Distance measures with heavy aggregation operators

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/authorsrights Author's personal copy Applied Mathematical Modelling 38 (2014) 3142–3153 Contents lists available at ScienceDirect Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm Distance measures with heavy aggregation operators José M. Merigó a,b,⇑, Montserrat Casanovas a, Shouzhen Zeng c a Department of Business Administration, University of Barcelona, Av. Diagonal 690, Barcelona 08034, Spain Manchester Business School, The University of Manchester, Booth Street West, M15 6PE Manchester, United Kingdom c College of Computer and Information, Zhejiang Wanli University, Ningbo 315100, China b a r t i c l e i n f o Article history: Received 20 July 2012 Received in revised form 4 November 2013 Accepted 29 November 2013 Available online 17 December 2013 Keywords: OWA operator Aggregation operator Distance measure Decision making a b s t r a c t The use of distance measures and heavy aggregations in the ordered weighted averaging (OWA) operator is studied. We present the heavy ordered weighted averaging distance (HOWAD) operator. It is a new aggregation operator that provides a parameterized family of aggregation operators between the minimum distance and the total distance operator. Thus, it permits to analyze an aggregation from its usual average (normalized distance) to the sum of all distances available in the aggregation process. We analyze some of its main properties and particular cases such as the normalized Hamming distance, the weighted Hamming distance and the OWA distance (OWAD) operator. This approach is generalized by using quasi-arithmetic means obtaining the quasi-arithmetic HOWAD (Quasi-HOWAD) operator and with norms obtaining the heavy OWA norm (HOWAN). Further extensions to this approach are presented by using moving averages forming the moving HOWAD (HOWMAD) and the moving Quasi-HOWAN (Quasi-HOWMAN) operator. The applicability of the new approach is studied in a decision making model regarding the selection of national policies. We focus on the selection of monetary policies. The key advantage of this approach is that we can consider several sources of information that are independent between them. Crown Copyright Ó 2013 Published by Elsevier Inc. All rights reserved. 1. Introduction Distance measures are very useful techniques that have been used in a wide range of applications [1–3]. The most common types of distance measures are the total distance and the normalized distance. The first one only considers the sum of all the individual distances considered and the normalized distance considers an average of the individual distances. A very popular distance is the Hamming distance (also known as the Manhattan distance depending on the problems considered). The Hamming distance [3] can be normalized in different ways depending on the interests of the specific problem considered. For example, it is possible to consider the normalized Hamming distance (NHD) and the weighted Hamming distance (WHD) that use the average and the weighted average (WA), respectively. Recently, several authors have suggested the use of the ordered weighted averaging (OWA) operator in the Hamming distance obtaining the ordered weighted averaging distance (OWAD) operator [4–6]. By using the OWA operator [7–9], we are able to provide a wide range of aggregation operators between the maximum and the minimum. Since its introduction, it has been receiving increasing attention. For example, Merigó and Casanovas extended this approach by using linguistic variables [10]. They also developed a generalization by using induced aggregation operators [11]. Furthermore, they also extended this approach by using the Euclidean distance [12] and the Minkowski distance [13]. Zeng and Su [14] ⇑ Corresponding author at: Manchester Business School, The University of Manchester, Booth Street West, M15 6PE Manchester, United Kingdom. Tel.: +44 1 613063451. E-mail addresses: [email protected], [email protected] (J.M. Merigó), [email protected] (M. Casanovas), [email protected] (S. Zeng). 0307-904X/$ - see front matter Crown Copyright Ó 2013 Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.11.036 Author's personal copy J.M. Merigó et al. / Applied Mathematical Modelling 38 (2014) 3142–3153 3143 studied the use of intuitionistic fuzzy sets in the OWAD operator They also considered imprecise environments with interval numbers, fuzzy information and linguistic generalized aggregation operators [15–17]. Merigó and Gil-Lafuente developed an application in human resource management [18] and in sport management [19]. Yager [20] generalized it by using norms. Merigó et al. [21,22] presented an extension by using similarity measures where the Hamming distance was included as a particular case. Xu and Xia [23] analyzed the use of hesitant fuzzy sets in the OWAD operator and Xu [24] considered fuzzy numbers. Merigó and Yager [25] studied distance measures with moving averages. An interesting extension is the heavy OWA (HOWA) operator [26,27]. The main advantage is that it allows the aggregation to move between the total operator