590 European J. International Management, Vol. 6, No. 5, 2012

Copyright © 2012 Inderscience Enterprises Ltd.

Decision making in the European Union under risk and uncertainty

José M. Merigó* Department of Business Administration, University of Barcelona, Av. Diagonal 690, 08034 Barcelona, Spain Email: jmerigo@ub.edu *Corresponding author

Clemente Lobato-Carral and Agustín Carrilero-Castillo Department of Business Administration, Universitat Politècnica Valencia, Camino Vera s/n, 46022 Valencia, Spain Email: clelocar@doctor.upv.es Email: agustincarrilero@gmail.com

Abstract: We study the use of aggregation operators in decision making at a European level. We introduce the induced probabilistic ordered weighted averaging – weighted averaging operator. It is a new aggregation operator that provides a unified framework between the probability, the weighted average and the ordered weighted averaging operator considering the degree of importance that each concept has in the aggregation. Moreover, this approach is able to deal with complex reordering processes by using order-inducing variables in order to assess complex attitudinal characters of the decision maker. We study some of its main properties and particular cases. We develop an application of the new approach in a European Union group decision making problem regarding the selection of the optimal inflation rate in the convergence criteria to be accomplished by the Members of the European Union.

Keywords: European Union; aggregation operators; OWA operator; group decision making; probability; weighted average; European Union decision making.

Reference to this paper should be made as follows: Merigó, J.M., Lobato-Carral, C. and Carrilero-Castillo, A. (2012) ‘Decision making in the European Union under risk and uncertainty’, European J. International Management, Vol. 6, No. 5, pp.590–609.

Biographical notes: José M. Merigó is an Assistant Professor at the Department of Business Administration in the University of Barcelona. He has a MSc and a PhD degree in Business Administration from University of Barcelona and a Bachelor Degree in Economics and a Master Degree in European Business Administration and Business Law from Lund University, Sweden. He has published more than 200 papers in journals, books and conference proceedings including journals such as Information Sciences,

Decision making in the European Union 591

Technological and Economic Development of Economy and International Journal of Intelligent Systems. He is currently interested in aggregation operators, decision making and uncertainty.

Clemente Lobato-Carral is a Part-time Professor at the Department of Business Administration in the Polytechnic University of Valencia (UPV). He has a PhD degree in Business Administration from UPV, a Bachelor degree in Industrial Organisation Engineering from UPV (Spain) and a Bachelor of Science in Mechanical Engineering from University of Teesside (England). His Doctorate’s specialisation is based in strategic tools applied to companies. He alternates his career as professor with senior executive positions in companies dedicated to logistics and transformation of the steel.

Agustín Carrilero-Castillo is a Professor of Service and Sectorial Marketing at ESIC Business and Marketing School, graduated in Business Administration from the University of Valencia. He has an MSc degree in Marketing and Commercial Management from ESIC and a PhD degree from the Polytechnic University of Valencia. He has held positions of senior executive in different sectors. At the moment, he is publishing articles in journals as well as national and international conferences, and he is collaborating in the writing of a book.

1 Introduction

Decision making models are very useful in a lot of disciplines (Figueira et al., 2005; Zavadskas and Turskis, 2011). Some of the most common techniques for dealing with decision making are the probability and the weighted average. Usually, they represent some kind of belief that the decision maker has about the problem, being the probability the objective perspective and the weighted average the subjective perspective. Another useful tool for decision making is the Ordered Weighted Average (OWA) (Yager, 1988). It provides a parameterised family of aggregation operators between the maximum and the minimum. Thus, it is able to underestimate or overestimate the initial results of the decision maker according to a degree of optimism or pessimism. The OWA operator can be extended by using order inducing variables that represent complex reordering processes forming the Induced OWA (IOWA) operator (Yager and Filev, 1999; Yager, 2003). Note that since its introduction the OWA and the IOWA operator have been studied by a lot of authors. For example, Beliakov et al. (2007), Xu and Da (2003), Yager and Kacprzyk (1997) and Yager et al. (2011) have provided complete overviews of the state of the art of the OWA operator. Merigó and Casanovas (2011a) and Merigó and Gil-Lafuente (2009) provided a general framework when dealing with induced aggregation operators. Merigó and Gil-Lafuente (2010) studied the use of distance measures in the OWA operator, Merigó and Casanovas (2011b) in the IOWA operator and Yager (2010) the use of norms. Wei (2011) analysed the use of harmonic means in the IOWA operator with uncertain environments assessed with fuzzy numbers. Chen and Zhou (2011) and Wu et al. (2009) developed an induced continuous aggregation operator in group decision making problems. Merigó and Gil-Lafuente (2011a) developed a fuzzy multi-person decision making process with induced aggregation operators. Tan and Chen (2010) studied the use of the Choquet integral in induced generalised aggregation operators and Xu and Xia (2011) the use of intuitionistic fuzzy sets.

