##
**Type-1 OWA operators for aggregating uncertain information with uncertain weights induced by type-2 linguistic quantifiers.**
*(English)*
Zbl 1187.68619

Summary: The OWA operator proposed by Yager has been widely used to aggregate experts’ opinions or preferences in human decision making. Yager’s traditional OWA operator focuses exclusively on the aggregation of crisp numbers. However, experts usually tend to express their opinions or preferences in a very natural way via linguistic terms. These linguistic terms can be modelled or expressed by (type-1) fuzzy sets. In this paper, we define a new type of OWA operator, the type-1 OWA operator that works as an uncertain OWA operator to aggregate type-1 fuzzy sets with type-1 fuzzy weights, which can be used to aggregate the linguistic opinions or preferences in human decision making with linguistic weights. The procedure for performing type-1 OWA operations is analysed. In order to identify the linguistic weights associated to the type-1 OWA operator, type-2 linguistic quantifiers are proposed. The problem of how to derive linguistic weights used in type-1 OWA aggregation given such type of quantifier is solved. Examples are provided to illustrate the proposed concepts.

### MSC:

68T37 | Reasoning under uncertainty in the context of artificial intelligence |

### Keywords:

aggregation; OWA operator; type-1 OWA operator; type-2 fuzzy sets; type-2 linguistic quantifiers; soft decision making
PDF
BibTeX
XML
Cite

\textit{S.-M. Zhou} et al., Fuzzy Sets Syst. 159, No. 24, 3281--3296 (2008; Zbl 1187.68619)

Full Text:
DOI

### References:

[1] | Bordogna, G.; Fedrizzi, M.; Pasi, G., A linguistic modelling of consensus in group decision making based on OWA operators, IEEE trans. systems man cybernet.—part A, 27, 126-133, (1997) |

[2] | Chiclana, F.; Herrera-Viedma, E.; Herrera, F.; Alonso, S., Induced ordered weighted geometric operators and their use in the aggregation of multiplicative preference relations, Internat. J. intelligent systems, 19, 233-255, (2004) · Zbl 1105.68095 |

[3] | Chiclana, F.; Herrera-Viedma, E.; Herrera, F.; Alonso, S., Some induced ordered weighted averaging operators and their use for solving group decision-making problems based on fuzzy preference relations, European J. oper. res., 182, 383-399, (2007) · Zbl 1128.90513 |

[4] | Cutello, V.; Montero, J., Hierarchies of aggregation operators, Internat. J. intelligent systems, 9, 1025-1045, (1994) |

[5] | Dong, W.M.; Wong, F.S., Fuzzy weighted averages and implementation of the extension principle, Fuzzy sets and systems, 21, 183-199, (1987) · Zbl 0611.65100 |

[6] | Dubois, D.; Prade, H., A review of fuzzy set aggregation connectives, Inform. sci., 36, 85-121, (1985) · Zbl 0582.03040 |

[7] | Fodor, J.; Roubens, M., Fuzzy preference modelling and multicriteria decision support, (1994), Kluwer Academic Publishers Dordrecht · Zbl 0827.90002 |

[8] | Fortin, J.; Dubois, D.; Fargier, H., Gradual numbers and their application to fuzzy interval analysis, IEEE trans. fuzzy systems, 16, 2, 388-402, (2008) |

[9] | Garibaldi, J.M.; Musikasuwan, S.; Ozen, T., The association between non-stationary and interval type-2 fuzzy setsa case study, (), 224-229 |

[10] | Garibaldi, J.M.; Ozen, T., Uncertain fuzzy reasoning: a case study in modelling expert decision making, IEEE trans. fuzzy systems, 15, 16-30, (2007) |

[11] | Guh, Y.-Y.; Hon, C.-C.; Lee, E.S., Fuzzy weighted average: the linear programming approach via charnes and Cooper’s rule, Fuzzy sets and systems, 117, 157-160, (2001) · Zbl 1032.91041 |

