Decision-making with distance measures and induced aggregation operators

Computers & Industrial Engineering 60 (2011) 66–76

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Decision-making with distance measures and induced aggregation operators q José M. Merigó ⇑, Montserrat Casanovas Department of Business Administration, University of Barcelona, Av. Diagonal 690, 08034 Barcelona, Spain

a r t i c l e

i n f o

Article history: Received 17 December 2009 Received in revised form 27 July 2010 Accepted 28 September 2010

Keywords: Decision-making OWA operator Distance measures Induced aggregation operators

a b s t r a c t In this paper, we present a new decision-making approach that uses distance measures and induced aggregation operators. We introduce the induced ordered weighted averaging distance (IOWAD) operator. IOWAD is a new aggregation operator that extends the OWA operator by using distance measures and a reordering of arguments that depends on order-inducing variables. The main advantage of IOWAD is that it provides a parameterized family of distance aggregation operators between the maximum and the minimum distance based on a complex reordering process that reflects the complex attitudinal character of the decision-maker. We studied some of IOWAD’s main properties and different particular cases and further generalized IOWAD by using Choquet integrals. We developed an application in a multi-person decision-making problem regarding the selection of investments. We found that the main advantage of this approach is that it is able to provide a more complete picture of the decision-making process, enabling the decision-maker to select the alternative that it is more in accordance with his interests. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction There is a wide range of methods for decision-making in the literature (Figueira, Greco, & Ehrgott, 2005; Gil-Aluja, 1999; Merigó, 2008, 2010; Merigó & Casanovas, 2009; Merigó, López-Jurado, Gracia, & Casanovas, 2009; Wei, 2009; Xu, 2009a, 2009b, 2009c; Yager, 2009a; Zarghami & Szidarovszky, 2009). A very useful technique for decision-making is the Hamming distance (Hamming, 1950) and, more generally, all the distance measures (Karayiannis, 2000; Kaufmann, 1975; Merigó, 2008; Merigó & Gil-Lafuente, 2007, 2008a, 2008b, 2009b, 2010; Szmidt & Kacprzyk, 2000). The main advantage of using distance measures in decision-making is that we can compare the alternatives of the problem with some ideal result (Gil-Aluja, 1999). Through this comparison, the alternative with the closest result to the ideal is the optimal choice. Usually, when using distance measures in decision-making, we normalize them by using the arithmetic mean or the weighted average (WA) obtaining the normalized Hamming distance (NHD) and the weighted Hamming distance (WHD), respectively. However, it is sometimes of interest to consider the possibility of parameterizing the results from the maximum distance to the minimum distance. In this case, the ordered weighted averaging (OWA) operator can be used (Yager, 1988). The OWA operator is a useful technique for aggregating information, providing a parameterized family of aggregation operators that includes the q

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⇑ Corresponding author. Tel.: +34 93 402 19 62; fax: +34 93 403 98 82. E-mail addresses: [email protected] (J.M. Merigó), [email protected] (M. Casanovas). 0360-8352/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2010.09.017

maximum, the minimum and the average, among others (Ahn, 2009; Beliakov, Pradera, & Calvo, 2007; Chiclana, Herrera-Viedma, Herrera, & Alonso, 2007; Emrouznejad, 2008; Liu, 2008; Wang, Luo, & Liu, 2007; Xu, 2005, 2009b; Yager, 1993, 2007, 2009a, 2009b, 2009c). The use of the OWA operator in different types of distance measures have been studied by several authors (Karayiannis, 2000; Merigó, 2008; Merigó & Gil-Lafuente, 2007, 2008a, 2008b, 2009b, 2010). Numerous authors have studied other developments concerning the OWA operator, refer, e.g., to Amin and Emrouznejad (2006), Beliakov et al. (2007), Cheng, Wang, and Wu (2009), Kacprzyk and Zadrozny (2009), Liu, Cheng, Chen, and Chen (2010), Merigó (2008, 2010), Merigó and Casanovas (2008, 2010a, 2010b), Merigó, Casanovas, and Martı´nez (2010), Merigó and Gil-Lafuente (2008c, 2009a) and Yager and Kacprzyk (1997). An interesting extension of the OWA operator is the induced OWA (IOWA) operator (Yager & Filev, 1999). IOWA differs in that the reordering step is not developed with the values of the arguments but can be induced by another mechanism such that the ordered position of the arguments depends upon the values of their associated order-inducing variables. The IOWA operator has received increasing attention in recent years (Chiclana et al., 2007; Herrera-Viedma, Chiclana, Herrera, & Alonso, 2007; Merigó, 2008; Merigó & Casanovas, 2009; Merigó & Gil-Lafuente, 2009a; Wei, Zhao, & Lin, 2010; Wu, Li, Li, & Duan, 2009; Yager, 2003). The aim of this paper is to present the use of the induced OWA (IOWA) operator in decision-making with distance measures. We formulate a more general model by using order-inducing variables in the reordering process of the OWA aggregation. We thus introduce a new aggregation operator: the induced ordered

J.M. Merigó, M. Casanovas / Computers & Industrial Engineering 60 (2011) 66–76

weighted averaging distance (IOWAD) operator. The IOWAD operator is an aggregation operator that provides a parameterized family of distance aggregation operators that ranges from the minimum to the maximum distance. The main advantage of the IOWAD operator is that it is able to deal with complex attitudinal characters (or complex degrees of orness) in the decision process by using order-inducing variables. In so doing, we are able to deal with more complex problems that are closer to real-world situations. In order to see the usefulness of the IOWAD operator, let us look into a real-world example in business decision-making. An important business decision, for example, is usually made by the company’s board of directors. The decision involves the attitudinal character of a group of persons that must be coordinated into one simple decision according to the group’s interests. Obviously, the attitudinal character of this example is much more complex than simply using the degree of optimism (degree of orness) of the company. Note that in this example, we analyze the attitudinal character (degree of orness) in group decisionmaking problems, but the actual analysis would be much more complex. Thus, a good method for analyzing this problem would be the use of order-inducing variables by using the IOWAD operator. We study basic properties of the IOWAD operator and we consider a wide range of particular cases: the NHD; the WHD; the ordered weighted averaging distance (OWAD) operator; the median-IOWAD; the olympic-IOWAD and the centered-IOWAD. We see that each particular case is useful for a certain situation according to the objectives of the decision-maker. Depending on the particular type of operator used, the results may differ. Note that it is possible to generalize the aggregation operator by using generalized and quasi-arithmetic means following the ideas of Merigó and Gil-Lafuente (2009a, 2009b). We also present a more general formulation by using mixture and infinitary operators such as the induced mixture distance (IMD) operator, the induced quasiarithmetic mixture distance (Quasi-IMD) operator and the infinitary IOWAD (1-IOWAD) operator. The main advantage of these generalizations is that they are able to provide a deeper representation of the specific problem considered that includes the IOWAD operator as a particular case. We also present an application of the new approach in a multi-person decision-making problem concerning the selection of investments. The main advantage of this model is that it gives a more complete view of the decision problem because it considers a wide range of distance aggregation operators according to the interests of the decision-maker. Moreover, by using several experts in the analysis, we obtain information that it is more robust because the opinion of several experts is always better than the opinion of one. For doing so, we introduce a new aggregation operator called the multi-person–IOWAD (MP–IOWAD) operator. We then study some of its main particular cases, such as the multi-person–OWAD (MP–OWAD) and the multi-person–WHD (MP–WHD). Note also that the IOWAD and the MP–IOWAD operator are applicable to a wide range of situations such as fuzzy set theory, operational research, statistics, economics and engineering. This method is also applicable to different decision-making problems such as in strategic decision-making, human resource management, product management and financial management. The main advantage of using distance measures is that we can compare the real-world information with ideal information and see which alternative better fits with the interests of the decision-maker. For example, in human resource selection, we can establish an ideal candidate that would perfectly fit the company and compare it with the real-world alternatives that we have in the market and select the candidate with closest results to the ideal one.