592 J.M. Merigó, C. Lobato-Carral and A. Carrilero-Castillo

Several studies have tried to use the weighted average and the probability in the OWA operator. For example, Torra (1997) developed a first model for using OWAs and weighted averages in the same formulation called the Weighted OWA (WOWA) operator. Later, Xu and Da (2003) developed another approach called the hybrid averaging operator and recently Merigó (2011a) has introduced the Induced OWA Weighted Average (IOWAWA). Some other studies have focused on the unification between the probability and the OWA in the same formulation including the concept of immediate probabilities (Yager et al., 1995; Engemann et al., 1996; Merigó, 2010) and the Probabilistic OWA (POWA) operator (Yager et al., 1995; Merigó, 2011b; Merigó and Wei, 2011).

In this paper, we introduce the Induced Probabilistic Ordered Weighted Averaging – Weighted Average (IPOWAWA) operator. It is a new aggregation operator that provides a unified framework between the probability, the weighted average and the OWA operator considering the degree of importance that each concept has in the aggregation. Moreover, this approach is able to deal with complex reordering processes by using order-inducing variables in order to assess complex attitudinal characters of the decision maker. We study some of its main properties and particular cases including the Induced Probabilistic OWA (IPOWA) operator, the IOWAWA operator, the PWA operator (Merigó, 2012), the IOWA operator, the Max-PWA and the Min-PWA operator. We also study some key measures with the IPOWAWA operator such as the entropy of dispersion, the variance, the covariance and a simple regression model.

We study its applicability and we see that it is very broad. We give special attention to European Union (EU) decision making problems. We analyse a wide range of situations such as decision making in EU law, EU politics, EU economics and EU science. We see that these decisions are very important because they involve high quantities of money. Thus, it becomes necessary the use of group decision making techniques in order to assess the information in a more complete way so we can reach the optimal decisions and obtain the highest benefits. For doing so, we introduce the Multi-Person IPOWAWA (MP-IPOWAWA) operator. It provides a framework for dealing with the opinion of several experts in the analysis.

We develop an illustrative application of the new approach in a group decision making problem regarding the selection of the optimal inflation rate that all the EU members should have when taking part of the European Monetary Union (EMU). We consider an environment that is conditioned by subjective and objective risk that it is assessed with weighted averages and probabilities. Furthermore, we assume that some part of the information depends on the attitudinal character of the decision maker that can be represented with the OWA operator in a context of complex uncertainty. We see that depending on the particular type of IPOWAWA operator used, the results may lead to different decisions.

This paper is organised as follows. In Section 2 we briefly review some basic preliminaries. Section 3 introduces the IPOWAWA operator. Section 4 analyses the applicability in EU decision making problems and Section 5 develops an illustrative example. Section 6 summarises the main conclusions of the paper.

Decision making in the European Union 593

2 Preliminaries

In this Section, we briefly review some basic concepts including the OWA, the IOWA, the OWAWA and the POWAWA operator.

2.1 The OWA operator

The OWA operator (Merigó, 2012) is an aggregation operator that provides a parameterised family of aggregation operators between the maximum and the minimum. It can be defined as follows:

Definition 1: An OWA operator of dimension n is a mapping OWA: Rn → R that has an associated weighting vector W of dimension n with

11n

jjw

==∑ and wj ∈ [0, 1], such

that:

( )1 21

, ,n

n j jj

OWA a a a w b=

= ∑… (1)

where bj is the j-th largest of the ai. The OWA operator is commutative, monotonic, bounded and idempotent and it

includes a wide range of particular cases (Yager, 1993; Merigó and Gil-Lafuente, 2009; Yager et al., 2011). It includes the classical methods for decision making under uncertainty as particular cases such as the optimistic and the pessimistic criteria, the Laplace criteria and the Hurwicz criteria.

2.2 The IOWA operator

The IOWA operator was introduced by Yager and Filev (1999) and it represents an extension of the OWA operator. Its main difference is that the reordering step is not developed with the values of the arguments ai. In this case, the reordering step is developed with order-inducing variables. The IOWA operator also includes as particular cases the maximum, the minimum and the average criteria. It can be defined as follows.

Definition 2: An IOWA operator of dimension n is a mapping IOWA: Rn × Rn → R that has an associated weighting vector W of dimension n with

11n

jjw

==∑ and wj ∈ [0, 1],

such that:

( )1 1 2 21

, , , , , ,n

n n j jj

IOWA u a u a u a w b=

= ∑… (2)

where bj is the ai value of the IOWA pair ⟨ui, ai⟩ having the j-th largest ui, ui is the order-inducing variable and ai is the argument variable.

Note that it is possible to distinguish between the Descending IOWA (DIOWA) operator and the Ascending IOWA (AIOWA) operator. For further reading on the IOWA, refer to Merigó and Gil-Lafuente (2009) and Yager et al. (2011).