[12] | Hanss, M., A nearly strict fuzzy arithmetic for solving problems with uncertainties, (), 439-443 |

[13] | Herrera, F.; Herrera-Viedma, E., Aggregation operators for linguistic weighted information, IEEE trans. systems man cybernet.—part A, 27, 646-656, (1997) |

[14] | Herrera, F.; Herrera-Viedma, E., Linguistic decision analysis: steps for solving decision problems under linguistic information, Fuzzy sets and systems, 115, 67-82, (2000) · Zbl 1073.91528 |

[15] | Herrera-Viedma, E.; Herrera, F.; Chiclana, F., A consensus model for multiperson decision making with different preference structures, IEEE trans. systems man cybernet.—part A, 32, 394-402, (2002) · Zbl 1027.91014 |

[16] | Herrera, F.; Herrera-Viedma, E.; Chiclana, F., A study of the origin and uses of the ordered weighted geometric operator in multicriteria decision making, Internat. J. intelligent systems, 18, 689-707, (2003) · Zbl 1037.68133 |

[17] | Herrera-Viedma, E.; Martinez, L.; Mata, F.; Chiclana, F., A consensus support system model for group decision-making problems with multigranular linguistic preference relations, IEEE trans. fuzzy systems, 13, 644-658, (2005) |

[18] | Hickey, T.; Ju, Q.; Van Emden, M.H., Interval arithmetic: from principles to implementation, J. ACM, 48, 5, 1038-1068, (2001) · Zbl 1323.65047 |

[19] | John, R.I.; Innocent, P., Modelling uncertainty in clinical diagnosis using fuzzy logic, IEEE trans. systems man cybernet.—part B, 35, 1340-1350, (2005) |

[20] | Kacprzyk, J.; Fedrizzi, M., A ‘soft’ measure of consensus in the setting of partial (fuzzy) preferences, European J. oper. res., 34, 316-325, (1988) |

[21] | Kao, C.; Liu, S.-T., Fractional programming approach to fuzzy weighted average, Fuzzy sets and systems, 120, 435-444, (2001) · Zbl 1103.90411 |

[22] | Kaufmann, A.; Gupta, M.M., Introduction to fuzzy arithmetic, theory and applications, (1985), Van Nostrand Reinhold New York · Zbl 0588.94023 |

[23] | Klir, G.J., Fuzzy arithmetic with requisite constraints, Fuzzy sets and systems, 91, 165-175, (1997) · Zbl 0920.04007 |

[24] | Klir, G.J.; Folger, T.A., Fuzzy sets, uncertainty and information, (1988), Prentice-Hall Singapore · Zbl 0675.94025 |

[25] | Llamazares, B., Choosing OWA operator weights in the field of social choice, Inform. sci., 177, 4745-4756, (2007) · Zbl 1284.91140 |

[26] | Mendel, J.M.; John, R.I.; Liu, F., Interval type-2 fuzzy logic systems made simple, IEEE trans. fuzzy systems, 14, 808-821, (2006) |

[27] | Meyer, P.; Roubens, M., On the use of the Choquet integral with fuzzy numbers in multiple criteria decision support, Fuzzy sets and systems, 157, 7, 927-938, (2006) · Zbl 1131.90390 |

[28] | Mich, L.; Fedrizzi, M.; Gaio, L., Approximate reasoning in the modeling of consensus in group decisions, (), 91-102 |

[29] | Mitchell, H.B.; Estrakh, D.D., A modified OWA operator and its use in lossless DPCM image compression, Internat. J. uncertain fuzziness knowledge based systems, 5, 429-436, (1997) · Zbl 1232.68139 |

[30] | Mitchell, H.B.; Estrakh, D.D., An OWA operator with fuzzy ranks, Internat. J. intelligent systems, 13, 69-81, (1998) · Zbl 1087.93516 |

[31] | Mizumoto, M.; Tanaka, K., Some properties of fuzzy sets of type 2, Inform. and control, 31, 312-340, (1976) · Zbl 0331.02042 |