67

This paper is organized as follows. In Section 2, we briefly review basic concepts that are used throughout the paper. In Section 3, we present the IOWAD operator. Section 4 analyzes different families of IOWAD operators. In Section 5, we develop an extension of IOWAD by using Choquet integrals. In Section 6, we present a method for multi-person decision-making with the IOWAD operator in investment decisions and Section 7 develops a numerical example of the new approach. Finally, we summarize the main conclusions of the paper in Section 8. 2. Preliminaries In this section, we briefly describe the Hamming distance, the OWA operator and the induced OWA operator. 2.1. The Hamming distance The Hamming distance (Hamming, 1950) is a useful technique for calculating the differences between two parameters, such as problems with two elements or two sets. The Hamming distance can be useful in fuzzy set theory, for example, when calculating distances between fuzzy sets, interval-valued fuzzy sets and intuitionistic fuzzy sets. To define the Hamming distance, we first define a distance measure. A distance measure must basically accomplish the following properties:    

Non-negativity: D(A1, A2) P 0. Commutativity: D(A1, A2) = D(A2, A1). Reflexivity: D(A1, A1) = 0. Triangle inequality: D(A1, A2) + D(A2, A3) P D(A1, A3).

For two sets, A = (a1, . . . , an) and B = (b1, . . . , bn), we can define the Hamming distance as follows: Definition 1. A normalized Hamming distance of dimension n is a mapping NHD: [0, 1]n  [0, 1]n ? [0, 1], such that:

NHDðA; BÞ ¼

! n 1X jai  bi j ; n i¼1

ð1Þ

where ai and bi are the ith arguments of the sets A and B respectively. Sometimes, when normalizing the Hamming distance, we prefer to give different weights to each individual distance. In this case, the distance is known as the weighted Hamming distance, which can be defined as follows: Definition 2. A weighted Hamming distance of dimension n is a mapping WHD: [0, 1]n  [0, 1]n ? [0, 1] that has an associated P weighting vector W of dimension n with W ¼ nj¼1 wj ¼ 1 and wj 2 [0, 1], such that:

WHDðA; BÞ ¼

n X

!

wi jai  bi j ;

ð2Þ

i¼1

where ai and bi are the ith arguments of the sets A and B respectively. Note that it is possible to generalize this definition to all real numbers by using Rn  Rn ? R. For the formulation used in fuzzy set theory, see, for example, Gil-Aluja (1998), Kaufmann (1975), Merigó (2008) and Szmidt and Kacprzyk (2000). 2.2. The OWA operator The OWA operator (Yager, 1988) provides a parameterized family of aggregation operators that include the maximum, the

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minimum and the average criteria as special cases. This operator can be defined as follows:

order-inducing variables. For two sets X = {x1, x2, . . . , xn} and Y = {y1, y2, . . . , yn}, IOWAD can be defined as follows:

Definition 3. An OWA operator of dimension n is a mapping OWA: Rn ? R that has an associated weighting vector W of dimension n P such that W ¼ nj¼1 wj ¼ 1 and wj 2 [0, 1], according to the following formula:

Definition 5. An IOWAD operator of dimension n is a mapping IOWAD: Rn  Rn  Rn ? R that has an associated weighting vector P W such that wj 2 [0, 1] and W ¼ nj¼1 wj ¼ 1, according to the following formula:

OWAða1 ; a2 ; . . . ; an Þ ¼

n X

wj bj ;

ð3Þ

IOWADðhu1 ; x1 ; y1 i; hu2 ; x2 ; y2 i; . . . ; hun ; xn ; yn iÞ ¼

j¼1

aðWÞ ¼

n X j¼1

wj



nj : n1

ð4Þ

Note that a(W) 2 [0, 1]. The more weight W is located close to the top, the closer a is to 1. In decision-making problems, the degree of orness is useful for representing the attitudinal character of the decision-maker by using it as the degree of optimism or pessimism. Different families of OWA operators are found by using different manifestations in the weighting vector, such as maximum, minimum and average criteria. For more information on other families, refer, for example, to Ahn (2009), Beliakov et al. (2007), Emrouznejad (2008), Liu (2008, 2010), Merigó (2008); Xu (2005) and Yager (1993, 2007, 2009a, 2009b). 2.3. The induced OWA operator The IOWA operator (Yager & Filev, 1999) is an extension of the OWA operator. The main difference is that the reordering step is not carried out with the values of the arguments ai. In this case, the reordering step is developed with order-inducing variables that reflect a more complex reordering process. The IOWA operator also includes as particular cases maximum, minimum and average criteria. The IOWA operator can be defined as follows: Definition 4. An IOWA operator of dimension n is a mapping IOWA: Rn  Rn ? R that has an associated weighting vector W of P dimension n with W ¼ nj¼1 wj ¼ 1 and wj 2 [0, 1], such that:

IOWAðhu1 ; a1 i; hu2 ; a2 i; . . . ; hun ; an iÞ ¼

wj bj ;

ð6Þ

j¼1

where bj is the jth largest ai. From a generalized perspective of the reordering step, it is possible to distinguish between the descending OWA (DOWA) operator and the ascending OWA (AOWA) operator. The OWA operator is commutative, monotonic, bounded and idempotent. The OWA operator aggregates the information according to the attitudinal character (or degree of orness) of the decision-maker (Yager, 1988). The attitudinal character is represented according to the following formula:



n X

n X

wj bj ;

ð5Þ

j¼1

where bj is the ai value of the IOWA pair hui, aii having the jth 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 (Merigó & Gil-Lafuente, 2009a). The IOWA operator is also monotonic, bounded, idempotent and commutative (Yager & Filev, 1999).