and the usual OWA operator. Recently, Merigó and Casanovas developed several extensions by using induced aggregation operators [28,29]. In this paper, we present the heavy OWA distance (HOWAD) operator. It is a distance aggregation operator that uses the OWA operator allowing the aggregation to be between the total (or absolute) distance and the normalized (or relative) distance. That is, it uses distance measures in the HOWA operator. Therefore, we get a more complete distance aggregation operator that includes the normalized and the total distance operator. Note that in this paper we use the Hamming distance as the distance measure. But it is also possible to use other distance measures such as the Euclidean distance or the Minkowski distance. We study some of its main properties and particular cases. We further generalize this approach by using generalized and quasi-arithmetic means obtaining the generalized HOWAD (GHOWAD) operator and the quasi-arithmetic HOWAD (Quasi-HOWAD) operator. Their main advantage is that they include a wide range of particular cases including the HOWAD operator and quadratic HOWAD (HOWQAD) operator. Moreover, we extend this approach by using norms [20] obtaining the HOWA norm (HOWAN) that include a wide range of particular cases including the addition OWA (A-OWA) and the multiplication OWA (M-OWA). We also extend this approach by using moving averages [25,30] obtaining the heavy ordered weighted moving averaging distance (HOWMAD). Thus, we can analyze the aggregation process in a dynamic way. Moreover, we generalize this model by using generalized aggregation operators and norms forming the quasi-arithmetic heavy ordered weighted moving averaging norm (QuasiHOWMAN) operator. We study the applicability of this new approach and we see that it is very broad because we can apply it in a wide range of fields including statistics, economics and engineering. We focus on a decision making problem regarding the selection of strategies. We see that the use of HOWAD operators permits to obtain a more complete representation of the aggregation process when the available information is independent. This paper is organized as follows. In Section 2 we briefly review some basic concepts. In Section 3 we present the HOWAD operator and study some of its main families. Section 4 presents several generalizations by using generalized aggregation operators, Section 5 by using norms and Section 6 with moving averages. Section 7 develops an illustrative example of the new approach. Finally, in Section 8, we summarize the main conclusions of the paper. 2. Preliminaries 2.1. The Hamming distance The Hamming distance [2] is a useful technique for calculating the distance between two elements, two sets, etc. In order to define the Hamming distance, first, we will define a distance measure. Basically, a distance measure has to 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 example, the weighted Hamming distance (WHD) between two sets X = (x1, x2, . . ., xn) and Y = (y1, y2, . . ., yn), can be defined as follows. Definition 1. A weighted Hamming distance of dimension n is a mapping WHD: Rn  Rn ? R that has an associated weighting vector W with wi e [0, 1] and the sum of the weights is 1, such that: dWHD ðX; YÞ ¼ n X wi jxi  yi j; ð1Þ i¼1 where xi and yi are the ith arguments of the sets X and Y. 2.2. The OWA operator The OWA operator was introduced by Yager [7]. It provides a parameterized family of aggregation operators between the maximum and the minimum. It can be defined as follows: Author's personal copy 3144 J.M. Merigó et al. / Applied Mathematical Modelling 38 (2014) 3142–3153 Definition 2. An OWA operator of dimension n is a mapping OWA: Rn ? R that has an associated weighting vector W of P dimension n with wj e [0, 1] and nj¼1 wj ¼ 1, such that: OWAða1 ; . . . ; an Þ ¼ n X wj bj ; ð2Þ j¼1 where bj is the jth largest of the ai and ai is the argument variable. Note that the OWA operator is commutative, monotonic, bounded and idempotent. Furthermore, it is possible to study a wide range of particular cases including the minimum, the maximum, and the average [7,31]. For further reading see, e.g. [8,9,32] 2.3. The heavy OWA operator The HOWA operator [26] is an extension of the OWA operator that allows the weighting vector to sum up to n. Thus, we are able to analyze the aggregation from the total operator to the usual OWA where the lower bound is the minimum. It can be defined as follows: Definition 3. A HOWA operator is a mapping HOWA: Rn ? R that has an associated weighting vector W with wj e [0, 1] and P 1 6 nj¼1 wj 6 n, such that: HOWAða1 ; . . . ; an Þ ¼ n X wj bj ; ð3Þ j¼1 where bj is the jth largest of the ai and ai is the argument variable. P Note that it is possible to formulate the HOWA operator in a more general way by using 0 < nj¼1 wj 6 n. Moreover, if we Pn allow wj e [0, 1], we get 0 < j¼1 wj < 1. Furthermore, if we allow wj e [1, 1], then, we can also obtain P 1 < nj¼1 wj < 1. Note that often in these situations, we will need to normalize the weighting vector. 