International Journal of Intelligent Systems and Applications (IJISA)
IJISA Vol. 7, No. 10, 8 Sep. 2015
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Aggregation Operators Review - Mathematical Properties and Behavioral Measures
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Index Terms
Aggregation, aggregation operators, behavioral measures of aggregation operators, intersection operators, OWA operators
Abstract
A problem that humans must face very often is that of having to add, melt or synthesize information, that is, combine together a series of data from various sources to reach a certain conclusion or make a certain decision. This involves the use of one or more aggregation operators capable to provide a collective preference relation. These operators must be chosen according to specific criteria taking into account the characteristic properties of each operator. Some conditions to be taken into account to identify them are the following: axiomatic strength, empirical setting, adaptability, numerical efficiency, compensation and compensation range, added behavior and scale level required of the membership functions. It is possible to establish a general list of possible mathematical properties whose verification might be desirable in certain cases: boundary conditions, continuity, not decreasing monotony, symmetry, idempotence, associativity, bisymmetry, self-distributivity, compensation, homogeneity, translativity, stability, ?-comparability, sensitivity and locally internal functions. For analyze the attitudinal character of the aggregation operator the following measures are studied: disjunction degree (orness), dispersion, balance and divergence. In this paper, a review of these issues is presented.
Cite This Paper
David L. La Red Martínez, Julio C. Acosta, "Aggregation Operators Review - Mathematical Properties and Behavioral Measures", International Journal of Intelligent Systems and Applications(IJISA), vol.7, no.10, pp.63-76, 2015. DOI:10.5815/ijisa.2015.10.08
Reference
[1]A. Pradera Gómez, Contribución al Estudio de la Agregación de Información en un Entorno Borroso, Tesis Doctoral, Universidad Politécnica de Madrid. 1999.
[2]L. Canós Darós & V. Liern Carrión, La Agregación de Información Para la Toma de Decisiones en la Empresa, XIV Jornadas de ASEPUMA y II Encuentro Internacional, Universidad de Extremadura, Espa?a, 2006.
[3]H. Legind Larsen, Efficient importance weighted aggregation between min and max, 9th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU’2002), Annecy, France, 2002.
[4]S. Cubillo, A. Pradera & E. Trillas, On Joining Operators and their and / or Behaviour, Proc. International Conference IPMU'98, 673-679, Paris, France, 1998.
[5]H. J. Zimmermann, Fuzzy sets theory and its application, Kluwer Academia Publishers, Boston / Dordrecht / London, 1991.
[6]F. Herrera, E. Herrera-Viedma, J. L. Verdegay, Direct approach processes in group decision making using linguistic OWA operators, Fuzzy Sets and Systems, 79, pp. 175-190, 1996.
[7]D. Dubois & H. Prade, A Review of Fuzzy Set Aggregation Connectives, Information Sciences 36. 85-121, 1985.
[8]G. Klir & B. Yuan, Fuzzy Sets and Fuzzy Logic. Theory and Applications, Prentice Hall PTR, 1995.
[9]B. Bouchon-Meunier, (editor), Aggregation and Fusion of Imperfect Information, Studies in Fuzziness and Soft Computing, Vol. 12, Physica-Verlag, 1998.
[10]J. Aczél, Lectures on Functional Equations and their Applications, Academic Press, New York, USA, 1966.
[11]D. Dubois, Modèles mathématiques de l' imprécis et de l’ incertain en vue d' applications aux techniques d' aide à la decision, Tesis Doctoral, Université Scientifique et Médicale de Grenoble, 1983.
[12]J. J. Dujmovié, Andness and Orsess as a Mean of Overall Importance, In: Proceedings of the IEEE World Congress on Computational Intelligence, Brisbane, Australia, June 10-15, pp. 83-88, 2012.
[13]J. J. Dujmovié & G. De Tré, Multicriteria Methods and Logic Aggregation in Suitability Maps, International Journal of Intelligent Systems 26 (10), 971-1001, 2011.
