Document Type: Research Paper


1 Singidunum University, 11000 Belgrade, SERBIA, Obuda University, 1034 Budapest, Hungary

2 Petroleum-Gas University of Ploiesti, Department of Computer Science, Information Technology, Mathematics and Physics, Bd. Bucuresti, No. 39, Ploiesti 100680, Romania

3 University "Alexandru Ioan Cuza", Faculty of Mathematics, Bd.Carol I, No. 11, Iasi, 700506, Romania


Intervals are related to the representation of uncertainty. In this sense, we introduce an integral of Gould type for an interval-valued multifunction relative to an interval-valued set multifunction, with respect to Guo and Zhang order relation. Classical
and specific properties of this new type of integral are established and several examples and applications from multicriteria decision making problems are provided.


[1] H. Agahi, R. Mesiar, Y. Ouyang, E. Pap and M. Strboja Berwald type inequality for Sugeno
integral, Appl. Math. Comput., 217(8) (2010), 4100{4108.
[2] H. Agahi, Y. Ouyang, R. Mesiar, E. Pap and M. ^Strboja, Holder and Minkowski type in-
equalities for pseudo-integral, Appl. Math. Comput., 217(21) (2011), 8630{8639.

[3] R. J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl., 12 (1965), 1{12.
[4] A. Aviles, G. Plebanek and J. Rodriguez, The McShane integral in weakly compactly gener-
ated spaces, J. Funct. Anal., 259(11) (2010), 2776{2792.
[5] G. Birkhoff, Integration of functions with values in a Banach space, Trans. Amer. Math.
Soc., 38(2) (1935), 357{378.
[6] A. Boccuto and A. R. Sambucini, A note on comparison between Birkhoff and Mc Shane
integrals for multifunctions, Real Analysis Exchange, 37(2) (2012), 3{15.
[7] A. Boccuto and A. R. Sambucini, A McShane integral for multifunctions, J. Concr. Appl.
Math., 2(4) (2004), 307{325.
[8] A. Boccuto, D. Candeloro and A. R. Sambucini, Henstock multivalued integrability in Ba-
nach lattices with respect to pointwise non atomic measures, Rendiconti Lincei Matematica
e Applicazioni, 26(4) (2015), 363{383.
[9] G. Bykzkan and D. Duan, Choquet integral based aggregation approach to software develop-
ment risk assessment, Inform. Sci., 180(3) (2010), 441{451.
[10] H. Bustince, J. Fernandez, R. Mesiar and J. Kalicka;, Discrete interval-valued Choquet
integral, Proceedings of the 6th International Summer School on Aggregation Operators(
AGOP)(2011), 23{27.
[11] D. Candeloro, A. Croitoru, A. Gavrilut and A. R. Sambucini, Atomicity related to non-
additive integrability, Rend. Circolo Matem. Palermo, 65(3) (2016), 435{449.
[12] B. Cascales and J. Rodriguez, Birkhoff integral for multi-valued functions, J. Math. Anal.
Appl., 297 (2004), 540{560.
[13] A. Croitoru and A. Gavrilut, Comparison between Birkhoff and Gould integral, Mediterr. J.
Math., 12 (2015), 329{347.
[14] A. Croitoru, A. Gavrilut and A. Iosif, Birkhoff weak integrability of multifunctions, International
Journal of Pure Mathematics, 2 (2015), 47{54.
[15] A. Croitoru and N. Mastorakis, Estimations, convergences and comparisons on fuzzy integrals
of Sugeno, Choquet and Gould type, Proceedings of the 2014 IEEE International Conference
on Fuzzy Systems (FUZ-IEEE)(2014), DOI 10.1109/FUZZIEEE.2014.689.1590, (2014), 1205{
[16] A. P. Dempster, Upper and lower probabilities induced by a multivalued mapping, Ann. Math.
Statist., 38 (1967), 325{339.
[17] A. Dinghas, Zum Minkowskischen Integralbegriff abgeschlossener Mengen, Math. Zeit., DOI:
10.1007/BF01186606, 66 (1956), 173{188.
[18] L. Drewnowski, Topological rings of sets, continuous set functions, integration, I, II,III, Bull.
Acad. Polon. Sci. Ser. Math. Astron. Phys., 20 (1972), 277{286.
[19] A. Gavrilut and A. Petcu, A Gould type integral with respect to a submeasure, An. St. Univ.
Al. I. Cuza Iasi, 53(2) (2007), 351{368.
[20] A. Gavrilut and A. Petcu, Some properties of the Gould type integral with respect to a
submeasure, Bul. Inst. Politehnic din Iasi, Sectia Mat. Mec. Teor. Fiz., 53(57)(5) (2007),
[21] A. Gavrilut, A Gould type integral with respect to a multisubmeasure, Math. Slovaca, 58
(2008), 43{62.
[22] A. Gavrilut, A generalized Gould type integral with respect to a multisubmeasure, Math.
Slovaca, 60 (2010), 289{318.
[23] A. Gavrilut, Fuzzy Gould integrability on atoms, Iranian Journal of Fuzzy Systems, 8(3)
(2011), 113{124.
[24] A. Gavrilut, Remarks of monotone set-valued multifunctions, Inform. Sci., 259 (2014), 225{
[25] A. Gavrilut, A. Iosif and A. Croitoru, The Gould integral in Banach lattices, Positivity, 19
(2015), 65-82.
[26] M. Grabisch, J. L. Marichal, R. Mesiar and E. Pap, Aggregation functions, Cambridge University
Press, 127, 2009.
[27] G. G. Gould, On integration of vector-valued measures, Proc. London Math. Soc., 15 (1965),