3. The heavy OWA distance operator 3.1. Main concepts The heavy OWA distance (HOWAD) operator is a distance aggregation operator that uses the HOWA operator. It allows the aggregation to be between the total distance and the normalized distance. Thus, we can provide analyze the aggregation process in a more general way from the minimum distance to the total operator. It can be defined as follows. Definition 4. A HOWAD operator of dimension n is a mapping HOWAD: Rn  Rn ? R that has an associated weighting vector P W with wi e [0, 1] and 1 6 nj¼1 wj 6 n, such that: HOWADðhx1 ; y1 i; hx2 ; y2 i; . . . ; hxn ; yn iÞ ¼ n X wj Dj ; ð4Þ j¼1 where Dj is the jth largest of the |xi  yi| value, and |xi  yi| is the argument variable represented in the form of individual distances. Example 1. Assume the following arguments in an aggregation process: X = (17, 14, 22, 19), Y = (23, 16, 17, 12) with the following order inducing variables U = (4, 8, 2, 5). Assume the following weighting vector W = (0.2, 0.3, 0.5, 0.5). If we calculate the distance between X and Y using the HOWAD operator, we get the following: HOWADðX; YÞ ¼ 0:2  j14  16j þ 0:3  j19  12j þ 0:5  j17  23j þ 0:5  j22  17j ¼ 8: A more general definition of the heavy aggregation is possible by allowing the weighting vector to range between 1 and 1. Thus, the weighting vector W becomes: 1 6 n X wj 6 1: j¼1 By allowing the weighting vector to range between 1 and 1, we can drastically under or over estimate the aggregation according to the weighting vector used. In general, if the weighting vector is high and tends to 1, then, we are drastically over estimating the results because they can increase a lot tending to 1. And if the weighting vector is negative (or tends to 0), we are drastically under estimating the results because they decreases a lot and may tend to 1. Note that the meaning about using a weighting vector that ranges from 1 to 1, may be different than the one explained here. A negative Author's personal copy J.M. Merigó et al. / Applied Mathematical Modelling 38 (2014) 3142–3153 3145 weighting vector is not normal in basic problems of the real life. However, we may think on complex situations where they become relevant. For example, we can assume a situation where the aggregation of one variable decreases the final result because it has a negative effect. This may occur when analyzing markets with competitors where some of them are focused on the destruction of the other competitors rather than focusing on their own benefits. The HOWAD operator is commutative and monotonic from the context of an OWA aggregation. A further interesting feature is that the HOWAD operator is bounded by the minimum distance and the total distance operator. Note that if the weighting vector ranges between 1 and 1, then the heavy aggregation is not bounded. The HOWAD operator is also commutative from the context of the distance measure, nonnegative and reflexive. These properties can be proved with the following theorems. Theorem 1 (Commutativity – distance measure). Assume f is the HOWAD operator, then: f ðhx1 ; y1 i; . . . ; hxn ; yn iÞ ¼ f ðhy1 ; x1 i; . . . ; hyn ; xn iÞ: ð5Þ Proof. It is straightforward and thus omitted. h Theorem 2 (Nonnegativity). Assume f is the HOWAD operator, then: f ðhx1 ; y1 i; . . . ; hxn ; yn iÞ P 0: ð6Þ Proof. It is straightforward and thus omitted. h Theorem 3 (Reflexivity). Assume f is the HOWAD operator, then: f ðhx1 ; x1 i; . . . ; hxn ; xn iÞ ¼ 0: ð7Þ Proof. It is straightforward and thus omitted. h Note that it is possible to distinguish between the descending HOWAD (DHOWAD) and the ascending HOWAD (AHOWAD) operator by using wj = wnjþ1 , where wj is the jth weight of the DHOWAD and wnjþ1 the jth weight of the AHOWAD operator. Another interesting transformation can be developed [31] by using wi = (1 + wi)/(n  1). Furthermore, we can also analyze situations with buoyancy measures [31]. In this case, we assume that wi P wj, for i < j. Note that it is also possible to consider a stronger case known as extensive buoyancy measure where wi > wj, for i < j. Additionally, we can also consider the contrary case, that is, wi 6 wj, for i < j, and the contrary case of the extensive measure wi < wj, for i < j. In the analysis of the weighting vector |W| of the HOWAD operator, we can use a similar methodology as the one used by Yager for the HOWA operator [26]. Therefore, we can define it as b(W) = (|W|  1)/(n  1). Since |W| e [1, n], then, b e [0, 1]. As it can be seen, if b = 1, we get the total distance operator and if b = 0, we get the OWA distance (OWAD) operator. Note that it is possible to look to the negation of b. Then, q = 1  b. If q = 0, we get the total distance operator and if q = 1, we get the OWAD operator. Once analyzed the magnitude of |W|, it is possible to study different measures for characterizing the weighting vector. The first measure, the attitudinal character, can be defined as follows: aðWÞ ¼  n  1 X nj wj : jWj j¼1 n  1 ð8Þ As it can be seen, a(W) e [0, 1]. Note that the formulation is the same than the HOWA operator because the arguments do not affect the result. The second measure, the entropy of dispersion, can be defined as: HðWÞ ¼    n 1 X wj : wj ln jWj jWj j¼1 ð9Þ Note that for the total operator, H(W) = ln n. A third measure that can be used, is the divergence of W:  2 n 1 X nj Div ðWÞ ¼ wj  aðWÞ : jWj j¼1 n1 ð10Þ If |W| = n, we get the divergence of the total distance and it is the same divergence than the average. That is, Div(W) = (1/12)[(n + 1)/(n  1)]. Note also that these three measures are reduced to the usual definitions when |W| = 1. Author's personal copy 3146 J.M. Merigó et al. / Applied Mathematical Modelling 38 (2014) 3142–3153 3.2. Families of HOWAD operators By using a different manifestation in the weighting vector, we are able to obtain a wide range of particular types of HOWAD operators. Following the methodology explained in [26,29], we can obtain the following cases.              