论文研究-一种不改变RBD拓扑结构的三态贝叶斯网络.pdf-资料库

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37 2 2017 2 Systems Engineering — Theory & Practice Vol.37, No.2 Feb., 2017 doi: 10.12011/1000-6788(2017)02-0486-10 : TB114.3; C94 Æ: A RBD , ( , 710072) (RBD) (BN) RBD !"#, % RBD ! BN &’. "(#$) “*+&” , “’-(./” 0+, 2345’-!-!./0 #$+!2, 01’-,#$0+232. 9;.4Æ)% ! BN 326>? (CPT) 89&’. @B%3CDE<= RBD BN 3, ! BN 3, 90 BN <)G?. *,AH)I?, JB ! BN .D RBD G , K1$ BN "#, N J2)1$KO. ; !; "#; *+&; ’-(./ A three-state Bayesian network without changing the topology of RBD WANG Yao, SUN Qin (School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China) Abstract An isomorphic three-mode Bayesian network (BN) method is proposed based on the topology of reliability block diagram (RBD). It overcomes the disadvantages to transform a RBD to an equivalent three-layer BN in terms of large topology distinction and combinational explosion problems. There are two modes — normal and failure in traditional RBD and three-layer models, in this paper the failure mode is further divided into two new modes: physical failure and normal but not work. Accordingly, a three-mode BN model with normal, physical failure and normal but not work is formulated to replace the original two-mode BN model. Then the conditional probability table (CPT) of each node in the three-mode BN is analyzed in further. Eventually, the equivalent three-layer BN and isomorphic three-mode BN are built for the RBD of the navigation mission of an aircraft, the two BNs are calculated and analyzed. The results demonstrate that the new three-mode BN model, which can overcome the combinational explosion problem, is an eﬀective method for system reliability analysis. Keywords Bayesian network; three-mode Bayesian network; combinational explosion; physic failure; normal without working 1 Q , , , : 2014-08-29 ST: (1989–), , , , , : , E-mail: wangyaorose @126.com; (1956–), , , , , : , E-mail: sunqin@n wpu.edu.cn. U: (2052013B003) Foundation item: Quality and Reliability Fundamental Project of China’s Ministry of Industry and Information Technology’s Twelfth Five-year Plan (2052013B003) VW !: , . RBD Æ [J]. , 2017, 37(2): 486–495. W !: Wang Y, Sun Q. A three-state Bayesian network without changing the topology of RBD[J]. Systems Engineering — Theory & Practice, 2017, 37(2): 486–495.

2 . # !! , !#"#"# [1−3]. , : RBD Æ 487 "#"$ (RBD) %&#"#"& ’#. $( )%&’$*’%(()+ *, )&%+ (BLOCK) ’*’&, [4,5]. , RBD "-’"#(, +,.- ’%*’ ()*; &+,.)-, .*"#(-0,. [6]. /1&2 04&, -+&+56 RBD -%&$77 [4,7−10], 2/3Petri 8RGGo- ﬂow 7Æ985 (BN) . & BN ./)-4, &6-6!, 7:18$79 67 [11−14]. ; RBD &6< BN &04", Torres-Toledano :’’9 RBD .1:< BN 0 [6]. 20;=#,/ RBD &="36-?0.<485: 1 4 RBD % BLOCK &,, 2 1.?%2@ 4, 3 4.?2@. AB2 0@Æ,(, Solano-Soto[5] 920,("#"-&, Trivedi[7] 92-?B ,(940, I9 RBD .1 BN J5 L= "36-?0, ?0@. >, (206B%& BN 36’?#. @L6 0 &< BN 736CAD D, 6 “” “/” C2@-, JF0 “ +<0” 2@01 BN 4’M2@. E.N@L:’<@ BN 76‘ A+,.F, .’ H. 1(a)(b) U1’+.V RBD < BN.