594 J.M. Merigó, C. Lobato-Carral and A. Carrilero-Castillo

2.3 The OWAWA operator

The Ordered Weighted Averaging Weighted Averaging (OWAWA) operator is an aggregation operator that unifies the weighted average and the OWA operator in the same formulation considering the degree of importance that each concept has in the analysis (Merigó, 2011a). It can be defined as follows:

Definition 3: An OWAWA operator of dimension n is a mapping OWAWA: Rn → R that has an associated weighting vector W of dimension n such that wj ∈ [0, 1] and

11n

jjw

==∑ , according to the following formula:

( )11

ˆ,n

n j jj

OWAWA a a v b=

= ∑… (3)

where bj is the j-th largest of the ai, each argument ai has an associated weight (WA) vi with

11n

iiv

==∑ and vi ∈ [0, 1], ˆ (1 )j j jv w vβ β= + − with β ∈ [0, 1] and vj is the weight

(WA) vi ordered according to bj, that is, according to the j-th largest of the ai. The OWAWA includes a wide range of particular cases (Merigó, 2011a). Especially,

when β = 0, we get the classical weighted average and if β = 1, we get the OWA operator.

2.4 The POWAWA operator

The Probabilistic OWA Weighted Average (POWAWA) operator is an aggregation operator that uses probabilities, weighted averages and OWAs in the same formulation. It unifies these three concepts considering the degree of importance we want to give to each case depending on the situation considered. It can be defined as follows:

Definition 4: A POWAWA operator of dimension n is a mapping POWAWA: Rn → R that has an associated weighting vector W of dimension n with wj ∈ [0, 1] and

11n

jjw

==∑ ,

such that:

( )11

ˆ,n

n j ji

POWAWA a a v b=

= ∑… (4)

where bj is the j-th largest of the ai, each argument ai has an associated weight vi with

11n

iiv

==∑ and vi ∈ [0, 1], a probability pi with

11n

iip

==∑ and pi ∈ [0, 1],

1 2 3ˆ j j j jv C w C v C p= + + , with C1, C2 and C3 ∈ [0, 1], 1 2 3 1C C C+ + = and vj and pj are the weights of vi and pi ordered according to bj, that is, according to the j-th largest of the ai.

3 The induced POWAWA operator

The Induced POWAWA (IPOWAWA) operator is an aggregation operator that extends the POWAWA operator by using order-inducing variables that represent complex

Decision making in the European Union 595

reordering processes of an aggregation. Furthermore, it unifies the IOWA, the Weighted Average (WA) and the Probabilistic Aggregation (PA) in the same formulation considering the degree of importance that each concept has in the aggregation. A further interesting issue of the IPOWAWA operator is that it provides a parameterised family of aggregation operators between the maximum and the minimum. Moreover, it also considers semi bounds based on the use of the probability (objective information), the weighted average (subjective information) and both of them (objective and subjective information). The IPOWAWA operator is defined as follows:

Definition 5: An IPOWAWA operator is a mapping IPOWAWA: Rn × Rn → R of dimension n, if it has an associated weighting vector W, with

11j

n

jw

==∑ and wj ∈ [0, 1], a

weighting vector V that affects the weighted average, with 1

1i

n

iv

==∑ and vi ∈ [0, 1],

and a weighting vector P that affects the probability, with 1

1i

n

ip

==∑ and pi ∈ [0, 1],

such that:

( )1 1 1 2 31 1 1

, , ,n n n

n n j j i i i ij i i

IPOWAWA u a u a C w b C v a C p a= = =

= + +∑ ∑ ∑… (5)

where bj is the ai value of the IPOWAWA pair ⟨ui, ai⟩ having the j-th largest ui, ui is the order inducing variable and C1, C2 and C3 ∈ [0, 1], with 1 2 3 1C C C+ + = .

Note that it is also possible to formulate the IPOWAWA operator unifying the three weighting vectors with a convex combination and adapting the reordering of the arguments of the probability and the weighted average to the OWA or vice versa. That is:

( )1 11

ˆ, , ,n

n n j jj

IPOWAWA u a u a v b=

= ∑… (6)

where bj is the ai value of the IPOWAWA pair ⟨ui, ai⟩ having the j-th largest ui, ui is the order-inducing variable, each argument ai has an associated weight vi with

11i

n

iv

==∑

and vi ∈ [0, 1], a probability pi with 1

1i

n

ip

==∑ and pi ∈ [0, 1], 1 2 3ˆ j j j jv C w C v C p= + + ,

with C1, C2 and C3 ∈ [0, 1], 1 2 3 1C C C+ + = , vj and pj are the weights of vi and pi ordered according to bj, that is, according to the order-inducing variables ui.

Note that if the weighting vector is not normalised, i.e. 1

1njj

W w=

= ≠∑ ,

11n

iiV v

== ≠∑ or

11n

iiP p

== ≠∑ , then, the IPOWAWA operator can be expressed as:

( ) 31 21 1 1 1

1 1 1

, , , ,n n n

j j i i i ij i i

CC CIPOWAWA u a u a w b v a p a

W V P= = =

= + +∑ ∑ ∑… (7)

Note also that it is possible to distinguish between the Descending IPOWAWA (DIPOWAWA) operator and the Ascending IPOWAWA (AIPOWAWA) operator by using 1ˆ ˆj n jv v − += , where ˆ jv is the j-th weight of the DIPOWAWA and 1ˆn jv − + the j-th weight of the AIPOWAWA operator.