3. The induced ordered weighted averaging distance operator The IOWAD operator is a distance measure that uses the IOWA operator in the normalization process of the Hamming distance. Thus, the reordering of the individual distances is developed with

where bj is the jxi  yij value of the IOWAD triplet hui, xi, yii having the jth largest ui, ui is the order-inducing variable and jxi  yij is the argument variable represented in the form of individual distances. In the following example, we present a simple numerical example showing how to use the IOWAD operator in an aggregation process: Example 1. Assume the following arguments in an aggregation process: X = (7, 30, 10, 15), Y = (4, 20, 6, 9) with the following order inducing variables U = (2, 6, 9, 7). Assume the following weighting vector W = (0.2, 0.2, 0.3, 0.3). If we calculate the distance between X and Y using the IOWAD operator, we get the following: IOWADðX; YÞ ¼ 0:2  j10  6j þ 0:2  j15  9j þ 0:3  j30  20j þ 0:3  j7  4j ¼ 5:9:

A fundamental aspect of the IOWAD operator is the reordering of the arguments based upon order-inducing variables. That is, the weights, rather than being associated with a specific argument as in the case with the usual Hamming distance, are associated with the position given by the order-inducing variables. This reordering introduces nonlinearity into an otherwise linear process. If D is a vector corresponding to ordered arguments bj, we call this the ordered argument vector. If WT is the transpose of the weighting vector, then the IOWAD operator can be presented as follows:

IOWADðhu1 ; x1 ; y1 i; hu2 ; x2 ; y2 i; . . . ; hun ; xn ; yn iÞ ¼ W T D:

ð7Þ

From a generalized perspective of the reordering step, it is possible to distinguish between descending (DIOWAD) and ascending (AIOWAD) orders. The weights of these operators are related by wj ¼ wnjþ1 , where wj is the jth weight of the DIOWAD operator and wnjþ1 the jth weight of the AIOWAD operator. Note that if the weighting vector is not normalized (Beliakov P et al., 2007), i.e., W ¼ nj¼1 wj – 1, then the IOWAD operator can be expressed as:

IOWADðhu1 ; x1 ; y1 i; hu2 ; x2 ; y2 i; . . . ; hun ; xn ; yn iÞ ¼

n 1 X wj bj : W j¼1

ð8Þ

Note that IOWAD(hu1, x1, y1i, hu2, x2, y2i, . . . , hun, xn, yni) = 0 if and only if xi = yi for all i 2 [1, n]. Note also that IOWAD(hu1, x1, y1i, hu2, x2, y2i, . . . , hun, xn, yni) = IOWAD(hu1, y1, x1i, hu2, y2, x2i, . . . , hun, yn, xni). An interesting issue arises when analyzing similarity measures, namely, the possibility of considering the dissimilarity measure. For the IOWAD operator, assuming that we are in the unit interval [0, 1], the dissimilarity measure is given by dissimilarityIOWAD = 1  IOWAD. As we can see, the dissimilarity is the dual of the IOWAD operator. Note that the dissimilarity can be studied in a similar way for all of the extensions mentioned in the rest of this paper. Other interesting generalizations can be developed following: Mesiar and Pap (2008), Spirkova (2008) and Torra and Narukawa (2010). Following Spirkova (2008), we can develop the function induced OWAD operator, which uses a generating function r for the

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order-inducing variables such that r: I ? R, being that I  R is a closed interval I = [a, b]. In this paper, we will use a more general representation by also using also a generating function for the arguments such that s: Rm ? R. This generating function expresses the formation of the arguments when a previous analysis exists, such as the use of a multi-person process where each argument is constituted by the opinion of m persons. Moreover, we will use a weighting function f for the weighting vector. Note that the use of a weighting function fi in the weighting vector of the weighted average is known as the Losonczi mean (Losonczi, 1971). If the function is equal for all weights f, then we get the simple Losonczi mean or the quasi-mixture operator (Spirkova, 2008). In this case, we directly extend the approach by obtaining the function induced mixture distance (IMD) operator as follows (note that we could also refer to it as the Losonczi-IOWAD (Lo-IOWAD) operator). In this definition, we refer to the arguments as two sets X = {x1, x2, . . . , xn} and Y = {y1, y2, . . . , yn}. Definition 6. An IMD operator of dimension n is a mapping IMD: [Rn  Rn  Rn ? R that has an associated vector of weighting functions f, r: I ? ]0, 1[, is a some positive continuous function, s: Rm ? R, such that:

IMDðhr o ðu1 Þ; sp ðx1 Þ; sq ðy1 Þi; . . . ; hr o ðun Þ; sp ðxn Þ; sq ðyn ÞiÞ Pn j¼1 fj ðsy ðbj ÞÞsy ðbj Þ ; ¼ Pn j¼1 fj ðsy ðbj ÞÞ

appear in the aggregation process. Note that j¼1 wj ¼ 1. By using, the IOWAD operator we get the infinitary IOWAD (1-IOWAD) operator as follows:

1-IOWADðhro ðu1 Þ; sp ðx1 Þ; sq ðy1 Þi; . . . ; hro ðun Þ; sp ðxn Þ; sq ðyn ÞiÞ ¼

1 X

wj bj ;

j¼1

ð11Þ However, note that the reordering process is much more complex, that is, we never know which argument is the largest argument because we have an unlimited number of arguments. This problem can be partially solved by using the order-inducing variables. For further reading about the usual OWA, see Mesiar and Pap (2008). Note that a similar extension could be developed by using the IMD operator, thus obtaining the 1-IMD operator, and by using generalized and quasi-arithmetic means, thus obtaining the 1-Quasi-IMD operator. The IOWAD operator is commutative, monotonic, bounded, idempotent, nonnegative and reflexive but it does not accomplish always the triangle inequality. These properties can be proved with the following theorems: Theorem 1 (Commutativity – OWA aggregation). Assume f is the IOWAD operator, then

ð9Þ

where sy(bj) is the jsp(xi)  sq(yi)j value of the IMD triplet hro(ui), sp(xi), sq(yi)i having the jth largest ro(ui); ui is the orderinducing variable; jsp(xi)  sq(yi)j is the argument variable represented in the form of individual distances; and o, p and q indicate that each order-inducing variable and each argument is formed by using a different function. Note that the IMD operator can be further generalized by using generalized and quasi-arithmetic means (Merigó & Gil-Lafuente, 2009a). The result is the induced generalized mixture distance (IGMD) operator and the induced quasi-arithmetic mixture distance (Quasi-IMD) operator. The Quasi-IMD operator can be defined as follows: Definition 7. A Quasi-IMD operator of dimension n is a mapping QIMD: Rn  Rn  Rn ? R that has an associated a vector of weighting functions f, r: I ? ]0, 1[, is some positive continuous function, s: Rm ? R, such that:

QIMDðhr o ðu1 Þ; sp ðx1 Þ; sq ðy1 Þi; . . . ; hro ðun Þ; sp ðxn Þ; sq ðyn ÞiÞ ¼ ! Pn j¼1 fj ðsy ðbj ÞÞgðsy ðbj ÞÞ Pn ; ¼ g 1 j¼1 fj ðsy ðbj ÞÞ