OWAD operator: When b = 0. Total distance operator: When b = 1. HOWA operator: If one of the sets is empty. Weighted Hamming distance: When the ordered position of the bj is the same than the ordered position of the ai and b = 0. Heavy weighted Hamming distance: When b – 0, and the ordered position of the bj is the same than the ordered position of the ai. Minimum distance: When wn = 1, wj = 0, for all j – n and b = 0. Push up allocation: We use wj = (1 ^ (|W|(j  1))) _ 0. Push down allocation: We use wnj+1 = (1 ^ (|W|(j  1))) _ 0. Uniform allocation: We use wj = |W|/n. Median type allocation: If n is even, we allocate the weights for j = 1 to a as wa+j = wa+1j = [1 ^ ((|W|2(j  1))/2)] _ 0. If n is odd, we allocate the weights for j = 1 to a as wa+1 = 1 and wa+1j = wa+1+j = [1 ^ ([(|W|  1)  2(j  1)]/2)] _ 0. Step-HOWAD allocation: Assuming b = Min[(K  1), (n  K)], we allocate the weights for j to b as wK = 1 and wK+j = wKj = [1 ^ ([(|W|  1)  2(j  1)]/2)] _ 0. Olympic-average allocation: We have to distinguish between two cases. s In the first case, where |W| < n  2m, we allocate the weight as wj = |W|/(n  2m) for j = m + 1 to n  m, and wj = 0 for j = 1 to m and for j = n  m + 1 to n. s In the second case, where |W| > n  2m, we allocate the weights as wj = 1 for j = m + 1 to n  m and wm+1j = wnm+j = [1 ^ ([(|W|  (n  2m))  2(j  1)]/2)] _ 0 for j = 1 to m. Arrow–Hurwicz allocation: Assuming that |W| = q and dimension n, we define the weights in two directions, push up and push down. First, we calculate xj = (1 ^ (kq  (j  1))) _ 0 for j = 1 to n and ŵnj+1 = (1 ^ ((1  k)q(j  1))) _ 0 for j = 1 to n. Then, we define the weights as wi = xi + ŵi. Note that xj = 0 for all j P kq + 1 P k|W| + 1 and ŵj = 0 for j < n(1  k)q 6 n  |W| + k|W| 4. Generalized aggregation operators with the HOWAD operator In this section we present several generalizations of the heavy aggregation operators by using generalized and quasiarithmetic means [32–35]. We focus on the quasi-arithmetic version because it includes the generalized mean as a particular case. We introduce the quasi-arithmetic HOWAD (Quasi-HOWAD) operator. The main advantage of this formulation is that it includes a wide range of particular cases. Thus, we are able to consider the aggregation process in a more complete way. It can be defined as follows. Definition 5. A Quasi-HOWAD operator of dimension n is a mapping QHOWAD: Rn  Rn ? R that has an associated weighting P vector W of dimension n with wj e [0, 1] and 1 6 nj¼1 wj 6 n, such that: QHOWADðhx1 ; y1 i; . . . ; hxn ; yn iÞ ¼ g 1 ! n X wj gðDj Þ ; ð11Þ j¼1 where Dj is the jth largest of the |xi  yi|, |xi  yi| is the argument variable represented in the form of individual distances being xi and yi the ith arguments of the sets X = (x1, x2, . . ., xn) and Y = (y1, y2, . . ., yn), and g(D) is a strictly continuous monotonic function. In the following, we briefly present some of the main particular cases of the Quasi-HOWAD operator.          If If If If If If If If If wi = 1/n, for all i, we obtain the quasi-arithmetic heavy averaging distance (Quasi-HAD) operator. g(D) = Dk, we get the generalized HOWA distance (GHOWAD) operator. g(D) = D, we get the usual HOWAD. g(D) = D2, we get the heavy ordered weighted quadratic averaging distance (HOWQAD). g(D) ? Dk, for k ? 0, we get the heavy ordered weighted geometric averaging distance (HOWGAD). g(D) = D1, we get the heavy ordered weighted harmonic averaging distance (HOWHAD). g(D) = D3, we get the heavy ordered weighted cubic averaging distance (HOWCAD). g(D) ? Dk, for k ? 1, we get the heavy maximum distance. g(D) ? Dk, for k ? 1, we get the heavy minimum distance. Note also that it is possible to consider a further generalization of this approach by using a weighting vector that ranges between 1 and 1, and by using infinitary aggregation operators. Author's personal copy J.M. Merigó et al. / Applied Mathematical Modelling 38 (2014) 3142–3153 3147 5. Norms in the HOWAD operator Norm aggregations provide a more general representation of the aggregation when dealing with distance measures because they allow including more complex operations in the analysis. A norm associates with some vector or tuple X = (x1, x2, . . ., xn) a unique non-negative scalar. A norm is a function f: Rn ? [0, 1) that has the following properties [20]: (1) f (x1, x2, . . ., xn) = 0 if and only if all xi = 0. (2) f (aX) = |a|f (X). (3) f (X) + f (Y) P f (X + Y), that is, the triangle inequality. Yager [20] suggested the use of norms in the OWA operator as: f ða1 ; a2 ; . . . ; an Þ ¼ Gðja1 j; ja2 j; . . . ; jan jÞ ¼ n X wj Dj ; ð12Þ j¼1 where Dj is the jth largest of the |ai|. Note that we can use norms to get a distance or a metric function assuming that if f is a norm then d(X, Y) = f (|X  Y|). Following this methodology, we can also use norms in the HOWAD operator forming the heavy OWA norm (HOWAN). It can be defined as follows. Definition 6. A HOWAN operator of dimension n is a mapping HOWAN: Rn  Rn ? R that has an associated weighting vector P W with wi e [0, 1] and 1 6 nj¼1 wj 6 n, such that: HOWANðjx1 ; y1 j; jx2 ; y2 j; . . . ; jxn ; yn jÞ ¼ n X wj Dj ; ð13Þ j¼1 where Dj is the jth largest of the |xi, yi|, and |xi, yi| is the argument variable represented in the form of individual norms. The main advantage of this formulation is that it includes a wide range of aggregations including those based on distance measures. Note that if HOWAN (X, Y) = f (|X  Y|), the HOWAN operator becomes the HOWAD operator. Furthermore, the HOWAN permits to establish a lot of internal operations between X and Y. For example:  If HOWAN (X, Y) = f (|[1 ^ (1  X + Y)|), we form the heavy OWA adequacy coefficient (HOWAAC) operator based on [21,22].  If HOWAN (X, Y) = f (min(X, Y)), we obtain the minimum HOWA operator and if f (max(X, Y)), the maximum HOWA operator.  If HOWAN (X, Y) = f (X + Y), we obtain the addition HOWA (A-HOWA) operator.  If HOWAN (X, Y) = f (X  Y), we obtain the subtraction HOWA (S-HOWA) operator.  If HOWAN (X, Y) = f (X  Y), we obtain the multiplication HOWA (M-HOWA) operator.  If HOWAN (X, Y) = f (X  Y), we obtain the division HOWA (D-HOWA) operator.  And so on. Next, we can also generalize the HOWAN operator by using generalized aggregation operators. By using quasi-arithmetic means, we introduce a strictly continuous monotonic function g(D) as follows: QHOWANðjx1 ; y1 j; jx2 ; y2 j; . . . ; jxn ; yn jÞ ¼ g 1 ! n X wj gðDj Þ ; ð14Þ j¼1 where Dj is the jth largest of the |xi, yi|, |xi, yi| is the argument variable represented in the form of individual norms and g(D) is a strictly continuous monotonic function. In this case we obtain the quasi-arithmetic HOWAN (Quasi-HOWAN) operator that includes the previous cases commented before including the quasi-arithmetic A-HOWA (Quasi-A-HOWA), the quasi-arithmetic M-HOWA (Quasi-M-HOWA), and so on. 6. Moving averages in the HOWAD operator The HOWAD operator and its generalizations can also be extended by using moving averages [25]. The main advantage is that we can represent the information in a dynamic way. If we use the moving average in the HOWAD operator we get heavy ordered weighted moving averaging distance (HOWMAD) operator. For two sets, X = {x1+t, x2+t, . . ., xm+t} and Y = {y1+t, y2+t, . . ., ym+t}, we can define it as follows. Definition 7. A HOWMAD operator of dimension m is a mapping HOWMAD: Rm  Rm ? R that has an associated weighting Pmþt vector W such that wj e [0, 1] and W = 1 6 j¼1þt wj 6 n, according to the following formula: Author's personal copy 3148 J.M. Merigó et al. / Applied Mathematical Modelling 38 (2014) 3142–3153 mþt X HOWMAD ðhx1þt ; y1þt i; . . . ; hxmþt ; ymþt iÞ ¼ wj Dj ; ð15Þ j¼1þt where Dj is the jth largest of the |xi  yi| value, |xi  yi| is the argument variable represented in the form of individual distances, m is the total number of arguments considered from the whole sample and t indicates the movement done in the average from the initial analysis. The HOWMAD operator can be further generalized by using norms obtaining the heavy ordered weighted moving averaging norm (HOWMAN). It can be defined in the following way. Definition 8. A HOWMAN operator of dimension m is a mapping HOWMAN: Rm  Rm ? R that has an associated weighting Pmþt vector W such that wj e [0, 1] and W = 1 6 j¼1þt wj 6 n, according to the following formula: HOWMAN ðjx1þt ; y1þt j; . . . ; jxmþt ; ymþt jÞ ¼ mþt X wj Dj ; ð16Þ j¼1þt where Dj is the jth largest of the |xi, yi|, |xi, yi| is the argument variable represented in the form of individual norms, m is the total number of arguments considered from the whole sample and t indicates the movement done in the average from the initial analysis. Furthermore, the HOWMAN operator can be generalized by using quasi-arithmetic means obtaining the quasi-arithmetic HOWMAN (Quasi-HOWMAN) operator. In this case, we add a function g(D) as follows: QHOWMAN ðjx1þt ; y1þt j; . . . ; jxmþt ; ymþt jÞ ¼ g 1 mþt X j¼1þt ! wj gðDj Þ ; ð17Þ where Dj is the jth largest of the |xi, yi|, |xi, yi| is the argument variable represented in the form of individual norms, m is the total number of arguments considered from the whole sample, t indicates the movement done in the average from the initial analysis and g(D) is a strictly continuous monotonic function. The Quasi-HOWMAN operator includes a wide range of particular cases by using different types of norms such as:  If QHOWMAN (X, Y) = f (|[1 ^ (1  X + Y)|), we get the quasi-arithmetic heavy ordered weighted moving averaging adequacy coefficient (Quasi-HOWMAAC) operator.  If QHOWMAN (X, Y) = f (min(X, Y)), we form the minimum HOWMA operator and if f (max(X, Y)), the maximum HOWMA operator.  