[14]J. Fodor & M. Roubens, On meaningfulness of means, Journal of Computational and Applied Mathematics, 103-115, 1995.
[15]G. Mayor, J. Su?er & P. Canet, Agregación multidimensional de números borrosos, Actas del VIH Congreso Espa?ol sobre Tecnologías y Lógica Fuzzy, Pamplona, Espa?a, 171-178, 1998.
[16]M. Mizumoto, Pictorial Representations of Fuzzy Connectives, Part II: cases of compensatory operators and self-dual operators, Fuzzy Sets and Systems 32, 45-79, 1989.
[17]H. T. Nguyen, V. Kreinovich & D. Tolbert, A measure of average sensitivity for fuzzy logics, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 2, No. 4, 361-375, 1994.
[18]H. T. Nguyen & E. A. Walker, A First Course in Fuzzy Logic, CRC Press, 1997.
[19]S. Ovchinnikov, Means on ordered sets, Mathematical Social Sciences 32, 39-56, 1996.
[20]S. Ovchinnikov, An Analytic Characterization of Some Aggregation Operators, International Journal of Intelligent Systems, Vol. 13, 59-68, 1998.
[21]C. Alsina, E. Trillas & L. Valverde, Sobre conectivos lógicos no distributivos para la teoría de los conjuntos borrosos, Pub. Mat. UAB 20. 69-72, 1980.
[22]C. Alsina, E. Trillas. & L. Valverde, On non-distributive logical connectives for fuzzy sets theory, Busefal 3, 18-29, 1983.
[23]C. Alsina, E. Trillas & L. Valverde, On some logical connectives for fuzzy sets theory, Journal of Mathematical Analysis and Applications, Vol. 93 (1), 15-26, 1983.
[24]R. Bellman & M. Giertz, On the analytic formalism of the theory of fuzzy sets. Inform. Sci. 5. 149-156, 1973.
[25]J. Fodor, Left-continuous t-norms in fuzzy logic: An Overview, Acta Polytechnica Hungarica, Journal of Applied Sciences, Budapest, Hungary, 1 (2), ISSN 1785-8860, 2004.
[26]M. J. Frank, On the simultaneous associativity of F(x,y) and x + y - F(x,y), Aequationes Mathematicae 19 (2-3), 194-226, 1979.
[27]M. M. Gupta & J. Qi, Theory of t-norms and fuzzy inference methods, Fuzzy Sets and Systems 40 (3), 431-450, 1991.
[28]C. H. Ling, Representation of Associative Functions, Publ. Math. Debrecen 12, 189-212, 1965.
[29]G. Mayor & J. Torrens, On a class of binary operations: non-strict Archimedean aggregation functions, Proc. 18th ISMVL, Palma de Mallorca, Espa?a, 54-59, 1988.
[30]A. L. Ralescu & D. A. Ralescu, Extensions of fuzzy aggregation, Fuzzy Sets and Systems 86, 321-330, 1997.
[31]B. Schweizer & A. Sklar, Associative functions and statistical triangle semigroups, Publ. Math. Debrecen 8, 169-186, 1961.
[32]V. Torra, The Weighted OWA Operator, International Journal of Intelligent Systems, 12, 153-166, 1997.
[33]G. Mayor & J. Martín, Funciones de agregación localmente internas, Actas del VIH Congreso Espa?ol sobre Tecnologías y Lógica Fuzzy, Pamplona, 355-35, 1998.
[34]R. R. Yager & A. Rybalov, Noncommutative self-identity aggregation, Fuzzy Sets and Systems 85, 73-82, 1997.
[35]T. Calvo & R. Mesiar, Weighted triangular norms-based aggregation operators, Fuzzy Sets and Systems, 137, pp. 3-10, 2003.
