论文研究-无偏灰色预测模型递推解法及其优化.pdf-资料库

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31 8 2011 8 Systems Engineering — Theory & Practice Vol.31, No.8 Aug., 2011 : 1000-6788(2011)08-1532-07 : N941.5 : A 1, 1, 1, 2 (1. , 210016; 2. , 321004) !"#%"#&’*., / 234 78 , : ;<=>23 74@A. BCD, F G B JKL<=>MN., OPQ, ;FKLJMN R4S, VXY [\ R4. _‘abde Æ# f ibj . " 23 ; 78 ; MN; 4 Recursive solution to unbiased grey model and its optimization SHI Bin1, LIU Si-feng1, DANG Yao-guo1, WANG Zheng-xin2 (1. College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China; 2. School of Economics and Management, Zhejiang Normal University, Jinhua 321004, China) Abstract As there are some jumping errors from the diﬀerential equation to diﬀerential equation in traditional grey modeling, this paper proposed recursive solution to unbiased grey model based on the unique expression of such models. It deduced prediction formulas of unbiased grey model under diﬀerent initial conditions with the recursive method. On this basis, further study of the optimization problem under two criteria has been done. The results show that both the optimal models under the same criteria have the same simulation and prediction values with higher precision than other methods. Finally, the eﬀectiveness and the practicality of the model are proved by a real project. Keywords unbiased grey model; recursive solution; optimization; forecast 1 #$ Æ, , , , %& , ’% , Æ . 20 80 [1] , “Æ ” , , , " #. ’ GM(1,1) , Æ %& , !" &!&%’ [2] , &()’#%&. , , ##*$%()* [3]. GM(1,1) Æ*,*, . [4] ’ GM(1,1) ’01,2, * + GM(1,1) , ,’"+0 6%0-$+, "0.,, #-7. GM(1,1) Æ*%2, 2/"%/. , :".’ GM(1,1) 1 0 + [5−6], 4 GM(1,1) -SSODMM[7] GM(1,1) -SOB[8]SSODW-GM(1,1)[9] GM (1, 1) -WELCP[10]5 [11] "%1( GM (1, 1) [12−13], ’+0 1,%0-22$2. 364:, 6 ./0: 2009-09-02 34: 957:=; (71071077); 98:5>9 (08AJY024); =?67:5; (200802870020); ;?@? 8=;9:@AB7>:@>9 (NZ2010006) 9;: ; (1973–), >, A, B?;@, 7:@CA, @C

8 ;, : HEDBIÆ?F@H 1533 A0 . H-, .. [11] *5, I + GM (1, 1) 2LB* a, b MJ GM(1,1) 1N . . [1] *0*, GM(1,1) 1N O “”, D# GM(1,1) 1N , O"Æ*$EF"&(. PH, KL BHS2*:. I2, Æ + GM(1,1) A 0 , * +JMM, $ T +JMK , L, M NN&UT1. 2 <=?@ABCEF O X (0) N1, +P1, k i=1 x(0)(i). *, x(1)(k) = X (0) = {x(0)(1), x(0)(2),··· , x(0)(n)}, X (1) = {x(1)(1), x(1)(2),··· , x(1)(n)}, . [5–6] , )" + GM(1,1) 2T : x(1)(k) = e−ax(1)(k − 1) + b a(1 − e−a), k = 2, 3,··· , n (1) *, −a 4W, b ’. MR β1 = e−a, β2 = b a(1 − e−a), (1) AP: (2) , + (U BGM). *RB, ˆβ = (β1, β2)T K % x(1)(k) = β1x(1)(k − 1) + β2, k = 2, 3,··· , n ST6%B. R ˆβ = (β1, β2)T +X, U +%STB *, ⎡ ⎢⎢⎢⎢⎣ Y = ˆβ = (BTB) ⎤ ⎥⎥⎥⎥⎦ , B = −1BTY ⎡ ⎢⎢⎢⎢⎣ x(1)(1) x(1)(2) 1 1 ... x(1)(n − 1) 1 ... x(0)(2) x(0)(3) ... x(0)(n) (3) ⎤ ⎥⎥⎥⎥⎦ . I XY-HS2, T L JMKM* + K . (U BGM − x(1)(1)) M: GH 1 O BY ˆβ 4%I, ˆβ = (BTB)−1BTY , ( ˆx(1)(1) = x(1)(1), U + ⎧⎨ ⎩ βk−1 x(1)(1) + 1 x(1)(1) + kβ2, (β1 − 1)βk−2 β2, 1 1 − βk−1 1 − β1 1 · β2, x(1)(1) + β2βk−2 1 1) ˆx(1)(k) = 2) ˆx(0)(k) = *, k = 2, 3,··· , n. IJ 1) (2) * k = 2, 3,· · ·, n, " x(1)(2) = β1x(1)(1) + β2 x(1)(3) = β1x(1)(2) + β2 ... x(1)(n) = β1x(1)(n − 1) + β2 β1 = 1 β1 = 1 β1 = 1 β1 = 1 , (4) (5) (6)