If QHOWMAN (X, Y) = f (X + Y), we get the quasi-arithmetic addition HOWA (Quasi-A-HOWA) operator.  If QHOWMAN (X, Y) = f (X  Y), we obtain the quasi-arithmetic subtraction HOWA (Quasi-S-HOWA) operator.  If QHOWMAN (X, Y) = f (X  Y), we form the quasi-arithmetic multiplication HOWA (Quasi-M-HOWA) operator.  If QHOWMAN (X, Y) = f (X  Y), we get the quasi-arithmetic division HOWA (Quasi-D-HOWA) operator. P  Note that the HOWAD and Quasi-HOWAD operator can also be reduced to the OWAD operator if nj¼1 wj ¼ 1. Thus, we can also obtain the OWA norm (OWAN), the addition OWA (A-OWA), the subtraction OWA (S-OWA), the multiplication OWA (M-OWA) and the division OWA (D-OWA).  By using moving averages when g(D) = D, we obtain the OWMA norm (OWMAN), the addition OWMA (A-OWMA), the subtraction OWMA (S-OWMA), the multiplication OWMA (M-OWMA) and the division OWMA (D-OWMA).  And so on. 7. Application in investment management In the following, we present an illustrative example of the new approach in a decision making problem [36–40] regarding the selection of investments. The motivation for using heavy aggregation operators in distance measures is the possibility of considering different aggregations between the normalized (or relative) Hamming distance and the absolute (or total) Hamming distance. Obviously, by doing this, we are allowing the weighting vector to be higher than one. Note that it is possible to consider other situations where we allow the weighting vector to be lower than one or higher than n depending on the problem considered and the degree of under or over estimation that we want to introduce in the analysis. Assume that a company that operates in Europe and North America is analyzing possible investments to carry out the next year and considers five possible alternatives. (1) (2) (3) (4) (5) A1: A2: A3: A4: A5: Invest in the Asian market. Invest in the South American market. Invest in the African market. Invest in all markets. Do not develop any investment. Author's personal copy 3149 J.M. Merigó et al. / Applied Mathematical Modelling 38 (2014) 3142–3153 After careful review of the information, the experts of the company establish the following general information regarding the investments. It is assumed that two general criteria affect the results:  Cr1: Expected world economic situation.  Cr2: Expected economic environment of the specific region considered. They summarize the information of the investments for each criteria in five general characteristics C = {C1, C2, C3, C4, C5}.      C1: C2: C3: C4: C5: Benefits in the short term. Benefits in the midterm. Benefits in the long term. Risk of the strategy. Other variables. The results provided by the first expert for each alternative and criteria are shown in Tables 1 and 2. With this information, we integrate the criteria in a global result for expert one by using a weighted average of the two criterions. It is assumed that both criterions have the same importance. Therefore, it is used the arithmetic mean in the analysis giving 50% importance to each criteria. The results are given in Table 3. The analysis for experts 2 and 3 is carried out following the same methodology as with expert 1. Therefore, first it is provided the individual information for each criteria and subsequently it is aggregated this information into a collective matrix that includes both of them. The results for expert 2 are provided in Tables 4–6 and for expert 3 in Tables 7–9. With this information, we construct the collective opinion of the group by aggregating the information using a weighted average U = (0.5, 0.25, 0.25). The results are shown in Table 10. According to the objectives of the company, the experts establish the following ideal investment. The results are shown in Table 11. Next, it is possible to develop different methods for selecting an investment based on different types of HOWAD operators. In this example, we will consider the total (or absolute) distance, the minimum distance, the NHD, the OWAD and the HOWAD operator. We assume the following weighting vector W = (0.1, 0.1, 0.2, 0.3, 0.3) and the following one when using heavy aggregations W⁄ = (0.2, 0.2, 0.4, 0.6, 0.6). The results are shown in Table 12. In order to compare this information between different alternatives, it is worth noting that each method leads to different results and decisions. From a general perspective, it is possible to analyze this situation with the results shown in Fig. 1. Table 1 Criteria 1 – expert 1. A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 40 100 50 70 40 80 20 50 20 60 50 70 30 60 20 10 50 20 50 70 90 80 60 70 30 Table 2 Criteria 2 – expert 1. A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 20 80 90 50 80 60 40 30 60 80 30 30 10 40 60 30 70 0 10 90 70 40 80 90 10 C1 C2 C3 C4 C5 30 90 70 60 60 70 30 40 40 70 40 50 20 50 40 20 60 10 30 80 80 60 70 80 20 Table 3 Global results – expert 1. A1 A2 A3 A4 A5 Author's personal copy 3150 J.M. Merigó et al. / Applied Mathematical Modelling 38 (2014) 3142–3153 Table 4 Criteria 1 – expert 2. A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 20 90 50 80 30 80 40 30 40 80 90 