?36?8?Vol.36, No.82016?8?Systems Engineering Theory ?;?A belief-rule-based inference method for modeling systems under uncertaintiesGUO Min(School of Economics and Management, Northern University of China, Taiyuan 030051, China)Abstract Belief-rule-based inference methodology using the evidential reasoning approach (RIMER) is a novel modeling methodology for systems under various uncertainties. In this paper, some basic charac- teristics of RIMER are analyzed. It is proved that RIMER is capable to model deterministic or stochasticsystems with sufficient accuracy. To handle more complex uncertainties with local or global unknown, an extended belief-rule-based inference framework is proposed. A numerical example (newsvendor inventory optimizing problem) is provided to illustrate the implementation process of the new RIMER approach and its validity and applicability.Keywords belief-rule-based inference; complex uncertainty; inventory1?,?: 1)?,?,?; 2)?,?,?;?,?,?1.?.?,?:1)?,?,?2.?.?,?,?.2)?,?,?.?,?34.?: 2015-06-01?:?(1969),?,?,?,?,?:?,?.?:?(70971046) Foundation item: National Natural Science Foundation of China (70971046)?:?.?J.?, 2016, 36(8): 19751982.?: Guo M. A belief-rule-based inference method for modeling systems under uncertaintiesJ. Systems Engineering Theory (?, 50%),?50%?“?”“?”?,?(incompleteness)?(unknown).2)?.?,?IF THEN?,?,?,?.?.?,?,?.2?2.1?,?,?,?,?,?6:R = .?U = Ui,i = 1,2,I?, A = A1,A2,AI?,?i?Ui?Ai = Aip,p = 1,2,|Ai|,?Aip?Ui?, |Ai|?Ai?.?“?” (, AND)?“?” (, OR)?. D = Dn,n =1,2,N?, N?. F?,?L?,?k?:Rk: IF A(k)1A(k)2A(k)ITHEN (D1,(k)1);(D2,(k)2);(DN,(k)N),?k,?(k)1,(k)2,(k)I.?A(k)i Ai, i = 1,2,I, (k)n?Dn?, n = 1,2,N,?n=1,2,N(k) n 1.?(k)i?Ui?I?,?k?Rk?8?:?1977?F?. 2.2?,?,?:1)?,?F?.?(?1 ?2 ?I) ,?i= (Aip,ip),p = 1,2,|Ai|?i?Ui?(?), ip?Ui?Aip?,?ip 0? p=1,2,|Ai|ip 1.?,?k?,?wk:?,?k?k.?“?” (, AND)?,? k=?i=1,2,I(k)i)(k) i.?(k) i=(k)i max j=1,2,I(k) j, (k) i= (ip|A(k)i= Aip),?k?Ui?A(k)i?Aip?(k)i?ip.?,?k?k?k,?k?:wk=kk?i=1,2,Lii.?k?(D1,(k)1);(D2,(k)2);(DN,(k)N)?wk,?. 2)?(ER?)8:?,?D-S?(mass)?,?k = 1,2,L,?:mn,k= wk(k)n,n = 1,2,N,mH,k= 1 ?n=1,2,Nmn,k= 1 wk?n=1,2,N(k)n, mH,k= 1 wk, mH,k= wk1 ?n=1,2,N(k)n,?mH,k= mH,k+ mH,k.?,?L?,?:m?n=?k=1,2,L(mn,k+ mH,k+ mH,k) ?k=1,2,L( mH,k+ mH,k),n = 1,2,N(1) m?H=?k=1,2,L( mH,k+ mH,k) ?k=1,2,L mH,k(2) m?H=?k=1,2,L mH,k(3)?,?:mn= Km?n(4) mH= K m?H(5) mH= K m?H(6)K =?n=1,2,N?k=1,2,L(mn,k+ mH,k+ mH,k) (N 1)?k=1,2,L( mH,k+ mH,k)1(7)?,?S(D) = (Dn,n),n = 1,2,N,?n=mn 1 mH, n = 1,2,N?Dn?,?(?D1 DN?)?H= 1 ? n=1,2,Nn.1978?36?3)?,?,?:u(S(D) =?n=1,2,Nu(Dn)n.?u()?,?,?u(Dn1) u(Dn), n = 2,3,N.?,?,?:uMax(S(D) =?n=1,2,N1u(Dn)n+ (N+ H)u(DN),uMin(S(D) =?n=2,3,Nu(Dn)n+ (1+ H)u(D1),?: uAvg(S(D) =uMin(S(D)+uMax(S(D) 2.3?3.1?2?,?,?,?.?1?,?.?(?).?y = y(x),?,?x,?y.?x?x1,x2,xk,xN,?xk?x?.?,?xk1,?i?Ui,?Aip,q,p = 1,2,|Ai|,q = p,|Ai|,?Aip,q?Aip,Aiq.?p = q?Aip,q?Aip,p= Aip.?D,?Dp,q,p = 1,2,N,q = p,N,?Dp,q? Dp,Dq.?p = q?Dp,q?Dp,p= Dp.2)?.?