1534 D C U V E S 31 L ˆx(1)(j − 1) Z+J ˆx(1)(j), j = 2, 3,··· , k, : ˆx(1)(k) = β1ˆx(1)(k − 1) + β2 = β1[β1ˆx(1)(k − 2) + β2] + β2 ... = βk−1 ˆx(1)(1) + (βk−2 1 + βk−3 1 1 + ··· + β1 + 1)β2 ˆx(1)(k) = βk−1 1 ˆx(1)(1) + 1 − βk−1 1 − β1 1 · β2 W β1 = 1 , W β1 = 1 , (7) (8) (9) (10) (11) (12) (14) (15) ˆx(1)(k) = ˆx(1)(1) + kβ2 ( ˆx(1)(1) = x(1)(1), ˆx(1)(k) K : 1 − βk−1 1 − β1 ⎧⎨ ⎩ βk−1 2) ’ ˆx(1)(k) ’+TU6: W β1 = 1 , x(1)(1) + 1 x(1)(1) + kβ2, ˆx(1)(k) = 1 · β2, β1 = 1, β1 = 1, k = 2, 3,··· , n ˆx(0)(k) = ˆx(1)(k) − ˆx(1)(k − 1) ˆx(1)(1) − βk−2 1 ˆx(1)(1) + 1 = βk−1 = (β1 − 1)βk−2 = (β1 − 1)βk−2 1 1 β2 ˆx(1)(1) + 1 − β1 ˆx(1)(1) + β2βk−2 1 · β2 − 1 − βk−2 1 − β1 1 · β2 1 1 − βk−1 1 − β1 · (βk−2 1 − βk−1 1 ) W β1 = 1 , ˆx(0)(k) = ˆx(1)(k) − ˆx(1)(k − 1) = β2 ( ˆx(1)(1) = x(1)(1), +K : x(1)(1) + β2βk−2 , ˆx(0)(k) = (13) GH 2 O BY ˆβ 4%I, ˆβ = (BTB)−1BTY , : “ L6”, ( ˆx(1)(n) = 1 β1 = 1, β1 = 1, k = 2, 3,· · ·, n (β1 − 1)βk−2 β2, 1 x(1)(n), U + (U BGM − x(1)(n)) M: 1) ˆx(1)(k) = 1 x(1)(n) + ⎧⎨ ⎩ βk−n x(1)(n) + (k − n)β2, (β1 − 1)βk−n−1 β2, 1 1 − βk−n 1 − β1 1 · β2, β1 = 1 β1 = 1 x(1)(n) + β2βk−n−1 1 , β1 = 1 β1 = 1 2) ˆx(0)(k) = *, k = 2, 3,··· , n. IJ X 1 , . #NY*JMVA0 W*, +K ZX β1, β2 -( x(1)(1)x(1)(n), [\$ (LKN%0. L, ZT’I %1. 3 <=?@ABL RIN U BGM − x(1)(1)U BGM − x(1)(n) ( ˆx(1)(1) = c1, ˆx(1)(n) = cn. YTN&U R c1, cn.

