1 Introduction
In many disciplines, decision makers have to deal with problems involving the aggregation of multicriteria evaluations and producing overall assessments within the application context. The ordered weighted averaging function (OWA) introduced by Yager Yager1988 is a fundamental aggregation function in decision making theory. One of the main motivations behind selecting the OWA functions for multicriteria aggregation is their flexibility in providing a general class of weighted aggregation functions bounded by the min and the max functions.
The determination of appropriate OWA weights is thus a very important object of study when applying OWA functions in the context of decision making. Among the various methods for obtaining OWA weights that can be found in the literature, namely Xu2005 ; Xu2003 ; Yager1993 , and more recently BeliakovBustinceSolaCalvo2016 ; CarlssonFuller2018 and Liu2011 , we can distinguish the following methodological categories: a) methods based on learning OWA weights from data, b) methods based on weightgenerating functions, and c) methods based on the characteristic measures of orness and disparity.
In this paper we focus on the problem of determining a special class of OWA functions based on the disparity measure under a given level of orness, since these two measures can lead to OWA weights associated with different attitudinal characters of decision makers, see MezeiBrunelli2017 ; Yager1988 . Depending on their weighting structure, OWA functions reflect different preferences of decision makers, from the optimistic to the pessimistic attitude. The attitudinal character of decision makers is measured by their orness, which takes values in the unit interval. The maximum (minimum) orness value is attained when decision makers are purely optimistic (purely pessimistic). On the other hand, the disparity evaluates the nonuniformity of the OWA weights. In the case of the arithmetic mean the disparity takes its minimal value.
In the literature several methods have been introduced to obtain the optimal weights by using the disparity measure. After the pioneering work of O’Hagan OHagan1988
on the maximal entropy method and the variancebased methods of Yager
YagerIJUFKBS1996 and Fuller and Majlender FullerMajlender2003 , Wang and Parkan WangParkan2005 proposed the minimax disparity method in which the objective is to minimize the maximum absolute difference between two adjacent weights. Liu Liu2007 proved the equivalence of the solutions of the minimum variance approach suggested by Fuller and Majlender FullerMajlender2003 and the minimax disparity model proposed by Wang and Parkan WangParkan2005 under a given level of orness. Extensions of disparitybased models for determining OWA weights are presented in AminEmrouznejad2006 ; EmrouznejadAmin2010 ; GongDaiHu2016 ; SangLiu2014 ; WangLuoHua2007. In this paper, we focus on the minimax disparity model since it has recently received a great deal of interest in the literature and is easy to solve due to its simple linear programming formulations.
The usual academic instances of the minimax disparity model focus on solving problems with small dimensions (). However, in applied operational research, optimization problems are often much more complex and require a heavy computational demand when there are hundreds or thousands of variables. In order to overcome the complexity of highdimensional problems, we consider the binomial decomposition framework, proposed by Calvo and De Baets CalvoDeBaets1998 , see also Bortot and Marques Pereira BortotMarquesPereira2014 and Bortot et al. BortotFedrizziMarquesPereiraThuy2017 ; BortotMarquesPereiraThuy2017 , which refers to the kadditive framework introduced by Grabisch Grabisch1997b ; Grabisch1997c ; Grabisch1997a . This framework allows us to transform the original problem, expressed directly in terms of the OWA weights, into a problem in which the weights are substituted by a new set of coefficients. In this transformed representation, we can consider only a reduced number of these coefficients, associated with the first kadditive levels of the OWA function, and we can set the remaining coefficients to zero. In this way the computational demand in highdimensional problems is significantly reduced.
The remainder of this paper is organized as follows. In Sect. 2 we briefly review the ordered weighted averaging functions and their representation in the binomial decomposition framework. Section 3 reviews the recent development of the minimax disparity model for determining OWA weights. In Sect. 4 we recall the minimax disparity model and reformulate it in terms of the coefficients in the binomial decomposition framework. We illustrate our approach for dimension . Finally, Sect. 5 contains some conclusive remarks.
2 OWA functions and the binomial decomposition framework
In this section we consider a point , with . The increasing reordering of the coordinates of x is denoted as . We now introduce the definition of the ordered weighted averaging function and its characterizing measures.
Definition 1
An Ordered Weighted Averaging (OWA) function of dimension is an averaging function
with an associated weighting vector
, such that and(1) 
Different OWA functions are classified by their weighting vectors. The OWA weights are characterized by two measures called orness and disparity. In the following part we review these two measures and their properties.