[32] | Ramik, J.; Rimanek, J., Inequality relation between fuzzy numbers and its use in fuzzy operation, Fuzzy sets and systems, 16, 123-138, (1985) · Zbl 0574.04005 |

[33] | Stahl, V., A sufficient condition for non-overestimation in interval arithmetic, Computing, 59, 4, 349-363, (1997) · Zbl 0894.65020 |

[34] | Sun Microsystems Inc., Interval Arithmetic in High Performance Technical Computing, White Paper, September 2002. |

[35] | Wang, Y.-M.; Parkan, C., A preemptive goal programming method for aggregating OWA operator weights in group decision making, Inform. sci., 177, 1867-1877, (2007) · Zbl 1147.91314 |

[36] | Wang, Z.; Yang, R.; Heng, P.-A.; Leung, K.-S., Real-valued Choquet integrals with fuzzy-valued integrands, Fuzzy sets and systems, 157, 2, 256-269, (2006) · Zbl 1080.28011 |

[37] | Xu, Z.S.; Da, Q.L., An overview of operators for aggregating information, Internat. J. intelligent systems, 18, 953-969, (2003) · Zbl 1069.68612 |

[38] | Yager, R.R., On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE trans. systems man cybernet., 18, 183-190, (1988) · Zbl 0637.90057 |

[39] | Yager, R.R., A general approach to criteria aggregation using fuzzy measures, Internat. J. man-machine study, 38, 187-213, (1993) |

[40] | Yager, R.R., Families of OWA operators, Fuzzy sets and systems, 59, 125-148, (1993) · Zbl 0790.94004 |

[41] | Yager, R.R.; Filev, D.P., Generalizing the modeling of fuzzy logic controllers by parameterized aggregation operators, Fuzzy sets and systems, 70, 303-313, (1995) |

[42] | Yager, R.R.; Filev, D.P.; Sadeghi, T., Analysis of flexible structured fuzzy logic controllers, IEEE trans. systems man cybernet., 24, 1035-1043, (1994) |

[43] | Yager, R.R.; Goldstein, L.S.; Mendels, E., FUZMAR: an approach to aggregating market research data based on fuzzy reasoning, Fuzzy sets and systems, 68, 1-11, (1994) |

[44] | Yang, R.; Wang, Z.; Heng, P.-A.; Leung, K.-S., Fuzzy numbers and fuzzification of the Choquet integral, Fuzzy sets and systems, 153, 1, 95-113, (2005) · Zbl 1065.28014 |

[45] | Zadeh, L.A., Fuzzy sets, Inform. and control, 8, 338-353, (1965) · Zbl 0139.24606 |

[46] | Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning-1, Inform. sci., 8, 199-249, (1975) · Zbl 0397.68071 |

[47] | A Zadeh, L., A computational approach to fuzzy quantifiers in natural languages, Comput. math. appl., 9, 149-184, (1983) · Zbl 0517.94028 |

[48] | Zadeh, L.A., From computing with numbers to computing with words—from manipulation of measurements to manipulation of perceptions, IEEE trans. circuits systems, 45, 105-119, (1999) · Zbl 0954.68513 |

[49] | Zhou, S.-M.; Gan, J.Q., Constructing accurate and parsimonious fuzzy models with distinguishable fuzzy sets based on an entropy measure, Fuzzy sets and systems, 157, 1057-1074, (2006) · Zbl 1092.68658 |

[50] | Zhou, S.-M.; Gan, J.Q., Constructing parsimonious fuzzy classifiers based on L2-SVM in high-dimensional space with automatic model selection and fuzzy rule ranking, IEEE trans. fuzzy systems, 15, 398-409, (2007) |

[51] | Zimmermann, H.-J., Fuzzy sets, decision making and expert systems, (1987), Kluwer Academic Publishers Boston |

[52] | Zimmermann, H.-J.; Zysno, P., Latent connectives in human decision making, Fuzzy sets and systems, 4, 37-51, (1980) · Zbl 0435.90009 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.