488 37 －－ 1－ 1－－－－ 2.2 : BN Me>‘ a 1 EbcHIK BN d !9T> CPT LC!&< BN 736D 2&. 6!, &W 1(a)(b) "N, F0 4D< BN J RBD Z.[. G ,.=H%*’D0, F2T 1(b) "NT O1 36 X1 ∼ X6, :X 6 A BLOCK 0\]6 RBD .< BN ,&YF. >, 6< BN ,&5L RBD =-?, &, L0? BN 7%". 6 CPT L!, (?, < BN 9’’?#: 1) V2]-? Oi m A BLOCK, # 3) "N, .?2-? 2 4 Oi m AO4, Oi CPT Pr(Oi|π(Oi)) 2m+1 AL, m , 85’’?#; 2) V236 n -?, # 4) "N, 2@4 Sys n AO4, CPTPr(Sys|π(Sys)) 2n+1 AL, n , 850’’?#. P&&C!D , @L 3 :’’<@ BN 7. 3 f BN Q R 3.1 BN h TW(X, &*$’36C “F4” 2@ (-): ^^F/.

2 , : RBD Æ 489 Q‘0F<0. @L9&CF4-Uab1 “^/” -> “+<0” -. F6 J%,&, “+<0” -’<Y (U: Q‘0*’<36+<0-). d e, &&36‘0*’36 3 2@: 2@^/2@+<02@, U (V 012 .?. F2, & 1(a) &V, X1 ^/-F X2 +<0-; X2 +<0-.^/-P" X3 +<0-, V>P, X6 CF4 -P" X7 +<0-; X7 CF4-4 Sys F4 (U: 01<&,7 BLOCK 4 Sys, Q “^/” -). P&&//J[*, %&V<@ BN (Z 1(c)), &Y4 X1 4 Sys C2@, F&(4P1[ “+<0” <@4. TU\, <@ BN RBD !6 ’, Æ=H0; FVOB< BN YF% BLOCK 0\]. TL !\, <@ BN &)’4-A’AO4, 4 CPT &L-1 9; F< BN & 4 O1 CPT L1 2m+1 = 256, m = 7, [ 9. IV&*’A m , 1(b) &< BN 4 O1 CPT L) U, -< BN ’’?#; m , 1(c) &<@ BN %4 CPT L*f1 9, "4EO’?#. <,\’, & RBD .1<@ BN 0b&& “+<0” -:’, =R , .6 RBD &1+.. ;X@L0A6 RBD 36+.J ([,.!+.), "‘ P-@, &,.Q4. :&7, ’36a+., I%,.’* $’Aa (,.&*’’1 2∼4 A); 9%,.h1’A BLOCK J, RBD "h1’A7+. (2 3(a)), ;X@L06&7 :X. Y Start Y Y{0,1} (a) Y X1 X2 … … Xn X1 X2 … … Xn Y End X1 X2 … … Xn Y Y{0,1,2} (b) Y X1 X2 … … Xn Sys Sys{0,2} (c) YSys a 2 BLOCK Y iYjZklm^_ a