50 10 60 30 30 50 20 40 50 10 70 70 80 10 C1 C2 C3 C4 C5 0 50 30 40 50 40 20 10 20 60 70 30 30 40 50 50 70 60 80 70 30 90 70 100 30 Table 5 Criteria 2 – expert 2. A1 A2 A3 A4 A5 Table 6 Global results – expert 2. A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 10 70 40 60 40 60 30 20 30 70 80 40 20 50 40 40 60 40 60 60 20 80 70 90 20 Table 7 Criteria 1 – expert 3. A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 60 40 10 70 50 50 10 80 40 60 90 70 30 90 50 30 50 10 70 70 40 90 50 60 10 C1 C2 C3 C4 C5 40 20 30 50 30 30 50 40 60 80 70 50 10 90 30 50 70 30 90 50 80 70 90 80 30 Table 8 Criteria 2 – expert 2. A1 A2 A3 A4 A5 Table 9 Global results – expert 3. A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 50 30 20 60 40 40 30 60 50 70 80 60 20 90 40 40 60 20 80 60 60 80 70 70 20 Author's personal copy 3151 J.M. Merigó et al. / Applied Mathematical Modelling 38 (2014) 3142–3153 Table 10 Information of the investments – collective results. A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 30 70 50 60 50 60 30 40 40 70 60 50 20 60 40 30 60 20 50 70 60 70 70 80 20 Table 11 Ideal investment. I C1 C2 C3 C4 C5 80 50 40 60 70 Table 12 Aggregated results. A1 A2 A3 A4 A5 Min Total NHD OWAD HOWAD 10 0 0 10 0 120 40 100 70 110 24 8 20 14 22 18 5 14 12 15 36 10 28 24 30 Fig. 1. Comparison between the minimum and the total distance. Table 13 Ranking of the investments. Ranking Ranking Min A2 = A3 = A5 A1 = A4 OWAD A2 A4 A3 A5 A1 Total A2 A4 A3 A5 A1 HOWAD A2 A4 A3 A5 A1 NHD A2 A4 A3 A5 A1 As we can see, for the minimum, three alternatives provide the lowest distance while for the total distance, A2 gives the lowest results and therefore is the optimal choice. A further interesting issue is to consider a ranking of the investments depending on the aggregation operator used. The results are shown in Table 13. As we can see, depending on the aggregation operator used, the ranking of the investments may be different. However, in this example it seems clear that A2 is the optimal choice. 8. Conclusions We have presented the HOWAD operator. It is a new distance aggregation operator (or distance measure) that includes the total (or absolute) distance and the normalized distance in the same formulation. Thus, we are able to provide a more Author's personal copy 3152 J.M. Merigó et al. / Applied Mathematical Modelling 38 (2014) 3142–3153 complete formulation of the aggregation process that includes a wide range of aggregations from the minimum distance to the total distance. We have studied some of its main properties and particular cases. Furthermore, we have considered the use of deeper generalizations by using generalized and quasi-arithmetic means, norms and moving averages. By using generalized means we have obtained the GHOWAD operator and by using quasi-arithmetic means, the Quasi-HOWAD operator. Their main advantage is that they include a wide range of particular cases including the quadratic HOWAD and the cubic HOWAD operator. The use of norms has permitted us to obtain a more general framework. We have called it the HOWAN operator. And by using moving averages we have formed the HOWMAD and the Quasi-HOWMAN operator as a dynamic representation of the previous approaches. Note that it is possible to apply it in a wide range of applications including statistics, decision making, engineering, physics and biology. We have focused on a decision making problem regarding the selection of strategies. The main advantage of this approach is that we can use independent information in the analysis. That is, we can aggregate the information from the minimum to a result higher than the maximum that has its bound in the total operator when all the information is independent. In future research, we expect to develop further advances by adding new characteristics in the analysis such as the use of Choquet integrals [41], weighted averages [42] and probabilities [43,44] in the analysis and more complex representations. We will also consider other potential applications in statistics and in other research disciplines. Acknowledgments We would like to thank the anonymous referees for valuable comments that have improved the quality of the paper. Support from the European Commission through the project PIEF-GA-2011-300062 is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] J. Gil-Aluja, The Interactive Management of Human Resources in Uncertainty, Kluwer Academic Publishers, Dordrecht, 1998. R.W. Hamming, Error-detecting and error-correcting codes, Bell Syst. Tech. J. 29 (1950) 147–160. A. Kaufmann, Introduction to the Theory of Fuzzy Subsets, Academic Press, New York, 1975. N. Karayiannis, Soft learning vector quantization and clustering algorithms based on ordered weighted aggregation operators, IEEE Trans. Neural Networks 11 (2000) 1093–1105. J.M. Merigó, A.M. Gil-Lafuente, New decision-making techniques and their application in the selection of financial products, Inf. Sci. 180 (2010) 2085– 2094. Z.S. Xu, J. Chen, Ordered weighted distance measure, J. Syst. Sci. Syst. Eng. 17 (2008) 432–445. R.R. Yager, On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE Trans. Syst. Man Cybern. B 18 (1988) 