Definition 2
Consider an OWA function with an associated weighting vector such that , two characterizing measures called orness and disparity are defined as
(2) 
Yager Yager1988 introduced the orness measures to evaluate the level of similarity between the OWA function and the or (maximum) operator. On the other hand, the disparity measure, as proposed by Wang and Parkan WangParkan2005 , is defined as the maximum absolute difference between two adjacent weights. Its value shows how unequally multicriteria evaluations are taken into account in the aggregation process. Both OWA characterizing measures are bounded in the unit interval. Three special OWA weighting vectors are , and . For these vectors we have the orness equal to and disparity equal to , respectively.
In the following we recall the binomial decomposition of the OWA functions as in Bortot and Marques PereiraBortotMarquesPereira2014 and Bortot et al. BortotFedrizziMarquesPereiraThuy2017 .
Definition 3
The binomial OWA functions , with , are defined as
(3) 
where the binomial weights , are null when , according to the usual convention that when , with .
Theorem 2.1
[Binomial decomposition] Any OWA function can be written uniquely as
(4) 
where the coefficients , are subject to the following conditions,
(5) 
The detail proof of Theorem 2.1 is given in Bortot and Marques Pereira BortotMarquesPereira2014 .
The binomial decomposition (4) expresses the linear combination between the OWA weights and the coefficients alpha and can be written as the linear system
(6) 
where the binomial weights and the coefficients are subjected to the conditions (5).
We notice that the coefficient matrix of the linear system is composed by the binomial weights , with the first weights being positive and nonlinear decreasing, and the last weights equal to zero. Hence there always exists a unique vector of coefficients alpha satisfying the linear system which is triangular and whose coefficient matrix is full rank and invertible.
3 The minimax disparity model for determining OWA weights
The determination of appropriate OWA weights is a very important study when applying OWA functions in the context of decision making. In this section, we briefly review the recent development of the specific class of OWA functions whose weights are determined by the minimax disparity methods.
In 2005 Wang and Parkan WangParkan2005 revisited the maximum entropy method introduced by O’Hagan OHagan1988 and proposed the minimax disparity procedure to determine the OWA weights in the convex optimization problem
Min.  (7)  
s.t. 
where stands for the orness of the weighting vector.
The objective function is nonlinear due to its formulation encompassing the absolute difference between two adjacent weights. In order to overcome this nonlinearity, the authors introduced a new variable called to denote the maximum absolute difference between two adjacent weights. This expression is rewritten equivalently by two inequations and . The original optimization problem is thereby reformulated into the linear programming problem
Min.  (8)  
s.t.  
where stands for the orness of the weighting vector.
The formulation (8) is easy to solve in practice due to its linearity. Many researchers, therefore, revisited this method and suggested numerous extensions AminEmrouznejad2006 ; EmrouznejadAmin2010 ; GongDaiHu2016 ; SangLiu2014 ; WangLuoHua2007 . In particular Liu Liu2007 proved the equivalence of the solutions of the minimax disparity model and the minimum variance method suggested by Fuller and Majlender FullerMajlender2003 .
4 The minimax disparity model and the binomial decomposition
In Sect. 3, we have reviewed minimax disparity methods for determining OWA weights. The empirical results in those methods are carried out in the small dimensions . In reallife problems, we usually encounter largescale optimization problems. One of the challenges that clearly emerges from the above methods in largescale problems is that the optimization problems formulated directly in terms of the OWA weights require high computational resources. Our objective is to improve the minimax disparity methods for determining OWA weights in highdimensional problems.
In order to reduce the prohibitive complexity of the optimization problems, we recall the work of Bortot and Marques Pereira BortotMarquesPereira2014 . According to the authors, any OWA function can be decomposed uniquely into a linear combination of the binomial coefficients and the binomial OWA functions , and can be written as , as described in Sect. 2. The positive aspect of the binomial decomposition framework is the kadditive levels in which a certain number of coefficients are used in the process of OWA weights determination while the remaining coefficients are set to zero. When the information of the coefficients alpha is available, we can reconstruct the OWA weights due to their onetoone correspondence. In this way the computational demand in highdimensional problems is significantly reduced.
We now transform the minimax disparity model (8) into a problem in which the weights are substituted by a set of coefficients ,
Min.  (9)  
s.t.  
where stands for the orness of the weighting vector. Notice that the first and the last constraints in our model (9) correspond to the boundary and the monotonicity conditions of the OWA weighting vector with .
