½ 26 ½ 2 2011 4 Journal of Systems Engineering Vol. 26 No. 2 Apr. 2011 Ô DEA-AHP º žâ, Ê, ( ï Ä Ò, ï 400044) ý : ǑÅ Ã Á ÃǑ, ÂÔ (DMU), Ä DEA AHP º, ÅÍÅ ½ Ë, Í»ÆÝ Ñ Á Ñ, Ò Í Ô DEA- AHP º, û Á â ½ ñ Ǒ ½Å. ³ : í (DEA); ß( AHP ); ³ Å; ñ : N94; C931 ÞÁ : A : 1000 5781(2011)02 0262 07 Interactive DEA- AHP model and its application XU Guang-ye, DAN Bin, XIAO Jian (College of Economics and Business Administration, Chongqing University, Chongqing 400044, China) Abstract: In order to study the influence of every index of multiple inputs and outputs problems on efficiency and attain the goal of increasing the relative effectiveness of decision making units(dmus), based on analyzing the advantage and disadvantage of data envelopment analysis(dea) and analytical hierarchical process( AHP ) models, this paper presents an interactive DEA- AHP model, which can completely rank for all DMUs. Additionally, the new model can relatively comprehensively reflect the weights among all DMUs and among index s attributes, and rationally discriminate the power of every index for efficiency. In this paper, an example about the evaluation of 6 Chinese commercial banks was used to explain the application of the model. Key words: efficiency evaluate 1 data envelopment analysis (DEA); analytical hierarchical process ( AHP ); pairwise comparison matrix; í (data envelopment analysis, DEA)àß ß(analytical hierarchical process, ß AHP ), Ö àß Ô, ÁË [1 4]. DEA ºǑ» (decision making unit, DMU) Ǒ :, DMU ; üâ Í ç Ô, Ǒ Í ç DMU ; Í ù Á ÃǑ, Â Ô DMU DEAàß Åèù ë [5]. î AHP ºÄÙí Ù, Çð ðè Æî, ½ ñ Ë Ã µ «, ð AHP àß è Ú [6]., Wang [1] ³»ÒÅDEA º AHP º Ô èå. Sinuany-stern [2] Äê DEAàß Åè DEA- AHP º SHMàß, Å AHPàß þú : 2008 12 27; ú : 2009 04 14. á : Á á (70671011); ÅÔ â á (CDJXS11022202)
½ 2 žâ : Ô DEA-AHP º 263. ð SHMàß±Ú û AHPàß : ñ à ³ Å. Ã, ÅÍÅ ½ Ë, Í»ÆÝ Ñ Á Ñ, Ò Í Ô DEA- AHP º., (ù ) Á, è Đ è (ù ) Á, (ù ) Á DEA º.,  ñ Ñ Ñ ³ Å. AHPàß Ç ñ, Ð ÍÅ ½ Ë î, Ñ Ñ. è àß SHMàß ½³, Ù Å àß ½. Á 6 â ½ ñ Ǒ ½Å. 2 Ô DEA- AHP º 2.1 DEA º n DMU j (j = 1,2,,n) m X j = (x 1j,x 2j,x mj ) T 0, s Y j = (y 1j,y 2j,y sj ) T 0, DMU 0 ÈǑ ñ.  