[36]H. Dyckhoff & W. Pedrycz, Generalized Means as Model of Compensative Operators, Fuzzy Sets and Systems 14, 143-154, 1984.
[37]E. Trillas, S. Cubillo & J. L. Castro, Conjunction and disjunction on ([0,1], ≤), Fuzzy Sets and Systems 72, 155-165, 1995.
[38]J. M. Fernández-Salido & S. Murakami, Extending Yager’s orness concept for the OWA aggregators to other mean operators, Fuzzy Sets and Systems, 139, pp. 515-542, 2003.
[39]T. Calvo & R. Mesiar, Aggregation operators: ordering and bounds, Fuzzy Sets and Systems, 139, pp. 685-697, 2003.
[40]J. C. Fodor, J. L. Marichal & M. Roubens, Characterization of some aggregation functions arising from MCDM problems, en Bouchonmeunier, B.; Yager, R. R. & Zadeh, L. A. (eds.), Fuzzy logic and soft computing, Series: Advances in fuzzy systems-Applications and theory, 4. World Scientific Singapore, pp. 194-201, 1995.
[41]R. Smolíkova & M. P. Wachowiak, Aggregation operators for selection problems, Fuzzy Sets and Systems, 131, pp. 23-24, 2002.
[42]D. Dubois & H. Prade, Weighted minimum and maximum operations in fuzzy set theory, Information Sciences 39, 205-210, 1986.
[43]R. R. Yager, On Ordered Weighted Averaging Operators in Multicriteria Decisionmaking, IEEE Transactions on Systems, Man and Cybernetics 18, 183-190, 1988.
[44]R. R. Yager & A. Kelman, Fusion of Fuzzy Information With Considerations for Compatibility, Partial Aggregation and Reinforcement, International Journal of Approximate Reasoning 15, 93-122, 1996.
[45]R. Mesiar & M. Komorníková, Triangular Norm-Based Aggregation of Evidence under Fuzziness, En Bouchon-Meunier, B. (editor): Aggregation and Fusion of Imperfect Information, Studies in Fuzziness and Soft Computing, Vol. 12, Physica-Verlag, 11-35, 1998.
[46]D. Filev & R. R. Yager, On the issue of obtaining OWA operators weights, Fuzzy Sets and Systems 94, 157-169, 1998.
[47]F. Herrera, E. Herrera-Viedma & F. Chiclana, A study of the origin and uses of the ordered weighted geometric operator in multicriteria decision making, International Journal of Intelligent Systems 18, 689-707, 2003.
[48]X. W. Liu, Some properties of the weighted OWA operator, IEEE Transactions on Systems, Man and Cybernetics, 36, 1, 118-127, 2006.
[49]B. Llamazares, Choosing OWA operator weights in the field of Social Choice, Information Sciences: an International Journal, 177, 21, 4745-4756, 2007.
[50]J. I. Peláez & J. M. Do?a, LAMA: A Linguistic Aggregation of Majority Additive Operator, International Journal of Intelligent Systems 18, 809-820, 2003.
[51]J. I. Peláez & J. M. Do?a, Majority Additive-Ordered Weighting Averaging: A New Neat Ordered Weighting Averaging Operators Based on the Majority Process, International Journal of Intelligent Systems 18, 469-481, 2003.
[52]J. I. Peláez & J. M. Do?a, A majority model in group decision making using QMA-OWA operators, Int. J. Intell. Syst. 21, 193-208, 2006.
[53]J. I. Peláez, J. M. Do?a & A. Mesas, Majority multiplicative ordered weighting geometric operators and their use in the aggregation of multiplicative preference relations, Mathware and Soft Computing, 12, 107-120, 2005.
[54]J. I. Peláez, J. M. Do?a & A. M. Gil, Application of Majority OWA operators in Strategic Valuation of Companies, Proc Int. Eurofuse Workshop, New Trends in Preference Modelling, 2006.