8 ;, : HEDBIÆ?F@H 1535 O[ c1, cn, 61,ZP]1,0(2Z-%ST#^T%, &UT, : n ⎧⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ c1 = k=2 n k−1 1 1−β1 β2(k−1) [x(1)(k)− 1−β n k=2 x(1)(k) n−1 + ( n 1 k=2 n t=2 min c S2 1 = ·β2]βk−1 1 , β1 = 1, [x(1)(t) − ˆx(1)(t)]2 (16) n k=2 ⎧⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ cn = k−n 1 n [x(1)(k)− 1−β n 1−β1 β2(k−n) k=2 x(1)(k) n−1 + ( n 1 k=2 ·β2]βk−n 1 , β1 = 1, 2 + 1)β2, β1 = 1, 2 + 1)β2, β1 = 1. , ,JMK ˆx(0)(k)|c1 ˆx(0)(k)|cn, U ˆx(0)(k)|c1 = ˆx(0)(k)|cn. [. GH 3 &UT, O U BGM − x(1)(1)U BGM − x(1)(n) ( ˆx(1)(1) = c1, ˆx(1)(n) = cn IJ W β1 = 1 , ˆx(0)(k)|c1 = ˆx(0)(k)|cn PH]. W β1 = 1 , ˆx(0)(k)|c1 = (β1 − 1)βk−2 n 1 · β2] · βk−1 1 + β2 · βk−2 1 x(1)(k) · βk−1 (1 − βk−1 1 ) · βk−1 1 + β2 n k=2 β2(k−1) 1 (β1 − 1) = = = (β1 − 1) (β1 − 1) k=2 n k=2 n k=2 n k=2 1 1 k=2 n [x(1)(k) − 1−βk−1 1−β1 β2(k−1) n n β2(k−1) n 1 + β2 k=2 k=2 1 1 + β2 βk−1 1 x(1)(k) · βk−1 n β2(k−1) 1 k=2 x(1)(k) · βk n n β2k 1 k=2 1 + β2 k=2 n k=2 βk 1 · βk−2 1 · βk−1 1 , · β2] · βk−n 1 1 1 k=2 n [x(1)(k) − 1−βk−n 1−β1 β2(k−n) n (1 − βk−n n β2(k−n) n 1 + β2 k=2 k=2 1 1 1 + β2 βk−n 1 · βk−n−1 1 x(1)(k) · βk−n n β2(k−n) 1 k=2 x(1)(k) · βk n β2k 1 1 + β2 k=2 n k=2 βk 1 · βk−1 1 , x(1)(k) · βk−n n k=2 n k=2 n k=2 (β1 − 1) (β1 − 1) (β1 − 1) = = = ˆx(0)(k)|cn = (β1 − 1)βk−n−1 1 k=2 1 + β2 · βk−n−1 n ) · βk−n 1 + β2 β2(k−n) 1 k=2 · βk−n−1 1 % ˆx(0)(k)|c1 = ˆx(0)(k)|cn. k=2 O[ c1, cn, 61,0(2Z-%ST#^T%, [x(0)(t) − ˆx(0)(t)]2 (17) n t=2 min c S2 0 =