CCR-I º Ý º [7], Đ ½k Á, Áî, X j = (x kj ) T (k I), I = {1,2,,m}Ǒ Á. Ó CCR I k º ( )] Min [θ 0k ε s k + s s + r θk 0,s k,s+ r r=1 s.t. λ j x kj + s k = θ0 k x k0,k I λ j y rj s + r = y r0, r = 1 λ j,s k,s+ r 0,j = 1,2,,n,  CCR-O º Ý º [7], Đ ½t Á, Áî, Y j = (y tj ) T (t O), O = {1,2,,s}Ǒ Á. Ó CCR O t º Max ϕ 0 t,s i0,s+ t0 s.t. [ ϕ 0 t + ε ( m s i0 + s + t0 i=1 λ j y tj s + t0 = ϕ 0 t y t0,t O λ j,s i0,s + t0 0,j = 1,2,,n )] λ j x ij + s i0 = x i0, i = 1,2,,m ê â εðè æ Î, ÒµǑðè Ù û Ù Ǒε = 10 5. 2.2 Ñ ³ Å Ä º(1)(2), Ñ ³ Å, Æù. 1 ½ k Á ³ Å. a k pq Ä ½ k Á, p q ( p,q {1,2,,n}), Ä Ǒ a k pq = θ p k /θq k, ak qp = 1/a k pq, a k pp = 1 (3) a k pq > 1, Ä p ³ q Ô, à ; ak pq = 1, Ä p (1) (2)
264 ½ 26 q. à ³ Å 1 a k 12 a k 1n a A k = k 21 1 ak 2n a k n1 a k n2 1 2 ½ t Á ³ Å. b k pq Ä ½ t Á, p q, Ä Ǒ b t pq = 1/ϕp t 1/ϕ q t = ϕ q t/ϕ p t,b t qp = 1/b t pq,b t pp = 1 (5) b t pq > 1, Ä p ³ q Ô, à ; b t pq = 1, Ä p q. à ³ Å 1 b t 12 b t 1n b B t = t 21 1 bt 2n b t n1 b t n2 1 3 Á Ñ ³ Å. k 1,k 2 I, ýk 1 Á ½ k 2 Á Ǒ c k1k 2 = 1 ( ) 1 1 n θ j k 1 θ j k n 2, c k2k 1 = 1/c k1k 2, c k1k 1 = 1 (7) c k1k 2 > 1, Ä ½ k 1 Á³½k 2 Á, ýk 2 Á³½k 1 Á ; c k1k 2 = 1, Ä. à ³ Å 1 c 12 c 1m c C = 21 1 c 2m c m1 c m2 1 4 Á Ñ ³ Å. t 1,t 2 O, ý t 1 Á ½ t 2 Á Ǒ d t1t 2 = 1 n 1 φ j t 1 ( 1 n 1 φ j t 2 (4) (6) (8) ) 1,d t2t1 = 1/d t1t2,d t1t1 = 1 (9) d t1t 2 > 1, Ä ½ t 1 Á³½ t 2 Á, ý t 2 Á³½ t 1 Á ; d t1t 2 = 1,Ä. à ³ Å Ǒ 1 d 12 d 1s d D = 21 1 d 2s d s1 d s2 1 5 Ñ ³ Å. e io Ä Á Á ( e oi Ä Á Á ), Ä e io = 1 mn m k=1 θ j k /( 1 sn s t=1 φ j t (10) 1 ), e oi = 1/e io (11)
½ 2 žâ : Ô DEA-AHP º 265 e io > 1, Ä Á³ Á, à Á³ Á ; e io = 1, Ä. à ³ Å E = [ 1 e io e oi 1 2.3 AHPàß Ë Ä Ñ ³ Å, AHPàß, Æ Æå. Ù Ô á ðµ, å ÁÇÞàß:, ¾ Å ǑÁÇÞ ½. à 1) ñ Å ÇǑ ω i = [ ω A 1,ω A 2,,ω A m], ω o = [ ω B 1,ω B 2,,ω B s ω A k,k = 1,2,,m;ωB t,t = 1,2,,s ÇǑ ³ Å Ak, B t. 2) Ñ Å Ñ Å ÇǑ W i = [ω i 1,ωi 2,,ωi m ]T, W o = [ω o 1,ωo 2,,ωo s ]T (14) ω i k,k = 1,2,,m;ω o t,t = 1,2,,s ÇǑ ³ Å C ½ k ½ ³ Å D ½t ½. 3) (13) (14) Ç ñ 4 ÅǑ ω 1, ω 2 ÇǑ ³ Å E ½è, ½. 5) (15) (16) Ç ñ ] ] (12) (13) ω io = [ω i W i,ω o W o ] (15) W io = [ω 1,ω 2 ] T (16) ω = [ω io W io, ω io W io ] (17) (17) Ç ÂÍ Ð ñ ³., Ä (13)æ (16), ÙÝ Á Ñ. Ä Å, Ëìõ ³ Òëë, èí õ ÂÔ. 3 SHMàß ³ Sinuany-stern [2]  ÅÍÅ Ë è DEA- AHP º SHMàß, ¾, Ä àß SHMàß ³, Ù àß ½. Ǒ àß SHMàß ð Ä DEA º, ¾ AHP º ½, àß Ò ð ³ SHMàß : 1) àß ÚCharnes [7], SHMàß Í Ú. ÙDEA û ð: Ǒ Á Ý Paretoµ. è, Á ( m + s ), Paretoµ á Ç, DEA Ç. ñ î, è Charnes Ǒ n max {mn,2(m + s)} (18) SHMàß, DEA º, Á Á,
