 x 112]/7(49 - 1)] = -1.
However, as addition of ranks is not allowed, also a subtraction,
the difference D, is not permitted, nor the multiplication or the
division.
Most people will understand the ordinal problem better 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.
Arbitrary Methods to go from an Ordinal Scale to a Cardinal Scale
1) Arithmetical Progression: 1, 2, 3, 4, 5 ...
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.0] = 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.0] = 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 1965) 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. They all stress
acceleration or dis-acceleration processes but are not aware of a
possible trend break. The full multiplicative method with its huge
numbers illustrates this trend break the best as shown in next Table A3.
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.
Table A1. Ordinal versus cardinal:
comparing the price of one commodity 1
Ordinal Cardinal
1
2
3
4
A 5 6.03$
6 6.02$
7 6.01$
B 8 6$
Table A2. Negative rank order correlations
Items Expert 1 Expert 2 D [D.sup.2]
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
-
[summation] 112
Table A3. Ranking of scenarios for the Belgian regions by the
Full-Multiplicative Method at the year 1996
1 Scenario IX Optimal Economic Policy in Wallonia 203,267
and Brussels
2 Scenario X Optimal Economic Policy in Wallonia 196,306
and Brussels even agreeing on the
Partition of the National Public Debt
3 Scenario VII Flanders asks for the Partition of 164,515
the National Public Debt
4 Scenario VIII No Solidarity at all 158,881
5 Scenario II Unfavorable Growth Rate for Flanders 90
6 Scenario IV an Unfavorable Growth Rate for 87
Flanders and at that moment asks
also for the Partition of the
National Public Debt
7 Scenario III Partition of the National Public Debt 54
8 Scenario I the Average Belgian 51
9 Scenario V Average Belgian but as compensation 49
Flanders asks for the Partition
of the National Public Debt
10 Scenario O Status Quo 43
11 Scenario VI Flanders asks for the Partition of 42
the National Public Debt
Source: Brauers, Ginevicius (2010)
doi: 10.3846/20294913.2012.734692
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Willem Karel M. Brauers (1), Alvydas Balezentis (2), Tomas
Balezentis (3)
(1) Vilnius Gediminas Technical University, Lithuania and Faculty
of Applied Economics, University of Antwerp, Prinsstraat 13, 2000,
Belgium
(2) Mykolas Romeris University, Ateities g. 20, LT-08303 Vilnius,
Lithuania
(3) Lithuanian Institute of Agrarian Economics, V. Kudirkos g. 18,
LT-03105, Vilnius
E-mails: (1) willem.brauers@ua.ac.be (corresponding author); (2)
a.balezentis@gmail.com; (3) t.balezentis@gmail.com
Received 10 December 2010; accepted 02 January 2012
(1) Also the Treaty of Maastricht, the starting point of the EURO,
correlates the Government Budget Deficit to GDP (Belgian Federal
Department of Finance, Documentatieblad 2008, 29).
(2) Also the World Bank correlates the Current Account Deficit in
the Balance of Payments to GDP (World Bank 2011).
Willem K. M. BRAUERS. Doctor honoris causa Vilnius Gediminas
Technical University, was graduated as: Ph.D. in economics (Un. of
Leuven), Master of Arts (in economics) of Columbia Un. (New York),
Master in Economics, Master in Management and Financial Sciences, Master
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 in English, Deutch and
French.
Alvydas BALEZENTIS. PhD (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 specialist at the Lithuanian Institute of
Agrarian Economics. 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, fuzzy logics, benchmarking
methods.