[28] C. Guo and D. Zhang, On set-valued fuzzy measures, Inform. Sci., 160 (2004), 13{25.
[29] S. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, vol. I, Theory. Mathematics
and its Applications, Kluwer Academic Publishers, Dordrecht, 419 (1997).
[30] M. Hukuhara, Integration des applications mesurables dont la valuer est un compact convexe,
Funkcialaj Ekvacioj, 10 (1967), 205-223.
[31] L. C. Jang, A note on the monotone interval-valued set function de ned by the interval-valued
Choquet integral, Commun. Korean Math. Soc., 22 (2007), 227{234.
[32] L. C. Jang, , A note on convergence properties of interval-valued capacity functionals and
Choquet integrals, Inform. Sci., 183 (2012) 151-158.
[33] L. C. Jang, Interval-valued Choquet integrals and their applications, J. Appl. Math. Comput.,
16 (2004), 429{445.
[34] L. C. Jang, On properties of the Choquet integral of interval-valued functions, J. Appl. Math.,
ID 492149, doi:10.1155/2011/492149, (2011).
[35] L. C. Jang, The application of interval-valued Choquet integrals in multicriteria decision aid,
J. Appl. Math. & Computing, 20(1-2) (2006), 549{556.
[36] L. S. Li and Z. Sheng, The fuzzy set-valued measures generated by fuzzy random variables,
Fuzzy Sets and Systems, 97 (1998), 203{209.
[37] E. Pap, Null-additive Set Functions, Kluwer Academic Publishers, Dordrecht, 1995.
[38] A. Precupanu and A. Croitoru, A Gould type integral with respect to a multimeasure I/II,
An. St. Univ. "Al.I. Cuza" Iasi, 48 (2002), 165{200 / 49(2003), 183{207.
[39] A. Precupanu, A. Gavrilut and A. Croitoru, A fuzzy Gould type integral, Fuzzy Sets and
Systems, 161 (2010), 661{680.
[40] A. Precupanu and B. Satco, The Aumann-Gould integral, Mediterr. J. Math., 5 (2008), 429{
[41] J. Sipos, Integral with respect to a pre-measure, Math. Slovaca, 29 (1979), 141{155.
[42] F. N. Sofi an-Boca, Another Gould type integral with respect to a multisubmeasure, An. Stiint.
Univ. "Al.I. Cuza" Iasi, 57 (2011), 13{30.
[43] G. Shafer, A Mathematical Theory of Evidence, Princeton University Press, Princeton, 1976.
[44] N. Spaltenstein, A De nition of Integrals, J. Math. Anal. Appl., 195 (1995), 835{871.
[45] K. Weichselberger, The theory of interval-probability as a unifying concept for uncertainty,
Int. J. Approx. Reason., 24 (2000), 149{170.
[46] L. A. Zadeh, Probability measures of fuzzy events, J. Math. Anal. Appl., 23 (1968), 421{427.