266 ½ 26 ì», Í Ú (18). àß ÍÅ Ý, m + s Ǒ. à ñ î Ú (18), àßèù Ú. ðè SHMà ß µ, Æð ³ Å Ý ½, ç Í ½ ñ, àß èù íå ³ Å Ý ½ ì» Ý. 2) àß³ SHMàß DEA º ßÞ, ðµ. SHMàß DEA º, Ù è ºÄ ½ ë, àß ³Ù DEA º, ±³ DEA º ßÞ., Ð º Ç Û, ñ î, Â Ú (18), à àßç Ù SHMàß. 3) àß ÍÅ ½ Ë, ÍÅ Â Ñ, SHMàß± ÍÆÝ. SHMàß Å Á ³ Å, àð ñ. àß Å ³ Å, û Å AHPàß, Å ñ, Å Á Ñ. ³, à ß Ǒ  Ҳ, ³ SHMàß ¾. 4) àß³ SHMàß ½ Ë Ò. Ò ð àß DEA º Đ Å îđ Å º, SHMàß DEA ºĐ ð Đ è º. Ò ÆÝ àß ³ Å. ǑÅ è àß³ SHMàß ½ Ë Ò, Ô Ä SHMàß 1 ³, Æ Þ[2], Ôà ÂÄ1. Ä 1 º SHMàß ³ Table 1 Comparison between new model and SHM DMU CCR SHM Ë º Ë A 1.000 0 0.153 002 3 0.220 4 2 B 1.000 0 0.209 948 1 0.223 1 1 C 1.000 0 0.162 917 2 0.185 8 4 D 1.000 0 0.141 403 5 0.188 4 3 E 0.510 2 0.142 773 4 0.084 4 5 F 0.403 2 0.098 400 6 0.050 0 6 G 0.333 3 0.076 374 7 0.044 1 7 H 0.028 1 0.015 178 8 0.003 7 8 Ä 1 ÙÝ D CCR E CCR, SHMàß Â±ð E D., Ð A C Û Å½è ß, г ³ ðç Û, ê ½è Ö ³, à A Ìð Ù C. Ù Å Ã, àß Â ³ Ò, Ǒ  Ò. 4 º ½ ǑÝã à ±, ³ Á Ä Ë,, ½ ½à ñ ë ûþ. ÁË DEA ñ â ½, Ñ [8 10]. Þ ÁüÂ, Á â ½ þ ìõ, Ô Çü Ó â ä Ǒ, âþ ÇĐ Çä ûú Ǒ. Ä2ðÐ Á6 â ½ þ 2007 ³. CCR º 6 â ½ ½ñ, ÙÝ ÅÍ ½Ǒ ßǑ, û Í ². Â Ô DEA- AHP º ½ ñ ± ², Ç ñ Â Ä 3. ÐÄ3 è½ Óñ Û 2007¾ü ½
½ 2 žâ : Ô DEA-AHP º 267, ½ ½, ½, Í ½ «, ½ «. ð ð Å ñ è Ë, û Í ². ǑÅ èæý º, Á Ä Ç Â ½. Ä 2 Á6 â ½2007 Table 2 The data of Chinese commercial bank for 6 commercial banks in 2007 Á ½ ¾ü ½ ½ ½ Í ½ ½ Ó 1 282.00 1 518.00 1 827.00 1 274.17 1 361.49 1 574.73 â 37.50 47.40 65.40 35.43 48.34 58.99 64.10 68.20 70.30 61.54 64.46 86.27 ä 358.00 374.00 405.05 321.50 359.60 382.63 âþ 35.70 44.20 61.50 35.49 45.66 57.92 ÇĐ 158.00 243.00 363.00 105.33 150.14 124.08 Çä 63.00 178.00 249.00 103.74 104.26 89.88 ûú 8.10 8.60 8.30 8.26 7.32 9.16 0.57 2.80 6.30 4.30 11.28 1.16 Ä 3 Ù º ñ  Table 3 The results of efficiency assessment base on new model Á ½ ¾ü ½ ½ ½ Í ½ ½ Ó (0.254 5) 0.167 7 0.167 7 0.167 7 0.167 7 0.161 7 0.161 7 â (0.253 0) 0.168 7 0.168 7 0.168 7 0.168 7 0.160 1 0.165 1 (0.245 0) 0.144 7 0.179 1 0.179 1 0.179 1 0.164 2 0.153 7 ä (0.247 5) 0.159 5 0.172 4 0.172 4 0.172 4 0.1507 0.172 4 âþ (0.266 8) 0.162 1 0.162 2 0.169 9 0.169 9 0.165 9 0.169 9 ÇĐ (0.189 5) 0.182 1 0.221 5 0.239 2 0.128 4 0.133 9 0.094 9 Çä (0.189 9) 0.105 5 0.236 2 0.238 8 0.184 2 0.135 3 0.100 0 ûú (0.252 9) 0.174 8 0.168 6 0.158 3 0.179 3 0.151 8 0.167 2 (0.100 9) 0.110 8 0.226 6 0.150 1 0.449 3 0.044 1 0.019 1 Óñ 0.156 9 0.183 4 0.181 3 0.183 2 0.149 0 