183–190. R.R. Yager, J. Kacprzyk, The Ordered Weighted Averaging Operators: Theory and Applications, Kluwer Academic Publishers, Norwell, 1997. R.R. Yager, J. Kacprzyk, G. Beliakov, Recent Developments on the Ordered Weighted Averaging Operators: Theory and Practice, Springer-Verlag, Berlin, 2011. J.M. Merigó, M. Casanovas, Decision making with distance measures and linguistic aggregation operators, Int. J. Fuzzy Syst. 12 (2010) 190–198. J.M. Merigó, M. Casanovas, Decision making with distance measures and induced aggregation operators, Comput. Ind. Eng. 60 (2011) 66–76. J.M. Merigó, M. Casanovas, Induced aggregation operators in the Euclidean distance and its application in financial decision making, Expert Syst. Appl. 38 (2011) 7603–7608. J.M. Merigó, M. Casanovas, A new Minkowski distance based on induced aggregation operators, Int. J. Comput. Intell. Syst. 4 (2011) 123–133. S. Zeng, W.H. Su, Intuitionistic fuzzy ordered weighted distance operator, Knowledge Based Syst. 24 (2011) 1224–1232. S.Z. Zeng, J.M. Merigó, W.H. Su, The uncertain probabilistic OWA distance operator and its application in group decision making, Appl. Math. Model. 37 (2013) 6266–6275. S.Z. Zeng, W.H. Su, Linguistic induced generalized aggregation distance operators and their application to decision making, Econ. Comput. Econ. Cybern. Stud. Res. 46 (2012) 155–172. S.Z. Zeng, W.H. Su, A. Le, Fuzzy generalized ordered weighted averaging distance operator and its application to decision making, Int. J. Fuzzy Syst. 14 (2012) 402–412. J.M. Merigó, A.M. Gil-Lafuente, OWA operators in human resource management, Econ. Comput. Econ. Cybern. Stud. Res. 45 (2011) 153–168. J.M. Merigó, A.M. Gil-Lafuente, Decision-making in sport management based on the OWA operator, Expert Syst. Appl. 38 (2011) 10408–10413. R.R. Yager, Norms induced from OWA operators, IEEE Trans. Fuzzy Syst. 18 (2010) 57–66. J.M. Merigó, A.M. Gil-Lafuente, Decision making techniques in business and economics based on the OWA operator, SORT – Stat. Oper. Res. Trans. 36 (2012) 81–101. J.M. Merigó, A.M. Gil-Lafuente, J. Gil-Aluja, Decision making with the induced generalized adequacy coefficient, Appl. Comput. Math. 2 (2011) 321–339. Z.S. Xu, M. Xia, Distance and similarity measures for hesitant fuzzy sets, Inf. Sci. 181 (2011) 2128–2138. Z.S. Xu, Fuzzy ordered distance measures, Fuzzy Optim. Decis. Making 11 (2012) 73–97. J.M. Merigó, R.R. Yager, Generalized moving averages, distance measures and OWA operators, Int. J. Uncertainty Fuzziness Knowledge Based Syst. 21 (2013) 533–559. R.R. Yager, Heavy OWA operators, Fuzzy Optim. Decis. Making 1 (2002) 379–397. R.R. Yager, Monitored heavy fuzzy measures and their role in decision making under uncertainty, Fuzzy Sets Syst. 139 (2003) 491–513. J.M. Merigó, M. Casanovas, Induced and heavy aggregation operators with distance measures, J. Syst. Eng. Electron. 21 (2010) 431–439. J.M. Merigó, M. Casanovas, Induced and uncertain heavy OWA operators, Comput. Ind. Eng. 60 (2011) 106–116. R.R. Yager, Time series smoothing and OWA aggregation, IEEE Trans. Fuzzy Syst. 16 (2008) 994–1007. R.R. Yager, Families of OWA operators, Fuzzy Sets Syst. 59 (1993) 125–148. J.M. Merigó, A.M. Gil-Lafuente, The induced generalized OWA operator, Inf. Sci. 179 (2009) 729–741. G. Beliakov, A. Pradera, T. Calvo, Aggregation Functions: A Guide for Practitioners, Springer-Verlag, Berlin, 2007. J. Fodor, J.L. Marichal, M. Roubens, Characterization of the ordered weighted averaging operators, IEEE Trans. Fuzzy Syst. 3 (1995) 236–240. R.R. Yager, Generalized OWA aggregation operators, Fuzzy Optim. Decis. Making 3 (2004) 93–107. H.Y. Chen, L.G. Zhou, An approach to group decision making with interval fuzzy preference relations based on induced generalized continuous ordered weighted averaging operator, Expert Syst. Appl. 38 (2011) 13432–13440. P. Liu, A weighted aggregation operators multi-attribute group decision making method based on interval-valued trapezoidal fuzzy numbers, Expert Syst. Appl. 38 (2011) 1053–1060. Author's personal copy J.M. Merigó et al. / Applied Mathematical Modelling 38 (2014) 3142–3153 3153 [38] P. Liu, F. Jin, A multi-attribute group decision making method based on weighted geometric aggregation operators of interval-valued trapezoidal fuzzy numbers, Appl. Math. Model. 36 (2012) 2498–2509. [39] G.W. Wei, FIOWHM operator and its application to group decision making, Expert Syst. Appl. 38 (2011) 2984–2989. [40] L.G. Zhou, H.Y. Chen, Generalized ordered weighted logarithm aggregation operators and their applications to group decision making, Int. J. Intell. Syst. 25 (2010) 683–707. [41] J. Bolton, P. Gader, J.N. Wilson, Discrete Choquet integral as a distance metric, IEEE Trans. Fuzzy Syst. 16 (2008) 1107–1110. [42] Z.S. Xu, Q.L. Da, An overview of operators for aggregating information, Int. J. Intell. Syst. 18 (2003) 953–968. [43] J.M. Merigó, The probabilistic weighted average and its application in multi-person decision making, Int. J. Intell. Syst. 27 (2012) 457–476. [44] J.M. Merigó, G.W. Wei, Probabilistic aggregation operators and their application in uncertain multi-person decision making, Technol. Econ. Dev. Econ. 17 (2011) 335–351.