P1

ð10Þ

where g is a strictly continuous monotonic function; sy(bj) is the jsp(xi)  sq(yi)j value of the IMD triplet (hro(ui), sp(xi), sq(yi)i) having the jth largest ro(ui); ui is the order-inducing variable; jsp(xi)  sq(yi)j is the argument variable represented in the form of individual distances; and o, p and q indicates that each order-inducing variable and each argument is formed by using a different function. Note that if g(b) = bk, we get the IGMD operator. Following Merigó and Gil-Lafuente (2009a), we can obtain a wide range of particular cases of the Quasi-IMD operator. For example, if g(b) = b, we obtain the IMD operator. If g(b) = b2, the induced quadratic mixture distance (IQMD) operator. If g(b) ? b0, the induced geometric mixture distance operator (IGMD) and if g(b) = b1, the induced harmonic mixture distance (IHMD) operator. Another interesting extension uses infinitary aggregation operators (Mesiar & Pap, 2008). Here, we can represent an aggregation process where there are an unlimited number of arguments that

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ ¼ f ðhu1 ; c1 ; d1 i; . . . ; hun ; cn ; dn iÞ;

ð12Þ

where (hu1, x1, y1i, . . ., hun, xn, yni) is any permutation of the arguments (hu1, c1, d1i, . . ., hun, cn, dni). Theorem 2 (Commutativity – distance measure). Assume f is the IOWAD operator, then

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ ¼ f ðhu1 ; y1 ; x1 i; . . . ; hun ; yn ; xn iÞ:

ð13Þ

Theorem 3 (Monotonicity). Assume f is the IOWAD operator; if jxi  yij P jci  dij, for all ii, then

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ P f ðhu1 ; c1 ; d1 i; . . . ; hun ; cn ; dn iÞ:

ð14Þ

Theorem 4 (Bounded). Assume f is the IOWAD operator, then

minfjxi  yi jg 6 f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ 6 maxfjxi  yi jg:

ð15Þ

Theorem 5 (Idempotency). Assume f is the IOWAD operator; if jxi  yij = a, for all i, then

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ ¼ a:

ð16Þ

Theorem 6 (Non-negativity). Assume f is the IOWAD operator, then

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ P 0:

ð17Þ

Theorem 7 (Reflexivity). Assume f is the IOWAD operator, then

f ðhu1 ; x1 ; x1 i; . . . ; hun ; xn ; xn iÞ ¼ 0:

ð18Þ

Note that the IOWAD operator does not always accomplish the triangle inequality because we may find some special situations where f (hu1, x1, y1i, . . . , hun, xn, yni) + f (hu1, y1, z1i, . . . , hun, yn, zni) < f (hu1, x1, z1i, . . . , hun, xn, zni). In the following, we present a numerical example where we prove that the IOWAD operator does not always accomplish the triangle inequality.

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Example 2. Assume X = (8, 10, 20), Y = (6, 6, 10) and Z = (4, 2, 10). We assume the following weighting vector W = (0, 1, 0). We use the following order-inducing variables: U (X, Y) = (3, 7, 9), U(Y, Z) = (4, 6, 2) and U(X, Z) = (4, 6, 8).

IOWADðX; YÞ ¼ 0  j20  10j þ 1  j10  6j þ 0  j8  6j ¼ 4; IOWADðY; ZÞ ¼ 0  j6  2j þ 1  j6  4j þ 0  j10  10j ¼ 2; IOWADðX; ZÞ ¼ 0  j20  10j þ 1  j10  2j þ 0  j8  4j ¼ 8: As we can see, 4 + 2 < 8; and therefore, IOWAD(X, Y) + IOWAD (Y,Z) < IOWAD (X, Z). We have thus proved that the IOWAD operator does not always accomplish the triangle inequality. Note that the order-inducing variables that we have used include the OWAD operator (Merigó & Gil-Lafuente, 2007, 2010) as a particular case. Therefore, we have also proved with this example that the OWAD operator does not accomplish the triangle inequality. It is also possible to develop a similar formulation of the IOWAD operator by first reordering the arguments and then calculating the distances. However, this formulation does not accomplish the usual properties of distance measures, such as commutativity. This measure can be formulated in the following way:

gðhu1 ; x1 ; y1 i; hu2 ; x2 ; y2 i; . . . ; hun ; xn ; yn iÞ ¼

n X

wk ck ;

ð19Þ

k¼1

where ck is the jxk  ykj value of the triplet hui x, i, yii; xk and yk are the ith argument variables of the sets X = {x1, . . . , xn} and Y = {y1, . . . , yn}, respectively, having the kth largest ui; and ui is the order-inducing variable. Note that if the reordering k is equal to the reordering j, this measure becomes the IOWAD operator described in Eq. (6), which can be proven as follows: Theorem 8. Assume f is the IOWAD operator and g is the measure explained in Eq. (19). If k = j, then

case, it would mean replacing the tied arguments by their normalized Hamming distance. In the analysis of the order-inducing variables of the IOWAD operator, we should note that the values used can be drawn from any space, with having a linear ordering the only requirement. Therefore, it is possible to use different kinds of attributes for the order-inducing variables that permit us to, for example, mix numbers with words in the aggregations. Note also that in some situations it is possible to use the implicit lexicographic ordering associated with words such as the ordering of words in dictionaries (Yager & Filev, 1999). 4. Families of IOWAD operators By using a different manifestation of the weighting vector, we are able to obtain different types of IOWAD operators, such as the normalized Hamming distance (NHD), the weighted Hamming distance (WHD), the ordered weighted averaging distance (OWAD) operator, the step-IOWAD, the window-IOWAD, the medianIOWAD, the olympic-IOWAD and the centered-IOWAD. Remark 1. For example, the maximum distance, the minimum distance, the step-IOWAD, the NHD, the WHD and the OWAD are obtained as follows:  The maximum distance is found if wp = 1 and wj = 0, for all j – p, and up = max{ai}.  The minimum distance if wp = 1 and wj = 0, for all j – p, and up = min{ai}.  More generally, if wk = 1 and wj = 0 for all j – k, we get the stepIOWAD operator.  The NHD is formed when wj = 1/n for all i.  The WHD is obtained when the ordered position of ui is the same as ai.  The OWAD is found if the ordered position of ui is the same as the ordered position of the values of jxi  yij, for all i.

gðhu1 ; x1 ; y1 i; hu2 ; x2 ; y2 i; . . . ; hun ; xn ; yn iÞ ¼ f ðhu1 ; x1 ; y1 i; hu2 ; x2 ; y2 i; . . . ; hun ; xn ; yn iÞ:

ð20Þ

Another issue to consider is the different measures used in the OWA literature for characterizing the weighting vector. For example, we could consider the entropy of dispersion, the balance operator, the divergence of W and the degree of orness (Merigó, 2008; Yager, 1988). The entropy of dispersion is defined as follows:

HðWÞ ¼ 

n X

wj lnðwj Þ:

ð21Þ

j¼1

For the balance operator, we get

BALðWÞ ¼

 n  X n þ 1  2j wj : n1 j¼1

n X

wj

j¼1



nj  aðWÞ n1

2 :

ð23Þ

The degree of orness can be defined in the following way:

aðWÞ ¼

n X j¼1

wj

 nj : n1

Remark 3. Note that it is possible to present a general form of the olympic-IOWAD operator, considering that wj = 0 for j = 1, 2, . . . , k, n, n  1, . . . , n  k + 1; and for all others, wj ¼ 1=ðn  2kÞ, where k < n/2. Note that if k = 1, then this general form becomes the usual olympic-IOWAD. If k = (n  1)/2, then it becomes the medianIOWAD operator.