0.146 2, Ä 3 ½è ÙÝ, Ñ (Ó (0.254 5)) ( (0.245 0))³Ǒ1.0388, Ä Ñ ; Ñ ( âþ (0.266 8)) ( (0.100 9))³Ǒ2.6442, Ä Ñ Đ «., Ð Ä3 ½ û. ³ ÐÄ ½2, 3½ÙÝÓ â ½ Ó â ; н4, 5½ ½ ä ; ½6½ Đ ½ âþ ; н7, 8½ Ùݾü ½ ½ ÇĐ ä, Í ½ ½ ; н9, 10½ ½ ûú, Í ½ ³., ÐÄ3 6 û ½1 ½ Ñ ÃǑ ½. Ä Â, Ô DEA AHP º Ý ½ Ë î, ÍÅ ½ Ë ñ î, Ǒ üâ Ñ Â Ò í. ð º Ñ û, ð ½ ñ. 5 Ë ñ Ä, Ù DEA ºàÍ Ǒ, Ù Í Ë Ǒ Í Ã. î, DEA ºǑ Í Á à Ǒ ÂÔ. Ã, Ä DEA º AHP º,
268 ½ 26 ÅÍÅ ½ Ë, Í»ÆÝ Ñ Á Ñ, Ò Í Ô DEA- AHP º. Ä DEA- AHP º SHMàß ½³ Á â ½ ñ Ǒ ½, Ù Å àß ½. Đ Þ: [1] Wang Y M, Liu J, Elhag T M S. An integrated AHP /DEA methodology for bridge risk assessment[j]. Computers & Industrial Engineering, 2008, 54(3): 513 525. [2] Sinuany S Z, Mehrez A, Hadad Y. An AHP /DEA methodology for ranking decision making units[j]. International Transactions in Operational Research, 2000, 7(2): 109 12 4. [3] ÝÙï, Û. Ù AHP DEA Ëß[J]., 2004, 19(5): 543-547. Yan Huahui, Cui Jinchuan. Multifactor sequencing method based on AHP and DEA[J]. Journal of Systems Engineering, 2004, 19(5): 543 547. ( in Chinese) [4] èù, É, ÝÍ. á AHP Á DEA º[J]., 1999, 14(4): 330-333. Wu Yuhua, Zeng Xiangyun, Song Jiwang. A DEA model with AHP restraint cone[j]. Journal of Systems Engineering, 1999, 14(4): 330 333. ( in Chinese) [5] Cooper W W, Seiford L M, Zhu J. Handbook on Data Envelopment Analysis[M]. Boston: Kluwer Academic Publishers, 2004. [6] William H. Integrated analytic hierarchy process and its applications : A literature review[j]. European Journal of Operational Research, 2008, 186(1): 211 228. [7] ï. í [M]. :, 2004. Wei Quanling. Data Envelopment Analysis[M ]. Beijing: Science Press, 2004. ( in Chinese) [8] ðá,,. ÙDEAàß Á â ½Ò [J]. Á Ò, 2006, 14(5): 52 61. Chi Guotai, Yang De, Wu Shanshan. The research on overall efficiency of Chinese commercial banksbased on DEA approach[j]. Chinese Journal of Management Science, 2006, 14(5): 52 61. ( in Chinese) [9] LinT T, Lee C C, Chiu T F. Application of DEA in analyzing a bank s operating performance[j]. Expert Systems with Applications, 2009, 36(5): 8883 8891. [10] Kao C, Liu S T. Stochastic data envelopment analysis in measuring the efficiency of Taiwan commercial banks[j]. European Journal of Operational Research, 2009, 196(1): 312 322. ß¼: žâ (1983 ), Á, Ë, µ, àå: í Ò; Email: xuguangye520@163.com; Ê (1966 ), Á, ï,,, µ ã, àå: Ò; (1975 ), Á, ï, µ, àå: Ò.