596 J.M. Merigó, C. Lobato-Carral and A. Carrilero-Castillo

The IPOWAWA operator is monotonic, bounded and idempotent. It is idempotent because if ai = a, for all i, then, IPOWAWA (⟨u1, a1⟩,…,⟨un, an⟩) = a. It is monotonic because if ai ≥ a, for all i, then, IPOWAWA (⟨u1, a1⟩,…,⟨un, an⟩) ≥ IPOWAWA (⟨u1, s1⟩,…,⟨un, sn⟩). It is bounded because the IPOWAWA aggregation is delimitated by the minimum and the maximum.

Next, we present some of the main particular cases of the IPOWAWA operator. By analysing the coefficients C, we get the following cases:

• If C1 = 1, we get the IOWA operator.

• If C2 = 1, we get the WA.

• If C3 = 1, we get the PA.

• If C1 = 0, we form the Probabilistic Weighted Average (PWA).

• If C2 = 0, we form the IPOWA operator.

• If C3 = 0, we form the IOWAWA operator.

• The more of C1, C2 or C3 to 1, the more we use the IOWA, the WA and the PA, respectively. And the more of C1, C2 and C3 to 0, the more we use the PWA, the IPOWA and the IOWAWA, respectively.

By analysing the weighting vector W, we can move the aggregation, according to the order-inducing variables from the minimum to the maximum. Especially, we find the Max-PWA and the Min-PWA operator as follows:

1 2 31 1

- n n

i i i i ii i

Max PWA C Max a C v a C p a= =

= + +∑ ∑ (8)

1 2 31 1

- n n

i i i i ii i

Min PWA C Min a C v a C p a= =

= + +∑ ∑ (9)

Note that a lot of other interesting boundary cases could be obtained by analysing the Max-PWA and the Min-PWA. For example, if C2 = 0, we can construct the Max-PA and the Min-PA aggregation as follows:

1 31

- n

i i ii

Max PA C Max a C p a=

= + ∑ (10)

1 31

- n

i i ii

Min PA C Min a C p a=

= + ∑ (11)

And if C3 = 0, we can form the Max-WA and the Min-WA aggregation obtaining subjective semi bounds:

1 21

- n

i i ii

Max WA C Max a C v a=

= + ∑ (12)

1 21

- n

i i ii

Min WA C Min a C v a=

= + ∑ (13)

Decision making in the European Union 597

Further situations could be considered by using other types of aggregation operators such as mixture operators and norm aggregations.

Note that we could study different measures for characterising the weights, such as the degree of orness, the entropy of dispersion, the divergence of W or the balance operator (Merigó and Casanovas, 2009; Merigó, 2012). For example, the entropy of dispersion can be formulated as follows:

1 2 31 1 1

ˆ( ) ln( ) ln( ) ln( )n n n

j j i i i ij i i

H V C w w C v v C p p= = =

⎛ ⎞= − + +⎜ ⎟

⎝ ⎠∑ ∑ ∑ (14)

As we can see, if C1 = 1, we get the Yager entropy (Yager, 1988) and if C2 = 1 or C3 = 1, the Shannon entropy (Shannon, 1948). If C1 = 0, we get the entropy of the PWA operator (Merigó, 2012), if C2 = 0, the entropy of the POWA operator (Merigó, 2011a; Merigó, 2011b; Merigó, 2011c) and if C3 = 0, the entropy of the IOWAWA operator (Merigó, 2011a; Merigó, 2011b; Merigó, 2011c).

Next, we can also consider the IPOWAWA operator in some key statistical measures such as the expected value, the variance, the covariance and a simple linear regression model. Note that Yager (1996, 2006) studied the use of the OWA operator in the variance and Merigó considered the variance and other related measures with the IOWAWA operator (2011a) and with the POWA operator (2011b). The expected value when using the IPOWAWA operator is formulated as shown in equation (5). The variance can be formulated as follows:

( ) 2 21 2 3

1 1 1

- ( ) ( )n n n

j j i i i ij i i

Var IPOWAWA X C w D C v x C p xμ μ= = =

= + − + −∑ ∑ ∑ (15)

where Dj is the (xi – µ)2 value of the IPOWAWA pair ⟨ui, ai⟩ having the j-th largest ui, ui is the order inducing variable, μ is the average (in this case, the IPOWAWA operator), C1, C2 and C3 ∈ [0, 1], with 1 2 3 1C C C+ + = , and

11j

n

jw

==∑ ,

11i

n

iv

==∑ and

11i

n

ip

==∑ .