490 37 3.2 RBD p BN cqd 3.1 "N, “+<0” 2@F0E BN 7U RBD a’. @6+.V -, P’)W+,.)- RBD <@ BN .0, [U> CPT %2L i C!. &, <@ BN J RBD U’, /XbU. )#<@ BN % 42@b(i . &)W)- RBD, <@ BN 7%42@b(i 02: 1) j BLOCK Y Q‘0 (Z 2(a)), 4 Y 1c@4, 8 “^/” 2@, U( “0”“1” .?; 2) j4 BLOCK Y ‘0 (Z 2(b)), 4 Y 1<@4, 8“^/” > “+ <0” 2@, U( “0”“1”“2” .?; 3) 6 RBD end R, %&.?2@ 4 Sys, Sys 1c@4, 8 “” . “ F4” C2@, U( “0”“2” .? (Z 2(c)). 2 "N, <@ BN 7c BLOCK ’’&,4.Y’ 4 Sys. &, 4 CPT Li 0Z# 1)∼3), 4 Sys CPT i 0Z# 4)5): 1) 6/d&V, 4 X1, X2,··· , Xn 1)W’2@J, 4 Y 82@ “1” L 1 1 − Ry, & Ry 1 RBD & BLOCK Y "#(. K Y Q‘04J, - (1) f&. (1) 2) K4 X1, X2,··· , Xn &364 Xi 82@ 0 J, 4 Y 82@ “0” L1 Ry, 82 Pr(Y = 1|X1, X2,··· , Xn) = 1 − Ry @ “2” L1 0.0; (2) 3) K4 X1, X2,··· , Xn P82@ 1 2 J, 4 Y 82@ “2” L1 Ry, 82@ “0” Pr(Y = 0|X1, X2,··· , Xi = 0,··· , Xn) = Ry Pr(Y = 2|X1, X2,··· , Xi = 0,··· , Xn) = 0.0 0 ≤ i ≤ n L1 0.0. Pr(Y = 0|X1 = 0, X2 = 0,··· , Xn = 0) = 0.0 Pr(Y = 2|X1 = 0, X2 = 0,··· , Xn = 0) = Ry (3) (4) 4) K4 X1, X2,··· , Xn &364 Xi 82@ 0 J, 4 Sys 82@ “0” L1 1.0; Pr(Y = 0|X1, X2,··· , Xi = 0,··· , Xn) = 1.0 5) K4 X1, X2,··· , Xn 6$482@ 1 2 J, 4 Sys 82@ “2” L1 1.0. Pr(Y = 2|X1 = 0, X2 = 0,··· , Xi = 0,··· , Xn = 0) = 1.0 (5) - (1)∼(5) TW(P’<@ BN &%4 CPT LX0. 1(c) % CPT L K",2-, F2- (5) )’ Sys O42@8 1 2 J, 41 2 L# 1.0, d e Sys CPT & 46 2L,1 1.0, eFOL", 35 2L1 0.0; O - (4), Sys CPT 1 2L,1 1.0, L", 2 2L1 0.0. *X, O- (4) > (5) K"!4 Sys CPT % (Z 1(c)Pr(Sys|X7)). , ,- (1)∼(3) "! 1(c) $ 4 CPT %. , - (1)∼(5) @fm7.g (%#), 3.1 &* -(&"N, Æ985P-(Uf , CPT &%2’ L8XQ*. ;F, - (1)∼(5) ,<efP-@. 4 ]s 3(a) 1]7!’h2)+ RBD. 2 RBD 26 A BLOCK ()W BLOCK .? *$’/4), % BLOCK /LPT)i. 8 T = 10000h, U%:< BN <@ BN, ,&C7(2Wa, -D&@L:’<@ BN 7i"2g.

2 , : RBD Æ 491 1 1 0.000004 X1:1 2 0.000001 3 X3:1 Start 2 0.000004 X2:2 4 0.000003 X4:2 5 3 0.000001 X5:1 6 0.000003 X6:2 4 7 0.000004 X7: 8 0.000004 X8: 9 0.000003 X9: 10 0.000005 X10: 13 0.000001 12 0.000006 14 0.000005 X14: 11 0.000004 X12: X13: 15 0.000006 X15: 16 0.000003 17 0.000006 X16: X17: 6 X11: 19 5 0.000007 7 X18: 18 0.000008 0.000006 20 X19: X20: 21 0.000001 X21: 8 0.000004 22 X22: X23: 0.000006 23 25 0.000001 26 0.000002 X24: X25: X26: 24 0.000001 End (a) nhjoi RBD X1 X2 X3 O1 O2 …… …… …… …… X26 BN1 O384 BN2 Sys BN3 (b) k BN X3 X4 X5 X6 X7 X8 X9 X14 X15 X16 X17 X24 X25 X26 Sys X12 X19 X20 X21 X13 X22 X23 (c) RBD BN a 3 hdijktlubcHIKd X1 X2 X10 X11 X18