Table 1. The 22 objectives characterizing the Economies
of the European Union Member States
No. Indicator Period
A. Actual
1 Government Budget Deficit (a) 2010
2 Government Debt (a) 2010
3 Current account deficit (BoP) (a) 2010
4 GDP per capita in PPP (a) 2010
5 GDP growth rate (a) 2010
6 Inflation (a) 2010
7 Government bond yields (a) 2010
8 Employment rate (b) 2010
9 Unemployment rate (b) 2010
10 Tertiary education (b) 2010
B. Prospective
11 Median Age (b) 2010
12 Proportion of population aged 0-14 (b) 2010
13 Proportion of population aged 15-64 (b) 2010
14 Proportion of population aged 65 2010
and over (b)
15 GDP per capita (EU-15 = 100) (c) 2011
16 GDP growth rate (c) 2011
17 Government Budget Deficit (c) 2011
18 Government consolidated gross debt (c) 2011
19 GDP per capita (EU-15 = 100) (c) 2012
20 GDP growth rate (c) 2012
21 Government Budget Deficit (c) 2012
22 Government consolidated gross debt (c) 2012
No. Indicator Dimension
A. Actual
1 Government Budget Deficit (a) % of GDP
2 Government Debt (a) % of GDP
3 Current account deficit (BoP) (a) % of GDP
4 GDP per capita in PPP (a) in current
international dollar
5 GDP growth rate (a) per cent of GDP of
previous year
6 Inflation (a) per cent compared to
a reference period
7 Government bond yields (a) per cent
8 Employment rate (b) per cent of age
group 15-64
9 Unemployment rate (b) per cent of the
professional
population
10 Tertiary education (b) per cent of age
group 30-34
B. Prospective
11 Median Age (b) Median of total
population
12 Proportion of population aged 0-14 (b) per cent of total
population
13 Proportion of population aged 15-64 (b) per cent of total
population
14 Proportion of population aged 65 per cent of total
and over (b) population
15 GDP per capita (EU-15 = 100) (c) index number with
EU-15 = 100
16 GDP growth rate (c) % of GDP of previous
year
17 Government Budget Deficit (c) % of GDP
18 Government consolidated gross debt (c) % of GDP
19 GDP per capita (EU-15 = 100) (c) index number with
EU-15 = 100
20 GDP growth rate (c) per cent of GDP of
previous year
21 Government Budget Deficit (c) % of GDP
22 Government consolidated gross debt (c) % of GDP
No. Indicator Optimum
A. Actual
1 Government Budget Deficit (a) MIN
2 Government Debt (a) MIN
3 Current account deficit (BoP) (a) MIN
4 GDP per capita in PPP (a) MAX
5 GDP growth rate (a) MAX
6 Inflation (a) MIN
7 Government bond yields (a) MIN
8 Employment rate (b) MAX
9 Unemployment rate (b) MIN
10 Tertiary education (b) MAX
B. Prospective
11 Median Age (b) MIN
12 Proportion of population aged 0-14 (b) MAX
13 Proportion of population aged 15-64 (b) MAX
14 Proportion of population aged 65 MIN
and over (b)
15 GDP per capita (EU-15 = 100) (c) MAX
16 GDP growth rate (c) MAX
17 Government Budget Deficit (c) MIN
18 Government consolidated gross debt (c) MIN
19 GDP per capita (EU-15 = 100) (c) MAX
20 GDP growth rate (c) MAX
21 Government Budget Deficit (c) MIN
22 Government consolidated gross debt (c) MIN
(a) The data come from IMF and World Bank (2011).
(b) EUROSTAT (2011).
(c) European Commission (2011).
Table 2. Data for the 22 objectives for the European Union Member
States (2010-2012)
1 2 3 4 5 6 7
MIN MIN MIN MAX MAX MIN MIN
Belgium 4.1 96.8 -1.04 32,600 2.2 2.3 3.9
Bulgaria 3.2 16.2 0.99 4,800 0.2 3.0 5.5
Czech R. 4.7 38.5 3.84 14,200 2.3 1.2 3.8
Denmark 2.7 43.6 -5.48 42,200 1.7 2.2 2.9
Germany 3.3 83.2 -5.71 30,300 3.7 1.2 2.8