In the following we report the empirical results with respect to dimension and kadditive level is equal to (Table 1). We notice that a number of coefficients alpha are used is significantly reduced in our model for the central orness values and . In these cases, two nonzero coefficients and are able to reconstruct the fulldimension set of OWA weights while the remaining coefficients are zero. In the remaining cases of orness values, a larger number of coefficients alpha is required to generate the OWA weights. As an example, we consider our proposed model with the orness value equal to . If the kadditive level increases from to , we obtain better objective values as expected (see Fig. 1). However it is evident that leads to the best tradeoff between the accuracy of the optimal value and the dimensionality reduction of the optimization problem.
Orness(w)  

Coefficients  
10  4.3  2.71  1.98  1.49  1  0.51  0.02  0  0  0  
45  5.4  1.93  0.98  0.49  0  0.49  0.98  0  0  0  
120  0  0  0  0  0  0  0  0  0  0  
210  0  0  0  0  0  0  0  3  0  0  
252  12.6  0  0  0  0  0  0  0  0  0  
210  16.8  0  0  0  0  0  0  9  0  0  
120  2.4  0  0  0  0  0  0  13.71  8.4  0  
45  9  1.29  0  0  0  0  0  9.64  13.5  0  
10  6.5  1.57  0  0  0  0  0  3.43  7.5  0  
1  1.4  0.5  0  0  0  0  0  0.5  1.4  1  
OWA weights w  
0  0  0  0  .05  0.10  0.15  0.20  0.27  0.43  1  
0  0  0  0.03  0.06  0.10  0.14  0.18  0.23  0.31  0  
0  0  0  0.05  0.07  0.10  0.13  0.16  0.19  0.20  0  
0  0.01  0.07  0.08  0.10  0.10  0.13  0.1  0.07  0  0  
0  0  0.06  0.09  0.10  0.10  0.11  0.11  0.10  0  0  
0  0  0.1  0.11  0.11  0.10  0.10  0.09  0.06  0  0  
0  0.07  0.14  0.13  0.12  0.10  0.08  0.07  0.01  0  0  
0  0.20  0.19  0.16  0.13  0.10  0.07  0.05  0  0  0  
0  0.31  0.23  0.18  0.14  0.10  0.06  0.02  0  0  0  
1  0.43  0.27  0.20  0.15  0.10  0.05  0  0  0  0  
1  0.12  0.04  0.02  0.01  0  0.01  0.02  0.04  0.12  1 
Moreover, in Table 1, we observe that the coefficients alpha associated with orness value and are respectively:

, which is associated with the OWA weighting vector ;

, which is associated with the OWA weighting vector ;

, which is associated with the OWA weighting vector .
5 Conclusions
This study proposes a new methodology for determining OWA weights in largescale optimization problems where the cost of computation of optimal weights is very high. Our model allows the optimization of the OWA weights to be transformed into the optimization of the coefficients in the binomial decomposition framework, considering the kadditive levels in order to reduce the complexity of the proposed model. The empirical result shows that a small set of the coefficients in the binomial decomposition can model a the fulldimensional set of the OWA weights.
However, there are still some issues that need to be addressed in future research. For instance, it is necessary to develop an algorithm to identify which kadditive level in the set gives the best trade off between accuracy and computational complexity according to the specific applications. It would also be interesting to perform a sensitivity analysis of the coefficients in the binomial decomposition with respect to the OWA weights.
Acknowledgements The author would like to thank Ricardo Alberto Marques Pereira and Silvia Bortot for their helpful remarks on the manuscript.