[55]J. I. Peláez, J. M. Do?a & J. A. Gómez-Ruiz, Analysis of OWA Operators in Decision Making for Modelling the Majority Concept, Applied Mathematics and Computation, Vol. 186, Pages 1263-1275, 2007.
[56]J. I. Peláez, J. M. Do?a & D. L. La Red, Analysis of the Majority Process in Group Decision Making Process, JCIS 2003 (7th Joint Conference on Information Sciences), ISBN N° 0-9707890-2-5, Proceedings, pp. 155-159, North Carolina, USA, 2003.
[57]J. I. Peláez, J. M. Do?a & D. L. La Red, Analysis of the Linguistic Aggregation Operator LAMA in Group Decision Making Process, 32 JAIIO (Argentine Conference on Computer Science and Operational Research) – ASAI 2003 (Argentine Symposium on Artificial Intelligence), ISSN N° 1666-1079, Argentina, 2003.
[58]J. I. Peláez, J. M. Do?a & D. L. La Red, Fuzzy Imputation Methods for Data Base Systems, Handbook of Research on Fuzzy Information Processing in Database, Hershey, PA, USA: Information Science Reference, 2008.
[59]J. I. Peláez, J. M. Do?a, D. L. La Red & A. Mesas, OWA Aggregation with Soft Majority Operators; 33 JAIIO (33° Jornadas Argentinas de Informática e Investigación Operativa) – ASIS 2004 (Primer Simposio Argentino de Sistemas de Información), Anales 2004 (publicación electrónica en CD), ISSN N° 1666-1141, Argentina, 2004.
[60]J. I. Peláez, J. M. Do?a, A. Mesas & D. L. La Red, Opinión de Mayoría en Toma de Decisiones en Grupo Mediante el Operador QMA-OWA, ESTYLF 2004 (XII Congreso Espa?ol Sobre Tecnologías y Lógica Fuzzy), Libro de Actas, págs. 449-454, ISBN N° 84-609-2160-3, Espa?a, 2004.
[61]J. M. Do?a, A. M. Gil, D. L. La Red & J. I. Peláez, A System Based on the Concept of Linguistic Majority for the Companies Valuation, EconoQuantum, Vol. 8 N. 2, Guadalajara, México, 121-142, 2011.
[62]C. H. Carlsson & R. Fuller, Fuzzy reasoning in decision making and optimization, Heidelberg: Springfield-Verlag, 2002.
[63]J. M. Merigó, M. Casanovas & L. Martínez, L. Linguistic aggregation operators for linguistic decision making based on the Dempster-Shafer theory of evidence, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 18, 287-304, 2010.
[64]M. O’Hagan, Aggregating template rule antecedents in real-time expert systems with fuzzy set logic, Proc. 22nd Annual IEEE Asilomar Conf. On Signals, Systems and Computers, Pacific Grove, CA, 1988.
[65]M. Marimin, M. Umano, I. Hatono & H. Tamura, Linguistic Labels for Expressing Fuzzy Preference Relations in Fuzzy Group Decision Making, IEEE Transactions on Systems, Man, and Cybernetics, 28(2): 205-218, 1998.
[66]L. A. Zadeh, The role of fuzzy logic in the management of uncertainty in expert systems, Fuzzy Sets and Systems, 11: 199-227, 1983.
[67]J. M. Do?a Fernández, Modelado de los procesos de toma de decisión en entornos sociales mediante operadores de agregación OWA, Tesis doctoral, Universidad de Málaga, Espa?a, 2008.
[68]R. Yager & D. P. Filev, Fuzzy logic controllers with flexible structures, Proc. Second Int. Cont. on Fuzzy Sets and Neural Networks, Izuka Japan, 317-320, 1992.
[69]R. R. Yager, Families of OWA operators, Fuzzy Sets and Systems, 59, 125-148, 1993.
[70]M. Grabisch, Fuzzy Integral in multicriteria decisión making, Fuzzy Sets and Systems 69, 279-298, 1995.