1536 D C U V E S 31 n &UT, : n c1 = k=2 (β1 − 1)2β2(k−2) 1 k=2 n ; cn = k=2 [x(0)(k) − β2βk−2 1 ](β1 − 1)βk−2 1 [x(0)(k) − β2βk−n−1 1 ](β1 − 1)βk−n−1 1 n k=2 (β1 − 1)2β2(k−n−1) 1 . [. GH 4 &UT, O U BGM − x(1)(1)U BGM − x(1)(n) ( ˆx(1)(1) = c1, ˆx(1)(n) = cn , ,JMK ˆx(0)(k)|c1 ˆx(0)(k)|cn, U ˆx(0)(k)|c1 = ˆx(0)(k)|cn. IJ n k=2 [x(0)(k) − β2βk−2 n k=2 1 1 ] · (β1 − 1) · βk−2 (β1 − 1)2 · β2(k−2) n x(0)(k) · βk n · βk−2 1 = k=2 1 1 x(0)(k) · βk−2 1 + β2 · βk−2 1 · βk 1 , β2k 1 k=2 ˆx(0)(k)|c1 = (β1 − 1)βk−2 n 1 (β1 − 1)2 n = k=2 (β1 − 1)2 · β2(k−2) 1 k=2 n k=2 ˆx(0)(k)|cn = (β1 − 1)βk−n−1 n 1 (β1 − 1)2 n % ˆx(0)(k)|c1 = ˆx(0)(k)|cn. k=2 = x(0)(k) · βk−n−1 1 k=2 (β1 − 1)2 · β2(k−n−1) 1 [x(0)(k) − β2βk−n−1 n k=2 1 1 ] · (β1 − 1) · βk−n−1 (β1 − 1)2 · β2(k−2) n x(0)(k) · βk n · βk−n−1 k=2 = 1 1 1 · βk 1 , β2k 1 k=2 + β2 · βk−n−1 1 4 PQS ‘, [NÆ\1, _ +1N M ’‘, T . [13] ‘]][JMM^. ^_ 1997–2004 a_aJaA 1. ’ 1997–2002 a_aJ U BGM − x(1)(1)U BGM − x(1)(n), ,N&UT1, *, ’ 2003 - 2004 a_ aJ%, b%‘‘]. T 1 1997–2004 UVXYZ[^ (_‘) 1997 d:bc 8.21 17.89 10.51 12.72 14.84 2002 1998 1999 2000 2001 9.52 2003 2004 21.22 26.79 1997–2002 a_aJ* U BGM − x(1)(1) , JMK * U BGM − x(1)(n) , JMK &U1 U BGM − x(1)(1) , JMK &U1 U BGM − x(1)(n) , JMK &U1 U BGM − x(1)(1) , JMK ˆx(0)(k) = 9.1619 × 1.1791k−2; ˆx(0)(k) = 20.8828 × 1.1791k−7; ˆx(0)(k) = 9.1679 × 1.1791k−2; ˆx(0)(k) = 20.8853 × 1.1791k−7; ˆx(0)(k) = 9.1682 × 1.1791k−2;

8 ;, : HEDBIÆ?F@H 1537 &U1 U BGM − x(1)(n) , JMK ˆx(0)(k) = 20.8858 × 1.1791k−7. 3 4 , c &U () TN10(c, dL, e JMK , \0(, aA 2. T 2 cdecfghjemn (_‘) ofHÆ U BGM − x(1)(n) U BGM − x(1)(1) ofHÆ ÆdIe ÆdIe ÆdIe ÆdIe 9.1619 10.8019 12.7355 15.0151 17.7028 20.8716 24.6076 9.1669 10.8077 12.7423 15.0232 17.7123 20.8828 24.6207 9.1679 10.8090 12.7438 15.0249 17.7144 20.8853 24.6237 9.1682 10.8091 12.7442 15.0254 17.7149 20.8858 24.6244 de 1998 1999 2000 2001 2002 2003 2004 9.52 10.51 12.72 14.84 17.89 21.22 26.79 *, 1998–2002 0(, 20032004 (. Me\022, aA 3 -A 4. Æj fefkÆ (gh) feikÆ (%) *, Zgh’2 = 1 5 T 3 cdpYZ[^ecfqr U BGM − x(1)(1) U BGM − x(1)(n) ofHÆ ofHÆ ÆdIe ÆdIe ÆdIe ÆdIe 0.2056 0.2068 1.79 6 1.78 ˆx(0)(i) − x(0)(i) ; Zgc’2 = 1 i=2 0.2071 1.79 6 |ˆx(0)(i)−x(0)(i)| . 5 i=2 x(0)(i) 0.2071 1.79 T 4 cdpYZ[^eghspqr (%) U BGM − x(1)(1) U BGM − x(1)(n) ofHÆ ofHÆ 2003 2004 1.6418 8.1462 1.5889 8.0968 1.5775 8.0862 1.5748 8.0837 A 3 - 4 W, \ +2 ]Æ0, 02!K, j ,Æ*)7. GM(1,1) "0+. . [13] , ÆJMM + 0j[P GM(1,1) , b0Mj . [13] 1. j6d, JMMI GM(1,1) * XY. 5 uv Æ JMMNÆ +-\1, ’$ 1A[, &UTN10(c. I* XYH S2, 0‘ GM(1,1) - +1N M"%h. H-, +JMM%K iH0 , [jJMMiHO2 , -’]U]+. dL, #Yk], UkjM" -nl, o$%4Wkh. wxz [1] mn. DIEDo [M]. mA: p:8=rs8, 2002. Deng J L. Grey Prediction and Decision Making[M]. Wuhan: Huazhong University of Science and Technology Press, 2002. [2] ql, un,

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