Bibliography
 (1) Amin, G.R., Emrouznejad, A.: An extended minimax disparity to determine the OWA operator weights. Comput. Ind. Eng. 50(3), 312–316 (2006)
 (2) Beliakov, G., Bustince Sola, H., Calvo, T.: A Practical Guide to Averaging Functions, Studies in Fuzziness and Soft Computing, vol. 329. Springer, Heidelberg (2016)
 (3) Bortot, S., Fedrizzi, M., Marques Pereira, R.A., Nguyen, T.H.: The binomial decomposition of generalized Gini welfare functions, the SGini and Lorenzen cases. Inform. Sciences (to appear)
 (4) Bortot, S., Marques Pereira, R.A.: The binomial Gini inequality indices and the binomial decomposition of welfare functions. Fuzzy Sets Syst. 255, 92–114 (2014)
 (5) Bortot, S., Marques Pereira, R.A., Nguyen, T.H.: The binomial decomposition of OWA functions, the 2additive and 3additive cases in n dimensions. Int. J. Intell. Syst. pp. 187– 212 (2018)
 (6) Calvo, T., De Baets, B.: Aggregation operators defined by korder additive/maxitive fuzzy measures. Int. J. Uncertain. Fuzz. 6(6), 533–550 (1998)
 (7) Carlsson, C., Fullér, R.: Maximal Entropy and Minimal Variability OWA Operator Weights: A Short Survey of Recent Developments. In: M. Collan, J. Kacprzyk (eds.) Soft Computing Applications for Group Decisionmaking and Consensus Modeling, pp. 187–199. Springer, Heidelberg (2018)
 (8) Emrouznejad, A., Amin, G.R.: Improving minimax disparity model to determine the OWA operator weights. Inform. Sciences 180(8), 1477 – 1485 (2010)
 (9) Fullér, R., Majlender, P.: On obtaining minimal variability OWA operator weights. Fuzzy Sets Syst. 136(2), 203–215 (2003)
 (10) Gong, Y., Dai, L., Hu, N.: An extended minimax absolute and relative disparity approach to obtain the OWA operator weights. J. Intell. Fuzzy Syst. 31(3), 1921–1927 (2016)
 (11) Grabisch, M.: Alternative representations of discrete fuzzy measures for decision making. Int. J. Uncertain. Fuzz. 5(5), 587–607 (1997)
 (12) Grabisch, M.: Alternative representations of OWA operators. In: R.R. Yager, J. Kacprzyk (eds.) Recent Developments in the Ordered Weighted Averaging Operators: Theory and Practice, Studies in Fuzziness and Soft Computing, vol. 265, pp. 73–85. Springer, Heidelberg (1997)
 (13) Grabisch, M.: korder additive discrete fuzzy measures and their representation. Fuzzy Sets Syst. 92(2), 167–189 (1997)
 (14) Liu, X.: The solution equivalence of minimax disparity and minimum variance problems for OWA operators. Int. J. Approx. Reason. 45(1), 68–81 (2007)
 (15) Liu, X.: A Review of the OWA Determination Methods: Classification and Some Extensions. In: R.R. Yager, J. Kacprzyk, G. Beliakov (eds.) Recent Developments in the Ordered Weighted Averaging Operators: Theory and Practice, pp. 49–90. Springer, Heidelberg (2011)
 (16) Mezei, J., Brunelli, M.: A closer look at the relation between orness and entropy of OWA function. In: M. Collan, J. Kacprzyk (eds.) Soft Computing Applications for Group Decisionmaking and Consensus Modeling, Studies in Fuzziness and Soft Computing, pp. 201–211. Springer, Heidelberg (2018)
 (17) O’Hagan, M.: Aggregating template or rule antecedents in realtime expert systems with fuzzy set logic. In: Signals, Systems and Computers, 1988. TwentySecond Asilomar Conference on, vol. 2, pp. 681–689. IEEE (1988)
 (18) Sang, X., Liu, X.: An analytic approach to obtain the least square deviation OWA operator weights. Fuzzy Sets Syst. 240, 103 – 116 (2014)
 (19) Wang, Y.M., Luo, Y., Hua, Z.: Aggregating preference rankings using OWA operator weights. Inform. Sciences 177(16), 3356–3363 (2007)
 (20) Wang, Y.M., Parkan, C.: A minimax disparity approach for obtaining OWA operator weights. Inform. Sciences 175(12), 20–29 (2005)
 (21) Xu, Z.: An overview of methods for determining OWA weights: Research Articles. Int. J. Intell. Syst. 20(8), 843–865 (2005)
 (22) Xu, Z.S., Da, Q.L.: An Overview of Operators for Aggregating Information. Int. J. Intell. Syst. 18(9), 953–969 (2003)
 (23) Yager, R.R.: On Ordered Weighted Averaging aggregation operators in multicriteria decision making. IEEE T. Syst. Man. Cyb. 18(1), 183–190 (1988)
 (24) Yager, R.R.: A general approach to criteria aggregation using fuzzy measures. Int. J. Man. Mach. Stud. 39(2), 187 – 213 (1993)
 (25) Yager, R.R.: On the inclusion of variance in decision making under uncertainty. Int. J. Uncertain. Fuzz. 04(05), 401–419 (1996)
Comments
There are no comments yet.