ð22Þ

For the divergence of W, we get

DIVðWÞ ¼

Remark 2. Another particular case is the olympic-IOWAD operator. This operator is obtained when w1 = wn = 0, and for all others, wj ¼ 1=ðn  2Þ. Note that if n = 3 or n = 4, the olympic-IOWAD becomes the median-IOWAD.



ð24Þ

where wj is the wj weight of the IOWAD aggregation ordered according to the values of arguments jxi  yij. A further issue is the problem of ties in the reordering process of the order-inducing variables. To solve this problem, we recommend the policy explained by Yager and Filev (1999), namely, replacing the tied arguments by their average. Note that in this

Remark 4. Additionally, it is also possible to present the contrary case of the general olympic-IOWAD operator. In this case, wj = (1/ 2k) for j = 1, 2, . . . , k, n, n  1, . . . , n  k + 1; and wj = 0, for all others, where k < n/2. Note that if k = 1, then we get the contrary case of the median-IOWAD. Remark 5. Another interesting family is the S-IOWAD operator based on the S-OWA operator (Yager, 1993). These operators can be subdivided in three classes: the ‘‘orlike”, the ‘‘andlike” and the generalized S-IOWAD operators. The generalized S-IOWAD operator is obtained when wp = (1/n)(1  (a + b)) + a and up = max{ai}, wq = (1/n)(1  (a + b)) + b and uq = min{ai} and wj = (1/n)(1  (a + b)) for j – p, q; where a, b 2 [0, 1] and a + b 6 1. Note that if a = 0, the generalized S-IOWAD operator becomes the ‘‘andlike” S-IOWAD, and if b = 0, it becomes the ‘‘orlike” S-IOWAD. Note also that if a + b = 1, we get the induced Hurwicz distance criteria.

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Remark 6. A further family that could be used is the centeredIOWAD operator, which is based on Yager (2007). An IOWAD operator can be defined as a centered aggregation operator if it is symmetric, strongly decaying and inclusive.  It is symmetric if wj = wj+n1.  It is strongly decaying when: i < j 6 (n + 1)/2, then wi < wj, and when i > j P (n + 1)/2, then wi < wj.  It is inclusive if wj > 0. Note that it is possible to consider a softening of the second condition by using wi 6 wj instead of wi < wj (softly decaying centered-IOWAD operator). Another particular situation of the centered-IOWAD appears if we remove the third condition (noninclusive centered-IOWAD). Remark 7. Using a similar methodology, we could develop numerous other families of IOWAD operators. For more information, refer to Ahn (2009), Beliakov et al. (2007), Chiclana et al. (2007), Emrouznejad (2008), Liu (2008), Merigó (2008), Merigó and Gil-Lafuente (2008a, 2008b, 2009a), Xu (2005), Yager (1993, 2009a, 2009b, 2009c). 5. Choquet integrals with distance measures and induced aggregation operators Following Bolton, Gader, and Wilson (2008), Choquet (1953), Mesiar (1995), Tan and Chen (2010), and Yager (2004), it is possible to develop an extension of the IOWAD operator by using the discrete Choquet integral. This method results in the induced Choquet distance integral aggregation (ICDIA) operator. Before presenting this new result, let us define the concept of fuzzy measure and the Choquet integral. The fuzzy measure (nonadditive measure) was introduced by Sugeno (1974) and it can be defined as follows: Definition 8. Let X be a universal set X = {x1, x2, . . . , xn} and P(X) the power set of X. A fuzzy measure on X is a set function on m: P(X) ? [0, 1] that satisfies the following conditions: (1) m(;) = 0, m(X) = 1 (boundary conditions) and (2) if A, B 2 P(X) and A # B, then m(A) 6 m(B) (monotonicity). The Choquet integral (Choquet, 1953) can be defined as follows in its discrete form: Definition 9. Let f be a positive real-valued function f: X ? R+ and m be a fuzzy measure on X. The (discrete) Choquet integral of f with respect to m is:

C m ðf1 ; f2 ; . . . ; fn Þ ¼

n X

fðiÞ ½mðAðiÞ Þ  mðAði1Þ Þ;

ð25Þ

ICDIAðhu1 ; x1 ; y1 i; hu2 ; x2 ; y2 i; . . . ; hun ; xn ; yn iÞ ¼

n X

bj ½mðAðiÞ Þ  mðAði1Þ Þ;

j¼1

ð26Þ

where bj is the jxi  yij value of the ICDIA triplet hui, xi, yii having the jth largest ui; ui is the order-inducing variable; the jxi  yij is the argument variable represented in the form of individual distances; A(i) = {x(1), . . . ,x(i)} i P 1; and A(0) = ;. This approach can be generalized by using both generalized and quasi-arithmetic means (Merigó & Gil-Lafuente, 2009a). For example, by using quasi-arithmetic means, we get the induced quasiarithmetic Choquet distance integral aggregation (Quasi-ICDIA) operator, which can be defined as follows: Definition 11. Let m be a fuzzy measure on X. An induced quasiarithmetic Choquet distance integral aggregation (Quasi-ICDIA) operator of dimension n is a function QICDIA: Rn  Rn  Rn ? R, such that: ! n X 1 QICDIAðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ ¼ g gðbj Þ½mðAðiÞ Þ  mðAði1Þ Þ ; j¼1

ð27Þ where g is a strictly continuous monotonic function; bj is the jxi  yij value of the ICDIA triplet hui, xi, yii having the jth largest ui, ui is the order-inducing variable; jxi  yij is the argument variable represented in the form of individual distances; A(i) = {x(1), . . . , x(i)} i P 1; and A(0) = ;. Note that this new approach can also be formulated in a more complete way by using a function for the formation of the orderinducing variables and the arguments as explained in Definition 6, that is, by using a Quasi-IMD triplet (hro(ui), sp(xi), sq(yi)i). A key feature in this new aggregation operator is that it includes a wide range of aggregation operators. For example:

 The quasi-arithmetic Choquet distance integral aggregation (Quasi-CDIA): When the ordered position of the order-inducing variables ui is the same than the ordered position of bj such that bj is the jth largest of the jxi  yij.  The induced generalized Choquet distance integral aggregation (IGCDIA): When g(b) = bk.  The generalized Choquet distance integral aggregation (GCDIA): When g(b) = bk and the ordered position of the order-inducing variables ui is the same than the ordered position of bj such that bj is the jth largest of the jxi  yij. We can also consider a wide range of families of all the previous cases following the methodology explained in Section 4 and in Merigó and Gil-Lafuente (2009a). For example, we could analyze the following cases:

i¼1

where () indicates a permutation on X such that f(1) P f(2) P    P f(n). In other words, f(i) is the ith largest value in the set {f1, f2, . . . , fn}; A(i) = {x(1), . . . , x(i)} i P 1; and A(0) = ;. In the following, we present the ICDIA aggregation as an extension of the Choquet integral that uses order-inducing variables and distance measures. Note that by using distance measures we are able to compare two sets of variables. Also by using orderinducing variables we can deal with complex reordering processes in the analysis. This operator can be defined as follows: Definition 10. Let m be a fuzzy measure on X. An induced Choquet distance integral aggregation (ICDIA) operator of dimension n is a function ICDIA: Rn  Rn  Rn ? R, such that:

 The induced Choquet distance integral aggregation (ICDIA): when g(b) = b.  The induced quadratic Choquet distance integral aggregation (IQCDIA): when g(b) = b2.  The induced harmonic Choquet integral aggregation: when g(b) = b1.  The Choquet distance integral aggregation (CDIA) (Bolton et al., 2008): when g(b) = b, and the ordered position of the orderinducing variables ui is the same as the ordered position of bj such that bj is the jth largest of the jxi  yij.  The quadratic Choquet distance integral aggregation (QCDIA): when g(b) = b2, and the ordered position of the order-inducing variables ui is the same as the ordered position of bj.

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 The harmonic Choquet distance integral aggregation: when g(b) = b1, and the ordered position of the order-inducing variables ui is the same as the ordered position of bj. We could also consider more complex situations by using interval numbers, fuzzy numbers, linguistic variables and more complex techniques. Note that the Quasi-ICDIA operator includes several other cases, such as the step-Quasi-ICDIA, the Olympic-Quasi-ICDIA, the median-Quasi-ICDIA, the centered-Quasi-ICDIA, the S-Quasi-ICDIA, and many more. However, we believe that those presented here are some of the most relevant. 6. Multiperson decision-making with the IOWAD operator The IOWAD operator is applicable in a wide range of situations, such as decision-making, statistics, engineering and economics. In summary, all of the studies that use the Hamming distance can be revised and extended by using this new approach. In this paper, we consider a decision-making application in the selection of investments by using a multi-person analysis. The main motivation for using the IOWAD operator in the selection of investments is that the decision-maker wants to decide according to a complex attitudinal character and needs to use the opinions of several persons (experts) to correctly assess the problem. This can be useful in numerous situations, such as when the board of directors of a company wants to make a decision. Obviously, the attitudinal character of the board of directors is highly complex because it involves the decisions of different persons, and their interests may be different. The process to follow in the selection of investments with the IOWAD operator in multi-person decision-making can be summarized as follows. Note that in the literature, we may find several other group decision making models, e.g., Alonso, Cabrerizo, Chiclana, Herrera, and Herrera-Viedma (2009), Cabrerizo, Alonso, and Herrera-Viedma (2009), Wei (2009) and Xu (2009a, 2009b, 2009c). Step 1: Let A = {A1, A2, . . . , An} be a set of finite alternatives, and C = {C1, C2, . . . , Cn}, a set of finite characteristics (or attributes), forming the matrix (xhi)mn. Let E = {E1, E2, . . ., Ep} be a finite set of decision makers. Let V = (v1, v2, . . ., vp) be the weighting vector Pp of the decision-makers such that k¼1 v k ¼ 1 and vk 2 [0, 1]. Each decision-maker provides their own payoff matrix   ðkÞ xhi . mn

Step 2: Fixation of the ideal levels of each characteristic to form the ideal investment (see Table 1) where P is the ideal investment expressed by a fuzzy subset, Ci is the ith characteristic to consider and yi 2 [0, 1]; i = 1, 2, . . ., n, is a number between 0 and 1 for the ith characteristic. Each decision-maker provides ðkÞ their own ideal investment yi . Step 3: Calculate the order-inducing variables (uhi)mn to be used in the payoff matrix for each alternative h and characteristic i. Calculate also the weighting vector W to be used in the IOWAD aggregation. Note that W = (w1, w2, . . ., wn) such that Pn j¼1 wj ¼ 1 and wj 2 [0, 1]. Step 4: Compare the ideal investment with the different alternatives considered using the IOWAD operator for each expert (person). In this step, the objective is to express numerically

Note that this aggregation process can be summarized using the following aggregation operator, we call the multi-person–IOWAD (MP–IOWAD) operator. Definition 12. An MP–IOWAD operator is an aggregation operator P that has a weighting vector V of dimension p with pk¼1 v k ¼ 1 and vk 2 [0, 1], and a weighting vector W of dimension n with P n j¼1 wj ¼ 1 and wj 2 [0, 1], such that:

       MP—IOWAD hu1 ; x11 ; . . . ; xp1 ; y11 ; . . . ; yp1 i; . . . ; hun ; x1n ; . . . ; xpn ; n  1  X yn ; . . . ; ypn i ¼ wj bj ; ð28Þ j¼1

where bj is the jxi -yij value of the MP–IOWAD triplet hui, xi, yii having the jth largest  k ui, k ui isk thek  order-inducing variable; and Pp   , x  y  is the argument variable projxi  yi j ¼ i i k¼1 v k xi  yi vided by each person (or expert) represented in the form of individual distances. The MP–IOWAD operator has properties similar to those explained in Section 3, such as the distinction between descending and ascending orders, quasi-mixtures operators, and so on. The MP–IOWAD operator includes a wide range of particular cases following the methodology explained in Section 4. Thus, we can find as special cases:  The multi-person–normalized Hamming distance (MP–NHD) operator.  The multi-person–weighted Hamming distance (MP–WHD) operator.  The multi-person–OWAD (MP–OWAD) operator.  The multi-person–OWA (MP–OWA) operator.  The multi-person–IOWA (MP–IOWA) operator.  The multi-person–WA (MP–WA) operator. Note that it is also possible to consider more complex situations by using different types of aggregation operators in the aggregation of the experts’ opinions because in Definition 12 we assumed that the experts’ opinions were aggregated by using the WA operator. However, it is also possible to use the OWA operator and the IOWA operator, among others. Moreover, it is possible to develop a similar model by using Choquet integrals obtaining the multiperson–ICDIA (MP–ICDIA) operator. 7. Illustrative example

Table 1 Ideal investment.