Obviously, once we have the variance-IPOWAWA operator, it is straightforward to obtain the standard deviation. For the covariance-IPOWAWA operator we get the following formulation:

( )1

ˆ- ,n

j jj

Cov IPOWAWA X Y v K=

= ∑ (16)

where Kj is the (xi − μ)(yi – ν) value of the IPOWAWA triplet ⟨ui, xi, yi⟩ having the j-th largest ui, ui is the order inducing variable, xi is the argument variable of the first set of elements X = x1, …, xn and yi the argument variable of the second set of elements Y = y1, …, yn, μ and ν are the average (in this case, the IPOWAWA operator) of the sets X and Y respectively, wj ∈ [0, 1] and

11n

jjw

==∑ , each argument (xi − μ)(yi – ν) has

an associated weight (WA) vi with 1 i

n

iv

=∑ = 1 and vi ∈ [0, 1] and a probabilistic weight

pi with 1

1i

n

ip

==∑ and pi ∈ [0, 1], 1 2 3ˆ j j j jv C w C v C p= + + with C1, C2 and C3 ∈ [0, 1],

and 1 2 3 1C C C+ + = and vj and pj are the weights of vi and pi ordered according to Kj, that is, according to the j-th largest ui.

598 J.M. Merigó, C. Lobato-Carral and A. Carrilero-Castillo

And when dealing with a linear regression model that aggregates the information with the IPOWAWA operator we get the following. First, we calculate β as follows:

( , )ˆ( )IPOWAWA

Cov IPOWAWA X YVar IPOWAWA X

β −=

− (17)

Next, we calculate the ˆ IPOWAWAα value as follows:

ˆ IPOWAWA IPOWAWA IPOWAWAy xα β= − (18)

where IPOWAWAx and IPOWAWAy are the average of the sets X = x1, …, xn and Y = y1, …, yn calculated by using an IPOWAWA operator. Once we have ˆ IPOWAWAα and ˆ

IPOWAWAβ , we can construct the linear regression model with the IPOWAWA operator as follows:

ˆˆh IPOWAWA IPOWAWA hy xα β= + (19)

Note that a lot of other extensions could be developed in statistics following the methodology shown above.

4 Decision making in the European Union with the IPOWAWA operator

The IPOWAWA operator is applicable in a wide range of problems including decision-making, statistics, engineering and economics. In summary, all of the studies that use the probability, the weighted average or the OWA can be revised and extended with this new approach.

In this paper we will focus on decision making applications at the European Union (EU) level. The creation of a new Super State such as the EU involves a lot of micro- and macro-decisions that have to be taken in order to create an efficient and productive state (Johnson, 2006; Farndale, 2010; Nugent, 2010; Bonnedahl, 2011; Hix and Hoyland, 2011). All the areas have to be considered since in all sections, decisions have to be taken regarding the general policies at the EU level such as in strategic management (Arendt and Brettel, 2010; Cassia and Colombelli, 2010; Krasniqi, 2010; Salavou, 2010; Sonmez and Moorhouse, 2010; Ullah et al., 2010; Visscher and Visscher-Voerman, 2010; Aspara et al., 2011; Liu and Hsu, 2011), innovation management (Zhang and Duan, 2010; Cavalcante et al., 2011; Goktan and Miles, 2011; Hotho and Champion, 2011; Huang et al., 2011) and in knowledge management (Lindblom and Tikkanen, 2010; Cegarra-Navarro et al., 2011). For example, the creation of EU law (Bishop, 2009; Chalmers and Davies, 2010) implies the adoption of a lot of decisions since for each regulation the EU has to decide on the appropriate rule that has to be imposed. As a summary, we present some general areas were new key decisions has to be taken in EU law.

• Constitution law: The Constitution of Europe represents fundamental regulations that affect the general principles of the EU.

• Common law.

• Competition law.

• Tax law.

Decision making in the European Union 599

• Business and banking law.

• Property law.

• Other regulations.

Note that decision making is a key concept that is usually used in all fields since all the actions and strategies taken implicitly involves a decision making process. Thus, we can also consider the following key areas where EU decisions are very important:

• EU decision making in politics.

- Decisions concerning international relations.

- Decisions in external policies.

- The establishment of a new EU democracy.

- The integration of the national political parties in the EU.

- Monetary, fiscal and commercial policies.

- Other policies.

• EU decision making in economics.

- EU decision making in macroeconomics: Decisions concerning the analysis of the consumption, savings, investments, debt, government spending, taxes, transactions, exportations, importations, the IS-LM-BP curve, the interest rate, the exchange rate, the inflation, the monetary supply, the monetary demand, the aggregate demand, the aggregate supply, the Phillips curve, the wages, the unemployment, Keynesian and classical models, growth theories and the Purchasing Power Parity (PPP).

- EU decision making in microeconomics: Decisions concerning the analysis of the individual households, consumer theory, production theory, general equilibrium, market failures, perfect competition, monopoly, oligopoly and asymmetric information.

- EU decision making in international economics including exportations, importations, comparative advantage, the standard trade model, economic of scale, international factor movements, exchanges rates, international economic policies and the global capital market.