[71]M. Grabisch, Fuzzy measures and integrals: a survey of applications and recent issues, in Dubois, D.; Prade, H. & Yager, R. (editors), Fuzzy Sets Methods in Information Engineering: a Guided Tour of Applications, J. Wiley & Sons, 1996.
[72]M. Grabisch, Fuzzy Integral as a Flexible and Interpretable Tool of Aggregation, in Bouchon-Meunier, B. (editor): Aggregation and Fusion of Imperfect Information, Studies in Fuzziness and Soft Computing, Vol. 12, Physica-Verlag, 51-72, 1998.
[73]A. R. De Soto & E. Trillas, Agregación de intervalos borrosos mediante el principio de extensión, Actas del VIII Congreso Espa?ol sobre Tecnologías y Lógica Fuzzy, Pamplona, Espa?a, 233-237, 1998.
[74]I. B. Turksen, Interval-valued fuzzy sets based on normal forms, Fuzzy Sets and Systems 20, 191-210, 1986.
[75]L. A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. SMC, SMC-3, 38-44, 1973.
[76]H. J. Zimmermann & P. Zysno, Latent connectives in human decision-making, Fuzzy Sets and Systems 4, 37-51, 1980.
[77]E. Trillas, A. Pradera & S. Cubillo, A mathematical model for fuzzy connectives and its application to operator's behavioural study, in Bouchon-Meunier, B.; Yager, R. R. & Zadeh, L. A. (eds): Information, Uncertainty, Fusion, Kluwer Scientific Publishers, 1999.
[78]C. Alsina, G. Mayor, M. S. Tomás & J. Torrens, A characterization of a class of aggregation functions, Fuzzy Sets and Systems 53, 33-38, 1993.
[79]J. Fodor & T. Calvo, Aggregation Functions Defined by t-Norms and t-Conorms, in Bouchon-Meunier, B. (editor). Aggregation and Fusion of Imperfect Information. Studies in Fuzziness and Soft Computing, Vol. 12, Physica-Verlag, 36-48, 1998.
[80]M. K. Luhandjula, Compensatory operators in fuzzy linear programming with multiple objectives, Fuzzy Sets and Systems 8, 245-252, 1982.
[81]G. Mayor, Contribució a 1' estudi de models matemàtics per a la lógica de la vaguetat, Tesis Doctoral, Universitat de les Illes Balears, Espa?a, 1984.
[82]G. Mayor, Sobre una classe d'operacions entre conjunts difusos: Operacions d'Agregació, Congrés Català de Lógica Matemàtica, Barcelona, Espa?a, 95-97, 1984.
[83]G. Mayor & E. Trillas, On the representation of some aggregation functions, Proc. 16th ISMVL, Blacksburg, VA, 110-114, 1986.
[84]G. Mayor & T. Calvo, On extended Aggregation Functions, Proc. IFSA'97, Praga, 281-285, 1997.
[85]R. R. Yager, On a class of weak triangular norm operators, Tech. Report #MII-1528, Machine Intelligence Institute, Iona College, New Rochelle, NY, USA, 1996.
[86]J. Fodor, R. R. Yager & A. Rybalov, Structure of Uninorms. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 411-427, 1997.
[87]R. R. Yager & A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems 80, 111-120, 1996.
[88]J. Dombi, Basic concepts for a theory of evaluation: The aggregative operator, European J. Oper. Res. 10, 282-293, 1982.
[89]E. Klement, R. Mesiar & E. Pap, On the relationship of associative compensatory operators to triangular norms and conorms, Int. Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 4, No. 2, 129-144, 1996.
[90]W. Silvert, Symmetric summation: A class of operations on fuzzy sets, IEEE Trans. Systems, Man Cybernet. 9, 659-667, 1979.
[91]C. E. Shannon & W. Weaver, A mathematical theory of communication, University of Illinois Press, Urbana, 1949.