Estonia -0.1 6.6 -3.76 10,700 2.3 2.7 5.4
Ireland 32.4 96.2 0.71 34,900 -0.4 -1.6 9.2
Greece 10.5 142.8 10.5 20,100 -4.5 4.7 13.2
Spain 9.2 60.1 4.56 23,100 -0.1 2.0 5.1
France 7 81.7 1.74 29,800 1.5 1.7 3.3
Italy 4.6 119 3.29 25,600 1.3 1.6 4.7
Cyprus 5.3 60.8 7.74 21,600 1.0 2.6 5.0
Latvia 7.7 44.7 -3.58 8,000 -0.3 -1.2 7.0
Lithuania 7.1 38.2 -1.85 8,400 1.3 1.2 5.1
Luxemburg 1.7 18.4 -7.82 79,500 3.5 2.8 3.1
Hungary 4.2 80.2 -2.06 9,700 1.2 4.7 7.3
Malta 3.6 68.0 4.17 14,800 2.7 2.0 4.4
Netherlands 5.4 62.7 -7.16 35,400 1.8 0.9 3.1
Austria 4.6 72.3 -2.71 34,100 2.3 1.7 3.3
Poland 7.9 55.0 4.47 9,300 3.8 2.7 5.9
Portugal 9.1 93.0 9.88 16,200 1.3 1.4 8.4
Romania 6.4 30.8 4.06 5,700 -1.3 6.1 7.1
Slovenia 5.6 38.0 0.84 17,300 1.4 2.1 4.3
Slovakia 7.9 41.0 3.44 12,100 4.0 0.7 4.2
Finland 2.5 48.4 -3.09 33,600 3.6 1.7 3.1
Sweden 0 39.8 -6.26 37,000 5.7 1.9 3.0
U. K. 10.4 80.0 3.18 27,400 1.4 3.3 3.3
8 9 10 11 12 13 14 15
MAX MIN MAX MIN MAX MAX MIN MAX
Belgium 62.0 8.3 44.4 40.9 16.9 65.9 17.2 106
Bulgaria 59.7 10.2 27.7 41.4 13.6 68.9 17.5 40.5
Czech R. 65.0 7.3 20.4 39.4 14.2 70.6 15.2 75
Denmark 73.4 7.4 47.0 40.5 18.1 65.6 16.3 110
Germany 71.1 7.1 29.8 44.2 13.5 65.8 20.7 109
Estonia 61.0 16.9 40.0 39.5 15.1 67.8 17.1 60.6
Ireland 60.0 13.7 49.9 34.3 21.3 67.4 11.3 111
Greece 59.6 12.6 28.4 41.7 14.4 66.7 18.9 75.9
Spain 58.6 20.1 40.6 39.9 14.9 68.3 16.8 91
France 64.0 9.8 43.5 39.8 18.5 64.9 16.6 97.2
Italy 56.9 8.4 19.8 43.1 14.1 65.7 20.2 92.8
Cyprus 69.7 6.3 45.1 36.2 16.9 70.0 13.1 88
Latvia 59.3 18.7 32.3 40.0 13.8 68.8 17.4 47.6
Lithuania 57.8 17.8 43.8 39.2 15.0 68.9 16.1 52.7
Luxemburg 65.2 4.5 46.1 38.9 17.7 68.3 14.0 248
Hungary 55.4 11.2 25.7 39.8 14.7 68.7 16.6 59.7
Malta 56.0 6.9 18.6 39.2 15.6 69.6 14.8 75
Netherlands 74.7 4.5 41.4 40.6 17.6 67.1 15.3 119
Austria 71.7 4.4 23.5 41.7 14.9 67.5 17.6 114
Poland 59.3 9.6 35.3 37.7 15.2 71.3 13.5 57.6
Portugal 65.6 12.0 23.5 40.7 15.2 66.9 17.9 70
Romania 58.8 7.3 18.1 38.3 15.2 69.9 14.9 40.9
Slovenia 66.2 7.3 34.8 41.4 14.0 69.5 16.5 79.6
Slovakia 58.8 14.4 22.1 36.9 15.3 72.4 12.3 69.3
Finland 68.1 8.4 45.7 42.0 16.6 66.4 17.0 105
Sweden 72.7 8.4 45.8 40.7 16.6 65.3 18.1 114
U. K. 69.5 7.8 43.0 39.6 17.5 66.0 16.5 101
16 17 18 19 20 21 22
MAX MIN MIN MAX MAX MIN MIN
Belgium 2.4 3.7 97 106 2.2 4.2 97.5
Bulgaria 2.8 2.7 18 41.6 3.7 1.6 18.6
Czech R. 2 4.4 41.3 76.1 2.9 4.1 42.9
Denmark 1.7 4.1 45.3 109 1.5 3.2 47.1
Germany 2.6 2 82.4 110 1.9 1.2 81.1
Estonia 4.9 0.6 6.1 62 4 2.4 6.9
Ireland 0.6 10.5 112 111 1.9 8.8 118
Greece -3.5 9.5 158 75.5 1.1 9.3 166
Spain 0.8 6.3 68.1 90.7 1.5 5.3 71
France 1.8 5.8 84.7 97.2 2 5.3 86.8
Italy 1 4 120 92.3 1.3 3.2 120
Cyprus 1.5 5.1 62.3 88.1 2.4 4.9 64.3
Latvia 3.3 4.5 48.2 49.2 4 3.8 49.4
Lithuania 5 5.5 40.7 54.8 4.7 4.8 43.6
Luxemburg 3.4 1 17.2 251 3.8 1.1 19
Hungary 2.7 -1.6 75.2 60.6 2.6 3.3 72.7
Malta 2 3 68 75.2 2.2 3 67.9
Netherlands 1.9 3.7 63.9 118 1.7 2.3 64
Austria 2.4 3.7 73.8 114 2 3.3 75.4
Poland 4 5.8 55.4 58.8 3.7 3.6 55.1
Portugal -2.2 5.9 102 67.8 -1.8 4.5 107
Romania 1.5 4.7 33.7 41.9 3.7 3.6 34.8
Slovenia 1.9 5.8 42.8 80.2 2.5 5 46
Slovakia 3.5 5.1 44.8 71.2 4.4 4.6 46.8
Finland 3.7 1 50.6 106 2.6 0.7 52.2
Sweden 4.2 -0.9 36.5 115 2.5 -2 33.4
U. K. 1.7 8.6 84.2 101 2.1 7 87.9
Table 3. The three Approaches of MULTIMOORA as applied for the
Economy of the EU Member States (a)
Member States Ratio S. Ref. Point Full Multipl. F.