- EU decision making in development economics such as the development process of Eastern Europe and its adaptation to the Western Europe, European regional development, the expansion of the EU to other less developed countries and the EU aid to the rest of the World.

- EU decision making concerning the agriculture, the manufacturing industry, the service sector and tourism issues.

• EU decision making in finance.

- The European Monetary Union (EMU) and the Euro: Decisions concerning the new monetary system and its integration with other countries.

600 J.M. Merigó, C. Lobato-Carral and A. Carrilero-Castillo

- Decision making at the European Central Bank (ECB) such as the selection of the optimal interest rates for the European economy.

- The European stock market and other issues concerning European financial integration.

- International finance from a European perspective.

• EU decision making in marketing: Decisions concerning product development, price determination, promotion and advertisement, distribution channels and segmentation strategies.

• EU decision making in management: Decisions regarding general management, production and strategic management and organisational issues.

• EU decision making in human resource management.

Moreover, we could also consider any other area where the EU should make a decision such as:

• EU decision making in mathematics and statistics: Decisions concerning key mathematical and statistical problems in the EU such as the Eurostat.

• EU decision making in physics and astronomy: Decisions concerning key physical problems in the EU such as the creation and development of the European Space Agency (ESA).

• EU decision making in chemistry.

• EU decision making in biology and medicine such as the type of health services given to the EU population.

• EU decision making in sports: Decisions concerning European sports and national teams such as the development of the unified European leagues in football, basketball and handball.

• EU decision making in arts and culture.

• EU decision making in TV, cinema, radio and newspaper.

• EU decision making in other problems.

In most of the cases, the decision is taken by a group of experts instead of a single individual. Thus, it is necessary to use a group decision process. We will present an EU multi-person decision making approach by using the IPOWAWA operator. Therefore, we will be able to deal with the opinion of several experts in the analysis and considering their attitudinal character and the subjective and objective information available in the problem. We focus on an application in an EU decision making problem concerning the levels allowed to the inflation rate of the EU Members and its future Members as stated in the convergence criteria. Note that the key advantage of using a multi-person decision making process is that we get more robust information since the knowledge of several experts is higher than the knowledge of only one of them. Thus, we are able to consider in a more efficient way the different alternatives and select the most appropriate one according to our interests. Note that this is especially relevant when key decisions has to be taken and a lot of money or similar is involved. Moreover, the use of order-inducing

Decision making in the European Union 601

variables permits to consider complex reordering processes that represent complex attitudinal characters. This is also important because different experts or countries may have different interests so the attitudinal character is not clearly defined and this has to be assessed in the analysis. In this paper we analyse the selection of the upper bounds that the inflation rate of a New Member State may have when entering the EMU.

The process to follow in the selection of the inflation rate in the EU convergence criteria with the IPOWAWA operator and by using a multi-person decision making process can be summarised as follows.

Step 1: Let A = A1, A2,…,Am be a set of finite alternatives and C = C1, C2,…,Cn a set of finite characteristics (or attributes), forming the matrix (xhi)m × n. Let E = E1, E2,…,Eq be a finite set of decision makers. Let V = (v1, v2,…,vq) be the weighting vector of the decision makers such that

11q

kkv

==∑ and vk ∈ [0, 1]. Each decision maker provides their

own payoff matrix (xhi(k))m × n.

Step 2: Calculate the order inducing variables (uhi)m × n to be used in the payoff matrix for each alternative h and characteristic i. Calculate also the weighting vector W, V and P to be used in the IPOWAWA aggregation. Note that W = (w1, w2,…,wn) such that

11n

jjw

==∑ and wj ∈ [0, 1], V = (v1, v2,…,vn) such that

11n

iiv

==∑ and vi ∈ [0, 1] and

P = (p1, p2,…,pn) such that 1

1nii

p=

=∑ and pi ∈ [0, 1].

Step 3: Use the weighted average (WA) to aggregate the information of the decision makers E by using the weighting vector Z = (z1, z2,…,zq). The result is the collective payoff matrix (ahi)m × n. Thus,

1

q khi k hik

x z x=

= ∑ .

Step 4: Calculate the aggregated results by using the IPOWAWA operator explained in equation (5). Consider different particular manifestations of the IPOWAWA operator by using different expressions in the weighting vector, as explained in Section 3.

Step 5: Adopt decisions according to the results found in the previous steps. Select the alternatives that provides the best results and establish a ranking of the alternatives from the most to the least preferred alternative.

Note that this aggregation process can be summarised using the following aggregation operator that we call the Multi-Person-IPOWAWA (MP-IPOWAWA) operator.