492 4.1 vÆc: BN h 37 O< BN 0# 1), 5 /’36=-?. O’ , && 8 ’,.U8’’A BLOCK, " <6X 8 ’,.’= BLOCK ’-?, F2-? {X1, X3, X5, X7, X9, X10, X12, X13, X14, X16, X17, X18, X19, X22, X24, X25, X26}. 8 ’,.f 36 384 -?, IW-? 17 A BLOCK. VO# 2), 1 26 A4: X1, X2,··· , X26, RBD & BLOCK )’’&,*. # 3) @L’-?"N, .?-?2@ 45 384 A: O1, O2,··· , O384, .85 2 4. I&)W Oi (1 ≤ i ≤ 384), O4 17 A (.?W-?536 17 A4), CPT c 218 = 262144 AL, ABO -(. VO# 4), 4 Sys O41 2 384 A 4, CPT c 2385 ≈ 1.146750× 10105 AL, -@Q023k>-, ’?#’. , %:< BN Z 3(b). 4.2 vÆc BN h 3 "N, a RBD %+ (BLOCK) SB*‘wy 6 4.1 4.2 &-, &< BN <@ BN (2Wa; >&<@ BN 7 i"2g. T< BN UN, 4 Sys O4- (1 384 A), f 85(1 O(m exp(384)), m = 411, m 1854A. sX< BN @’’?#. FT<@ BN UN, 4 X22 4 X23 O4A- (1 3 A), f 85(1 O(m exp(3)), m = 27, 85( ‘, ’?#4f. !UP’<@ BN "#(;EP>?IP- 0. 1) "#(-0: 6<@ BN &, ogOD@ [17,18] Sys = 0, VO-""#( R Sys = 0.725606. 26 Pr(Sys = 0) = Pr(Sys|π(Sys)) Pr(Xi|π(Xi)) S = {X1, X2,··· , X26} (6) &, π(Sys) > π(Xi)) U.? 3(c) &4 Sys > Xi O4. F24 X12 O414 X10 > X11. Pr(Sys|π(Sys) > Pr(Xi|π(Xi)) U.?4 Sys > Xi CPT, % CPT 7L8X S i=1

2 , : RBD Æ 493 @ 4.2 P’. Xk, CPT l0I)#& CPT &2@ZL2l00; CPT ?0I)#,?0IoGD@ S, 1pI0YL [17, 18]. 2) ;EP Pr(Sys = 1|Xi = 1) -0: 6<@ BN &, ogOD@ Sys = 1 > Xi = 1; o S = {X1, X2,··· , Xi−1, Xi+1,··· , X26}, VO- (6) "-Li Pr(Sys = 1, Xi = 1); 6<@ BN &, ogOD@ Xi = 1; o S = {X1, X2,··· , Xi−1, Xi+1,··· , X26, Sys}, VO- (6) "- Li Pr(Xi = 1); VOÆ9 [17,18], -;EPE: Pr(Sys = 1|Xi = 1) = 3) ?IP Pr(Xi = 1|Sys = 1) -0: Pr(Sys = 1, Xi = 1) Pr(Xi = 1) ;EP-0#Z, -Li Pr(Sys = 1, Xi = 1); -<"#( Pr(Sys = 1) = 1 − R Sys; VOÆ9 [14], -?IPE: Pr(Xi = 1|Sys = 1) = Pr(Sys = 1, Xi = 1) Pr(Sys = 1) . . ;EP>?IPEUZ. 1 45 . p 1 m BN {rs|tc RBD urs q BLOCK r Ri rpsp Pr(Sys = 1|Xi = 1) sp Pr(Xi = 1|Sys = 1) X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23 X24 X25 X26 0.960789 0.960789 0.990050 0.970446 0.990050 0.970446 0.960789 0.960789 0.970446 0.951229 0.960789 0.990050 0.941765 0.951229 0.941765 0.970446 0.941765 0.923116 0.932394 0.941765 0.990050 0.960789 0.941765 0.990050 0.990050 0.980199 s RBD 0.301773 0.301773 0.295632 0.281403 0.295632 0.281403 0.301773 0.301773 1.000000 0.301510 0.308460 1.000000 1.000000 0.314704 0.307817 1.000000 1.000000 1.000000 0.274787 0.274854 0.277223 0.315087 0.301251 1.000000 1.000000 1.000000 BN 0.301773 0.301773 0.295632 0.281403 0.295632 0.281403 0.301773 0.301773 1.000000 0.301510 0.308460 1.000000 1.000000 0.314704 0.307817 1.000000 1.000000 1.000000 0.274787 0.274854 0.277223 0.315087 0.301251 1.000000 1.000000 1.000000 BN 0.0431231 0.0431231 0.0107201 0.0303090 0.0107201 0.0303090 0.0431231 0.0431231 0.1077060 0.0535903 0.0440788 0.0362616 0.2122310 0.0559353 0.0653280 0.1077060 0.2122310 0.2801950 0.0677022 0.0583322 0.0100525 0.0450256 0.0639344 0.0362616 0.0362616 0.0721592 < BN 7’’?#, <&2P-. /1g<@ BN i", (J RBD 0-2"#(. J RBD 0-"#(#2:

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