P=

the distance between the ideal investment and the different alternatives considered. Note that it is possible to consider a wide range of IOWAD operators, such as those described in Sections 3 and 4. Step 5: Use the weighted average (WA) to aggregate the information of the decision-makers E by using the weighting vector V. The result is the collective  payoff matrix (jxhi – yhij)mn. Thus, P jxhi  yhi j ¼ pk¼1 v k xkhi  ykhi . Step 6: Calculate the aggregated results by using the IOWAD operator explained in Eq. (6). Consider different particular manifestations of the IOWAD operator by using different expressions in the weighting vector, as explained in Section 4. Step 7: Adopt decisions according to the results found in the previous steps. Select the alternative/s that provides the best result/s. Moreover, establish an ordering or a ranking of the alternatives from the most to the least preferred alternative to enable consideration of more than one selection.

C1

C2

...

Ci

...

Cn

y1

y2

...

yi

...

yn

In the following, we develop a brief illustrative example of the new approach in a multi-person decision-making problem concerning investment selection.

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Assume a decision-maker wants to invest money in a company. After analyzing the market, he considers five possible alternatives: (1) (2) (3) (4) (5) (6)

Invest Invest Invest Invest Invest Invest

in in in in in in

a a a a a a

chemical company called A1. food company called A2. computer company called A3. car company called A4. furniture company called A5. pharmaceutical company called A6.

A1 A2 A3 A4 A5 A6

After careful review of the information, the group of experts establishes the following general information about the investments. They summarize the information of the investments in six general characteristics C = {C1, C2, C3, C4, C5, C6}:      

C1: C2: C3: C4: C5: C6:

Table 3 Characteristics of the investments – expert 2. C2

C3

C4

C5

C6

0.6 0.7 0.7 0.6 0.7 0.6

0.8 0.6 0.6 0.7 0.8 0.4

0.5 0.8 0.8 0.5 0.7 0.8

0.6 0.6 0.7 0.6 0.7 0.7

0.4 0.7 0.8 0.8 0.6 0.6

0.8 0.7 0.8 0.7 0.8 0.7

Table 4 Characteristics of the investments – expert 3.

A1 A2 A3 A4 A5 A6

Benefits in the short term. Benefits in the mid term. Benefits in the long term. Risk of the investment. Difficulty of the investment. Other factors.

C1

C1

C2

C3

C4

C5

C6

0.7 0.7 0.6 0.6 0.7 0.4

0.6 0.6 0.5 0.7 0.8 0.5

0.6 0.7 0.8 0.7 0.6 0.9

0.6 0.6 0.5 0.5 0.7 0.7

0.4 0.6 0.8 0.8 0.6 0.6

0.7 0.7 0.8 0.6 0.8 0.6

C1

C2

C3

C4

C5

C6

0.9 0.8 0.8

0.9 0.9 0.9

1 1 1

0.9 1 1

0.8 0.9 0.9

0.9 0.8 0.8

C1

C2

C3

C4

C5

C6

12 17 16 14 22 15

6 13 14 17 25 13

24 9 12 20 27 11

17 12 10 12 14 17

30 7 8 9 16 19

14 4 6 6 18 21

Table 5 Ideal investment.

The group of company experts is constituted by three persons, each offering their own opinions regarding the results obtained with each investment. The results are shown in Tables 2–4. Note that the results are valuations (numbers) between 0 and 1. According to the objectives of the decision-maker, each expert establishes his own ideal investment. The results are shown in Table 5. To analyze the attitudinal character of the group of experts, we consider that they use order-inducing variables shown in Table 6, which represents the complex attitudinal character in the decision process. Note that in this example, the decision-maker assumes a different attitudinal character for each alternative because the results given by each alternative are not equal. The main advantage of using order inducing variables is that we can represent complex decision processes that include psychological factors such as time pressure, personal affects to each alternative and other related aspects. With this information, we can make an aggregation to make a decision. First, we aggregate the information of the three experts to obtain a unified payoff matrix represented in the form of individual distances between the available and ideal alternatives. We use the WA to obtain this matrix while assuming that V = (0.3, 0.3, 0.4). The results are shown in Table 7. It is now possible to develop different methods based on the IOWAD operator for the selection of an investment. We are able to provide a more complete picture to the decision-maker because we are able to consider different future scenarios. Due to our uncertainty, we do not know which scenario is the correct scenario. Therefore, the representation of different particular cases that could happen (from the minimum to the maximum) seems to be useful for gaining a complete picture of the different future situations. Thus, the decision-maker knows the results that can be obtained with each alternative and thus, select the one that

seems to be in closest accordance with his interests. Note that all of this analysis is done in uncertainty so we do not know the correct answer until the future becomes the present. In this example, we consider the maximum distance, the minimum distance, the NHD, the WHD, the step-IOWAD (k = 2), the induced Hurwicz distance criteria (a = 0.4), the OWAD, the AOWAD, the IOWAD, the AIOWAD, the median and the olympic-IOWAD operators. We assume the following weighting vector W = (0.1, 0.1, 0.2, 0.2, 0.2, 0.2). The results are shown in Tables 8 and 9. As we can see, for most of the cases the best alternative is A3 because it seems to be the one with the lowest distance to the ideal investment. However, for some particular situations, we may find another optimal choice. Therefore, it is of interest to establish an

Table 2 Characteristics of the investments – expert 1.

Table 7 Collective results in the form of individual distances.

A1 A2 A3 A4 A5 A6

C1

C2

C3

C4

C5

C6

0.7 0.8 0.5 0.6 0.9 0.8

0.8 0.6 0.4 0.7 0.8 0.3

0.6 0.9 0.8 0.6 0.4 0.7

0.7 0.7 0.3 0.7 0.7 0.7

0.5 0.6 0.8 0.8 0.7 0.6

0.9 0.7 0.8 0.6 0.8 0.7

E1 E2 E3

Table 6 Order-inducing variables.

A1 A2 A3 A4 A5 A6

A1 A2 A3 A4 A5 A6

C1

C2

C3

C4

C5

C6

0.16 0.1 0.23 0.23 0.07 0.25

0.18 0.3 0.4 0.2 0.1 0.49

0.43 0.21 0.2 0.39 0.43 0.19

0.34 0.34 0.47 0.38 0.27 0.27

0.44 0.24 0.07 0.07 0.24 0.27

0.04 0.13 0.03 0.2 0.03 0.17

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J.M. Merigó, M. Casanovas / Computers & Industrial Engineering 60 (2011) 66–76

Table 8 Aggregated results 1.

A1 A2 A3 A4 A5 A6

Maximum

Minimum

NHD

WHD

Step (k = 2)

Hurwicz

0.44 0.34 0.47 0.39 0.43 0.49

0.04 0.1 0.03 0.07 0.03 0.17

0.265 0.22 0.233 0.245 0.19 0.273

0.284 0.224 0.217 0.251 0.211 0.254

0.43 0.3 0.4 0.2 0.1 0.27

0.284 0.118 0.11 0.276 0.334 0.182

Table 9 Aggregated results 2.