Definition 6: A MP-IPOWAWA operator is a mapping MP-IPOWAWA: (Ωn)p × Ωn → Ω that has a weighting vector X of dimension q with

11q

kkx

==∑ and xk ∈ [0, 1] and a

weighting vector W of dimension n with 1

1njj

w=

=∑ and wj ∈ [0, 1], such that:

( ) ( )( )1 11 1 1

1

ˆ- , , , , , , , ,n

p pn n n j j

j

MP IPOWAW u a a u a a v b=∑… … … (20)

where bj is the ai value of the MP-IPOWAWA pair ⟨ui, ai⟩ having the j-th largest ui, ui is the order-inducing variable,

1

q ki k ik

a x a=

= ∑ , kia is the argument variable provided by

each person (or expert), each argument ai has an associated weight (WA) vi with

602 J.M. Merigó, C. Lobato-Carral and A. Carrilero-Castillo

11i

n

iv

==∑ and vi ∈ [0, 1] and a probability pi with

11i

n

ip

==∑ and pi ∈ [0, 1],

1 2 3ˆ j j j jv C w C v C p= + + , with C1, C2 and C3 ∈ [0, 1], 1 2 3 1C C C+ + = and vj and pj are the weights of vi and pi ordered according to bj, that is to say, according to the j-th largest of the ui.

The MP-IPOWAWA operator accomplishes similar properties than those explained in Section 3. Thus, we can find as special cases:

• The Multi-Person-arithmetic mean (MP-AM) operator.

• The Multi-Person-WA (MP-WA) operator.

• The Multi-Person-IPOWA (MP-IPOWA) operator.

• The Multi-Person-IOWA (MP-IOWA) operator.

• The Multi-Person-IOWAWA operator.

• The Multi-Person-POWA (MP-POWA) operator.

• The Multi-Person-POWAWA operator.

• The Multi-Person-OWA (MP-OWA) operator.

• The Multi-Person-PA (MP-PA).

• Etc.

Note that we can consider more complex situations by using different types of aggregation operators in the aggregation of the expert’s opinion instead of using the WA operator. And it is possible to consider a lot of other methodologies for decision making (Figueira et al., 2005; Zavadskas and Turskis, 2010; Yan, 2011; Merigó, 2011c; Merigó and Gil-Lafuente, 2011b).

5 Application in European Union group decision making

In this Section, we develop a brief illustrative example of the new approach in a multi-person decision making problem concerning the decision process of the European Union (EU) integration.

In this example, we assume that the EU leaders are planning the inflation rate of the new members. Today, according to the convergence criteria, the inflation rate of all the EU members has to be lower than the average of the three members with lowest inflation rate plus an additional 1.5%. Obviously, this is the theoretical requirement but in real situations, the policy is to consider all the particularities of each country in a case-by-case assessment. For the next period, they are considering the following five possible alternatives:

1 A1: Allow the inflation rate to be 3% higher than the average of the three EU members with lowest inflation rate.

2 A2: Allow the inflation rate to be 2.5% higher.

3 A3: Allow the inflation rate to be 2% higher.

Decision making in the European Union 603

4 A4: Do not make any change.

5 A5: Allow the inflation rate to be only 1% higher.

After careful review of the information, the group of experts of the EU establishes the following general information regarding the alternatives. They also assume that the benefits depend on the state of nature that occurs in the future. In this example, we assume five potential states of nature that may occur S = S1, S2, S3, S4, S5.

• S1: Negative growth rate.

• S2: Growth rate close to 0.

• S3: Low positive growth rate.

• S4: Positive growth rate.

• S5: High growth rate.

The group of experts of the EU is constituted by 1000 persons divided in three general groups, each offering their own single opinion regarding the results obtained with each strategy considered. The results are shown in Tables 1–3. Table 1 Group of experts 1

S1 S2 S3 S4 S5 A1 60 70 50 60 80 A2 40 40 80 70 60 A3 50 50 70 90 80 A4 30 60 70 80 50 A5 40 80 60 70 70

Table 2 Group of experts 2

S1 S2 S3 S4 S5 A1 40 60 50 60 70 A2 70 40 90 40 60 A3 50 60 70 90 70 A4 30 60 70 60 50 A5 40 80 50 70 60

Table 3 Group of experts 3

S1 S2 S3 S4 S5 A1 50 70 40 60 80 A2 70 40 80 40 60 A3 40 60 70 90 70 A4 30 50 80 80 50 A5 50 80 60 50 60

604 J.M. Merigó, C. Lobato-Carral and A. Carrilero-Castillo

To analyse the attitudinal character of the government, we consider that they use order-inducing variables which represent the complex attitudinal character in the decision process: U = (12, 20, 24, 18, 16). The main advantage of using order-inducing variables is that we can represent complex decision processes that include psychological factors such as time pressure and other related aspects.

