Group decision making method based on single valued neutrosophic Choquet integral operator∗

Single valued neutrosophic set (SVNS) depicts not only the incomplete information, but also the indeterminate information and inconsistent information which exist commonly in belief systems. The existing decision making methods for SVNS consider the case that the attributes are independent, and cannot handle the correlation among attributes. Based on the Choquet integral and the cosine similarity degree of single valued neutrosophic number, we propose an operator to aggregate single valued neutrosophic numbers (SVNNs), which can deal with the single valued neutrosophic information with connective attributes. By using the proposed single valued neutrosophic Choquet integral operator, an approach is given for the multi-attribute group decision making problems with SVNNs. An example is showed to illustrate the validity and applicability of the proposed method. Keywords multi-attribute group decision making, single valued neutrosophic sets, Choquet integral, aggregation operators Chinese Library Classification C934 2010 Mathematics Subject Classification 90B50 ÂvFÏ: 2017-03-28 * Ä7‘8: I[g,‰ÆÄ7 (No. 71371107) 1. ­H“‰ŒÆ+nÆ , ìÀFì 276800, College of Operations and Management, Qufu Normal University, Rizhao 276800, Shandong, China 2. 2ŒÆêƉÆÆ , ô€ 2 225002, College of Mathematical Science, Yangzhou University, Yangzhou 225002, Jiangsu, China † Ï&Šö E-mail: [email protected] 2Ï ÄuüŠ¥œ8ChoquetÈ©Žf +ûü{ 111 0 Ú ó gZadehJÑ 8Vg±5, 8nØ 2ïÄ. Atanassov3DÚ 8 Ä:þÏLO\šäáÝ1⁄4êJÑ †ú 8nØ. †ú 8UN ́L ˆ &E, gJѱ5 ׄuÐÚ2A^. †ú 8U?nØ & E%ØU?nØ(1⁄25Úؘ—&E. ~X, 3˜ N ̄ò õ‘ÀJ¥, kÀ‘: (!†Ø!Ø(1⁄2. duûüö‡< @Uåk, ûüöŒU3 (ÚØ(1⁄2ü‡À ‘¥gþ, ŒU¬ÓžÀJüö, aÑy ÚO(J: ( 'Ǐ 0.5, †Ø 'Ç  0.4,Ø(1⁄2 'Ǐ 0.3. w,†ú 8Ã{Lˆda&E.u ́Smarandache3† ú 8 Ä:þJÑ ¥œ8nØ. ¥œ8nØ3†ú 8Ä:þO\ Õá Ø (1⁄2Ý, ́ 8چú 8 ˜«í2. æ^¥œ8nØ, þã~f¥ ûü&EŒ ±£ãx(0.5, 0.3, 0.4). 3¥œ8nØ¥ûüöŒ±¦^ý¢§Ý!”ýÝÚØ(1⁄2§Ý5£ãé* ̄Ô μd, gJѱ5Úå 2 '5ÚïÄ. @Ï ¥œ8 ́lóÆ ÝJÑ , JuA^u¢Sûü¥. WangÚ Smarandache Äu¥œ8nØlEâ ÝJÑ ˜«üŠ¥œ8Vg, ¿?Ø Ùƒ'$Ž5KÚ5Ÿ. LiuÚTangJÑ Äu«m ¥œ8 \ 8(ŽfÚûü{. YeJÑ ÄuüŠ¥œ8 ڃqÝ û ü{, Wang JÑ ÄuüŠ¥œ8 MSMŽfÚTODIM õá5ûü{. ®k'uüŠ¥œ8ûü{ ïÄ̇Äá5ƒmƒpÕá ûü ̄K, 3 y¢ûü¥,á5ƒm~~3ˆ«'é'X,I‡Äμd&EüŠ¥œ8…á5m k'é'X õá5ûü ̄K.uChoquetÈ©ŽfŒ±Ä&Em ƒp'X, ©òÙí2 üŠ¥œ8œ/, |^üŠ¥œ8 {uƒqÝ' {, JÑ üŠ ¥œ8ChoquetÈ©Žf. TŽfØ=Ä á5m ­‡5,ӞŒ±‡Ná5m 'é'X,, éTŽf 5ŸÚAϜ/?1 ?Ø,¿3dŽfÄ:þJÑ ¦)õ á5+ûü ̄K Ž{. 1 üŠ¥œ89Ù$ŽÚ5Ÿ 1⁄2 1.1 X˜é–8, A = {x(TA(x), IA(x), FA(x))|x ∈ X}, K¡üŠ¥œ 8. TA(x), IA(x), FA(x)©OL«áu ý¢§Ý, Ø(1⁄2§Ýڔý§Ý, ÷v ∀x ∈ X, TA(x), IA(x), FA(x) ∈ [0, 1], 0 6 TA(x) + IA(x) +FA(x) 6 3.¡ (T (x), I(x), F (x))üŠ¥ œê, ¿òÙ{Px = (Tx, Ix, Fx). 1⁄2 1.2 é?¿¥œêxi = (Ti, Ii, Fi), xj = (Tj , Ij , Fj), ك'$Ž1⁄2ÂXe: (1) xi ⊕ xj = (Ti + Tj − TiTj , IiIj , FiFj); (2) xi ⊗ xj = (TiTj , Ii + Ij − IiIj , Fi + Fj − FiFj); (3) λxi = ( 1− (1− Ti), (Ii), (Fi) ) , λ > 0; (4) (xi) λ = ( (Ti) , 1− (1− Ii), 1− (1− Fi) ) , λ > 0; (5) xi Ö8xi = (1− Ti, 1− Ii, 1− Fi). þã$Ž5KäkXe5Ÿ: (1) xi ⊕ xj = xj ⊕ xi; (2) xi ⊗ xj = xj ⊗ xi;