1 Belgium -0.07166 0.3196606 4182584.6
2 Bulgaria -0.17729 0.5091396 310541.0781
3 Czech Republic -0.11831 0.4534052 2039033.258
4 Denmark 0.865113 0.2549106 88764673141
5 Germany 0.373762 0.335337 65568553083
6 Estonia 0.172228 0.4689265 1.05313E+12
7 Ireland -0.50645 0.7270675 12751986265
8 Greece -1.1709 0.710736 68949.84482
9 Spain -0.38842 0.4814092 582713.4039
10 France 0.11367 0.3719685 18481324.78
11 Italy -0.33378 0.4322718 3849003.582
12 Cyprus -0.0528 0.6053159 3072201.537
13 Latvia -0.3024 0.4873291 9.29662E+11
14 Lithuania -0.16221 0.4846028 1715231781
15 Luxemburg 1.609567 0.5091396 1.45714E+11
16 Hungary -0.26461 0.4757423 1046439986
17 Malta -0.21317 0.466408 2143188.821
18 Netherlands 0.75226 0.3005764 1.9199E+11
19 Austria 0.378763 0.3094369 49788752164
20 Poland -0.33806 0.4784686 419639.2501
21 Portugal -0.83554 0.6884153 256512.0942
22 Romania -0.38694 0.5030054 42127.3994
23 Slovenia 0.054059 0.4239422 7002669.453
24 Slovakia -0.29411 0.4593844 1376975.646
25 Finland 0.567699 0.3128448 70281186108
26 Sweden 1.716535 0.2896712 1.53242E+14
27 United Kingdom -0.00365 0.427989 4160130.904
Member States Rank RS Rank RP Rank MF
1 Belgium 12 6 10
2 Bulgaria 13 22 16
3 Czech Republic 11 13 15
4 Denmark 6 3 8
5 Germany 7 8 6
6 Estonia 4 16 2
7 Ireland 25 26 23
8 Greece 27 27 27
9 Spain 24 19 25
10 France 18 9 18
11 Italy 22 12 22
12 Cyprus 19 24 20
13 Latvia 15 21 12
14 Lithuania 10 20 11
15 Luxemburg 1 7 3
16 Hungary 20 17 5
17 Malta 17 15 17
18 Netherlands 5 2 7
19 Austria 8 4 9
20 Poland 16 18 19
21 Portugal 26 25 26
22 Romania 23 23 24
23 Slovenia 14 10 14
24 Slovakia 9 14 13
25 Finland 3 5 4
26 Sweden 2 1 1
27 United Kingdom 21 11 21
(a) The details of the calculations can eventually be given
by the authors. Table 2 forms the basis of these calculations.
Table 4. The final classification of MULTIMOORA for the
Economies of the EU Countries
MULTIMOORA Countries Rank RS Rank RP Rank MF
1 Sweden 2 1# 1#
2 Luxemburg 1# 7 3#
3 Finland 3# 5 4#
4 Netherlands 5# 2# 7#
5 Denmark 6# 3# 8
6 Germany 7# 8 6#
7 Austria 8# 4 9#
8 Estonia 4# 16 2#
9 Belgium 12 6# 10#
10 Slovakia 9# 14 13#
11 Slovenia 14 10# 14#
12 Lithuania 10# 20 11#
13 Czech Republic 11# 13 15#
14 France 18# 9# 18
15 Hungary 20 17# 5#
16 Latvia 15# 21 12#
17 Malta 17# 15# 17
18 Bulgaria 13# 22 16#
19 Poland 16# 18# 19#
20 Cyprus 19# 24 20#
21 U. K. 21# 11# 21#
22 Italy 22# 12# 22#
23 Romania 23# 23 24#
24 Spain 24# 19# 25
25 Ireland 25# 26 23#
26 Portugal 26# 25# 26#
27 Greece 27 27 27
If there is General Dominance in two of the three methods
the domination is indicated with a large bold figure.
Note: If there is General Dominance in two of the three methods
the domination is indicated with a #.