A1 A2 A3 A4 A5 A6

OWAD

AOWAD

IOWAD

AIOWAD

Median

Olympic

0.231 0.2 0.193 0.217 0.158 0.252

0.298 0.241 0.27 0.267 0.218 0.292

0.231 0.224 0.217 0.235 0.175 0.284

0.284 0.227 0.27 0.267 0.177 0.26

0.19 0.275 0.335 0.305 0.05 0.26

0.2425 0.2725 0.285 0.22 0.11 0.32

Ordering

Minimum

Ordering

Appendix A. Proofs of the theorems

A2 A4 A5 A1 A3 A6

OWAD

A5 A3 A2 A4 A1 A6

A3 = A5 A1 A4 A2 A6

AOWAD

A5 A2 A4 A3 A6 A1

NHD

A5 A2 A3 A4 A1 A6

IOWAD

A5 A3 A2 A1 A4 A6

WHD

A5 A3 A2 A4 A6 A1

AIOWAD

A5 A2 A6 A4 A3 A1

StepIOWAD Hurwicz

A5 A4 A6 A2 A3 A1

MedianIOWAD OlympicIOWAD

A5 A1 A6 A2 A4 A3

A3 A2 A6 A4 A1 A5

Acknowledgements We would like to thank the anonymous reviewers for valuable comments that have improved the quality of the paper. Support from projects JC2009-00189 and MC238206 is gratefully acknowledged.

Table 10 Ordering of the investments.

Maximum

operators. We have also extended the IOWAD by using Choquet integrals, thus obtaining the ICDIA and Quasi-ICDIA operators. We have analyzed an application of the new approach in a multi-person decision-making problem regarding the selection of investments. To do so, we have introduced the MP–IOWAD operator. We have seen that this approach provides better information for decision-making because it is able to consider a wide range of scenarios depending on the interests of the decision-maker. Moreover, by using order-inducing variables, it is possible to consider different scenarios according to complex attitudinal characters. We have also seen that, depending on the particular type of aggregation operator used, the results may lead to different decisions. In future research, we expect to develop further extensions of this approach by using other characteristics in the decision process, such as uncertain information (e.g., interval numbers, fuzzy numbers, and linguistic variables), weighted and generalized aggregation operators, and more complex structures. We will also consider other decision-making problems and other applications.

A5 A4 A1 A2 A3 A6

ordering of the investments for each particular case. Note that the best choice is the one with the lowest distance. The results are shown in Table 10. As we can see, depending on the particular type of distance aggregation operator used, the results may be different, thus leading to different decisions. Note that the main advantage of using the IOWAD operator is that we can use order-inducing variables that represent complex reordering processes in the aggregation in order to consider more complex information in the decisionmaking problem.

Proof of Theorem 1. Let

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ ¼

n X

wj bj ;

ð29Þ

wj ej :

ð30Þ

j¼1

f ðhu1 ; c1 ; d1 i; . . . ; hun ; cn ; dn iÞ ¼

n X j¼1

Because (hu1, x1, y1i, . . . , hun, xn, yni) is a permutation of (hu1, c1, d1i, . . . , hun, cn, dni), and we have jxi  yij = jci  dij, for all i, and then

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ ¼ f ðhu1 ; c1 ; d1 i; . . . ; hun ; cn ; dn iÞ:



Note that the commutativity of the IOWAD can also be studied from the context of a distance measure, which can be proved with the following theorem: Proof of Theorem 2. Let

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ ¼

n X

wj bj ;

ð31Þ

wj ej :

ð32Þ

j¼1

8. Conclusions

f ðhu1 ; y1 ; x1 i; . . . ; hun ; yn ; xn iÞ ¼

n X j¼1

We have presented a decision-making approach that uses distance measures and induced aggregation operators. This approach is based on the use of the IOWAD operator. IOWAD is an extension of the OWA operator that uses the Hamming distance and orderinducing variables in the reordering process. The main advantage of this operator is that it is able to consider complex attitudinal characters in the decision process. This is a key feature in decision-making because there are usually numerous factors that affect decisions, such as when the decision-maker is actually a group of persons like a company’s board of directors. To provide a more complete formulation, we have further generalized the IOWAD operator by using mixture operators, generalized and quasi-arithmetic means and infinitary aggregation

Because jxi  yij = jyi  xij, for all i, then

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ ¼ f ðhu1 ; y1 ; x1 i; . . . ; hun ; yn ; xn iÞ:



Proof of Theorem 3. Let

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ ¼

n X

wj bj ;

ð33Þ

wj ej :

ð34Þ

j¼1

f ðhu1 ; c1 ; d1 i; . . . ; hun ; cn ; dn iÞ ¼

n X j¼1

J.M. Merigó, M. Casanovas / Computers & Industrial Engineering 60 (2011) 66–76

Because jxi  yij P jci  dij, for all i, then

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ P f ðhu1 ; c1 ; d1 i; . . . ; hun ; cn ; dn iÞ:



Proof of Theorem 4. Let max{jxi  yij} = c and min{jxi  yij} = d; then

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ ¼

n X

wj bj 6

n X

j¼1

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ ¼

n X

j¼1

Pn

j¼1 wj

n X

wj ;

ð35Þ

j¼1

n X

wj bj P

j¼1

Because

wj c ¼ c

wj d ¼ d

j¼1

n X

wj :

ð36Þ

j¼1

¼ 1, we get

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ 6 c;

ð37Þ

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ P d:

ð38Þ

Therefore,

minfjxi  yi jg 6 f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ 6 maxfjxi  yi jg:



Proof of Theorem 5. Because jxi  yij = a, for all i, we have

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ ¼

n X

wj bj ¼

n X

j¼1

Because

Pn

j¼1 wj

wj a ¼ a

j¼1

n X

wj :

ð39Þ

j¼1

¼ 1, we get

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ ¼ a:



Proof of Theorem 6. Let

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ ¼

n X

wj bj ;

ð40Þ

j¼1

Because jxi  yij P 0, for all i, we obtain

f ðhu1 ; x1 ; y1 i; . . . ; hun ; xn ; yn iÞ P 0:



Proof of Theorem 7. Let

f ðhu1 ; x1 ; x1 i; . . . ; hun ; xn ; xn iÞ ¼

n X

wj bj ;

ð41Þ

j¼1

Because xi = xi, jxi  xij = 0, for all i, therefore, we get

f ðhu1 ; x1 ; x1 i; . . . ; hun ; xn ; xn iÞ ¼ 0:



Proof of Theorem 8. Let

gðhu1 ; x1 ; y1 i; hu2 ; x2 ; y2 i; . . . ; hun ; xn ; yn iÞ ¼ f ðhu1 ; x1 ; y1 i; hu2 ; x2 ; y2 i; . . . ; hun ; xn ; yn iÞ ¼

n X k¼1 n X

wk ck ;

ð42Þ

wj bj ;

ð43Þ

j¼1

Because k = j, ck = bj, and thus, jxk  ykj = jxi  yij, for all i, we obtain

gðhu1 ; x1 ; y1 i; hu2 ; x2 ; y2 i; . . . ; hun ; xn ; yn iÞ ¼ f ðhu1 ; x1 ; y1 i; hu2 ; x2 ; y2 i; . . . ; hun ; xn ; yn iÞ



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