With this information, we can aggregate it in order to make a decision. First, we aggregate the information of the three groups of experts to obtain a collective matrix. We use the WA to obtain this matrix by assuming that X = (0.2, 0.3, 0.5). The results are shown in Table 4. Table 4 Collective results

S1 S2 S3 S4 S5 A1 49 67 45 60 77 A2 64 40 83 46 60 A3 45 58 70 90 68 A4 30 55 75 74 50 A5 45 80 57 60 62

Now, we can develop different methods for establishing the weighting vector. In this example, we ask the people to give their preferences and we find the following absolute values. Note that each person has one vote that can be divided in different ways across the weighting vector and we assume that everybody is equally important. The results are presented in Table 5. Table 5 Absolute frequency for the OWA operator

w1 w2 w3 w4 w5 W 100 300 300 200 100

We now normalise the information so the weighting vector sums to 1. The results are shown in Table 6. Table 6 Relative frequency for the OWA operator

w1 w2 w3 w4 w5 W 0.1 0.3 0.3 0.2 0.1

Next, we develop a similar analysis for the weighted average and the probabilistic weighting vector. The results are shown in Tables 7–10. Table 7 Absolute frequency for the WA operator

v1 v2 v3 v4 v5 V 200 300 200 200 100

Table 8 Relative frequency for the WA operator

v1 v2 v3 v4 v5 V 0.2 0.3 0.2 0.2 0.1

Decision making in the European Union 605

Table 9 Absolute frequency for the probabilistic weighting vector

p1 p2 p3 p4 p5 P 100 200 400 200 100

Table 10 Relative frequency for the probabilistic weighting vector

p1 p2 p3 p4 p5 P 0.1 0.2 0.4 0.2 0.1

In this example, we consider the minimum, the maximum, the Min-PWA, the Max-PWA, the Probabilistic Aggregation (PA), the Weighted Average (WA), the IOWA operator, the PWA operator, the IPOWA, the IOWAWA and the IPOWAWA operator. We assume that C1 = 0.3, C2 = 0.4 and C3 = 0.3. The results are shown in Tables 11 and 12. Table 11 Aggregated results 1

Min Max Min-PWA Max-PWA PWA A1 45 77 53,74 63,34 57,48 A2 40 83 53,48 66,38 59,25 A3 45 90 64,25 73,75 66,78 A4 30 75 51,06 64,56 60,08 A5 45 80 56,99 67,49 62,12

Table 12 Aggregated results 2

PA WA IOWA IOWAWA IPOWA IPOWAWA A1 56 58,6 62,9 63,44 59,45 59,11 A2 62,8 56,6 52,5 54,84 57,65 57,23 A3 68,9 65,2 69,5 67,04 69,2 67,6 A4 63,8 57,3 59,2 58,11 61,5 59,82 A5 61,5 62,6 64,6 63,45 63,05 62,87

As we can see, A3 seems to be the optimal choice because in all the cases it provides the highest results. Furthermore, it is interesting to establish a ranking of the strategies in order to see other potential alternatives. The results are shown in Table 13. Table 13 Ranking of the alternatives

Ranking Ranking Min A1=A3=A5⎬A2⎬A4 WA A3⎬A5⎬A1⎬A4⎬A2 Max A3⎬A2⎬A5⎬A1⎬A4 IOWA A3⎬A5⎬A1⎬A4⎬A2

Min-PWA A3⎬A5⎬A1⎬A2⎬A4 IOWAWA A3⎬A5⎬A1⎬A4⎬A2 Max-PWA A3⎬A5⎬A2⎬A4⎬A1 IPOWA A3⎬A5⎬A4⎬A1⎬A2

PWA A3⎬A5⎬A4⎬A2⎬A1 IPOWAWA A3⎬A5⎬A4⎬A1⎬A2 PA A3⎬A4⎬A2⎬A5⎬A1

606 J.M. Merigó, C. Lobato-Carral and A. Carrilero-Castillo

As we can see from the ranking of the alternatives, each type of IPOWAWA operator may give different rankings leading to different decisions.

6 Conclusions

We have introduced the IPOWAWA operator in decision making problems in the EU. Its key advantage is that it unifies the probability, the weighted average and the IOWA operator in the same formulation and considering the degree of importance that each concept has in the aggregation. We have studied some of its main properties including a wide range of particular cases such as the IPOWA, the IOWAWA, the PWA, the Max-PWA and the Min-PWA operator. We have also extended this approach when dealing with group decision making problems by using a multi-person aggregation operator that we have called the MP-IPOWAWA operator.

We have studied the applicability of the IPOWAWA operator and we have seen that it can be applied in a wide range of fields. We have focused on applications produced in EU decision making problems. We have seen that in the EU it creates a lot of decisions and some of them are extremely important. Among others, we have seen that in the EU we find a lot of key decisions in EU law, EU politics, EU economics, EU finance and EU science. We have focused on an EU multi-person decision making problem concerning the fixation of the upper bounds of the inflation rate for the members of the EMU. By using the MP-IPOWAWA operator, we have been able to consider a wide range of scenarios and select the one in closest accordance to our interests.

In future research, we expect to develop further extensions of the IPOWAWA operator by using other formulations such as the use of generalised aggregation operators, distance measures and unified aggregation operators. We will also consider uncertain environments where the available information is imprecise and can be assessed with fuzzy numbers and linguistic variables. We will also consider other applications in EU decision making problems giving special attention to key decisions that involve high quantities of money such as the fixation of the European Constitution, key European regulations and key European economic policies.

Acknowledgements

Support from the University of Barcelona under project 099311 is gratefully acknowledged.

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