Ranking efficient DMUs using minimizing distance in DEA

Zia i, Shok ollah; Raissi, Sadigh A icle Ranking e icien DMUs using minimizing dis ance in DEA Jou nal o Indus ial Enginee ing In e na ional P o ided in Coope a ion wi h: Islamic Azad Uni e si y (IAU), Teh an Sugges ed Ci a ion: Zia i, Shok ollah; Raissi, Sadigh (2016) : Ranking e icien DMUs using minimizing dis ance in DEA, Jou nal o Indus ial Enginee ing In e na ional, ISSN 2251-712X, Sp inge , Heidelbe g, Vol. 12, pp. 237-242, h ps://doi.o g/10.1007/s40092-016-0141-2 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/157483 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h p://c ea i ecommons.o g/licenses/by/4.0/ ORIGINAL RESEARCH Ranking e icien DMUs using minimizing dis ance in DEA Shok ollah Zia i 1 •Sadigh Raissi 2 Recei ed: 24 July 2015 / Accep ed: 4 Janua y 2016 / Published online: 22 Janua y 2016 ÓThe Au ho (s) 2016. This a icle is published wi h open access a Sp inge link.com Abs ac In many applica ions, anking o decision making uni s (DMUs) is a p oblema ic echnical ask p o- cedu e o decision make s in da a en elopmen analysis (DEA), especially when he e a e ex emely e icien DMUs. In such cases, many DEA models may usually ge he same e iciency sco e o di e en DMUs. Hence, he e is a g owing in e es in anking echniques ye . The main pu pose o his pape is o o e come he lack o in easi- bili y and unboundedness in some DEA anking me hods. The p oposed me hod is o anking ex eme e icien DMUs in DEA based on exploi ing he lea e-one ou and minimizing dis ance be ween DMU unde e alua ion and i ual DMU. Keywo ds Da a en elopmen analysis (DEA) Ranking  E iciency Ex eme e icien In oduc ion Da a en elopmen analysis (DEA) was ini ia ed by Cha nes e al. (1978) as a me hod o assess ela i e e iciency o homogeneous decision making uni s wi h mul iple inpu s and mul iple ou pu s. Then, Banke e al. (1984) ex ended basic DEA models unde e u ns o scale. As ega ds, he mos models o DEA a e in oduced he mo e han one e icien DMU in e alua ing he ela i e e iciency DMUs, hus he in es iga ing ank o e icien DMUs is an in e - es ing esea ch opic. A DMU is called ex emely e icien i i canno be ep esen ed as a linea combina ion (wi h nonnega i e coe icien s) o he emaining DMUs (Coope e al. 2007). In da a en elopmen analysis, he e a e se e al me hods o anking o he ex eme e icien DMUs, e.g. AP (Ande sen and Pe e sen 1993) me hod, MAJ (Meh a- bian e al. 1999) me hod. Ande sen and Pe e sen p oposed a new p ocedu e o ank e icien DMUs. The AP me hod exhibi s he ank o a gi en DMU by emo ing i om he e e ence se and by compu ing i s supe e iciency sco e. Howe e , he AP model may be in easible in some cases. I is p o ed ha supe e icien DEA models a e in easible (see Th all 1996, Coope e al. 2007, Sei o d and Zhu 1999, Cha nes e al. 1989). Meh abian e al. (Cha nes e al. 1978) sugges ed as MAJ model o comple e anking e icien DMUs, bu hei app oach lacks in easibili y in some cases, oo. To o e come he d awbacks o he AP (Ande sen and Pe e sen 1993) and MAJ (Meh abian e al. 1999) models, Jahanshahloo e al. (2004a) p esen ed a me hod o ank he ex emely e icien DMUs in DEA models wi h cons an and a iable e u ns o scale using L 1 -no m. The p oposed model is a nonlinea p og amming o m which has he compu a ional complexi y in sol ing. A complex ea men was applied in Jahanshahloo e al. (2004a) o con e he nonlinea model in o a linea one which p o ides an app oxima ely op imal solu ion. Wu and Yan (2010) ha e also used an e ec i e ans o ma ion o con e he non- linea model in Jahanshahloo e al. (2004a) in o a linea model. Also Jahanshahloo e al. (2004b) ha e applied g adien line o anking e icien uni s. Rezai Bal e al. (2012) applied Tchebyche no m (L 1 -no m) in oduced in (B iec 1998; Ta a es e al. 2001) o comple e anking &Shok ollah Zia i [email p o ec ed] Sadigh Raissi [email p o ec ed] 1 Depa men o Ma hema ics, Fi oozkooh B anch, Islamic Azad Uni e si y, Fi oozkooh, I an 2 School o Indus ial Enginee ing, Islamic Azad Uni e si y, Sou h Teh an B anch, Teh an, I an 123 J Ind Eng In (2016) 12:237–242 DOI 10.1007/s40092-016-0141-2 e icien uni s. Ami eimoo i e al. (2005) in oduced a me hod o anking o ex eme e icien DMUs, based on dis ance. Hashimo o (1999) p oposed a supe e iciency DEA model wi h assu ance egion in o de o ank he DMUs comple ely. To gesen e al. (1996) sugges ed a me hod o anking e icien uni s, by hei impo ance as benchma ks o he ine icien uni s. Sex on e al. (1986) in es iga ed a anking me hod o DMUs based on a c oss- e iciency a io ma ix. The c oss-e iciency anking me hod compu es he e iciency sco e o each DMU ha de e mines a se o op imal weigh s using linea p og ams co esponding o each DMU. Then by aking he a e age o sco es o gi en DMU is ob ained he ank o ha DMU. Liu and Peng (2008) de e mined one common se o weigh s o anking e icien DMUs, ha DMUs a e anked acco ding o he e iciency sco e weigh ed by he common se o weigh s. Bal e al. (2008) sugges ed a DEA model o anking o DMUs based on de ining he coe icien o a ia ion o inpu –ou pu weigh s. Khodabakhshi and A ya ash (2012) p oposed a me hod o ank he e icien DMUs. Acco ding o hei me hod, i s he minimum and maximum e iciency alues o each DMU a e compu ed unde he assump ion ha he sum o e iciency alues o all DMUs is equal o uni y. Then, he ank o each DMU is de e mined in p opo ion o a combina ion o i s minimum and maximum e iciency alues. She y and Pakkala (2010) sugges ed a me hod o anking e icien uni s, which is c ea ed he a e age o he co esponding inpu s and ou pu s o all DMUs. Ea ly, Jahanshahloo and Fi oozi Shahmi zadi (2013) modi ied he model which was p oposed by Bal e al. (2008). They in oduced wo new models o anking e icien DMUs based on L 1 -no m and using mean o inpu –ou pu weigh s. Fo ou new me hod i does no need any addi ional cons ain s. In his pape , we sugges a new me hod o anking ex eme e icien DMUs. The es o he pape is o ganized as ollows. In ‘‘DEA models and anking models e iew’’, we e iew he concep o DEA amewo k. We e iew some anking me hods in ‘‘The p oposed anking model o e icien DMUs’’, ‘‘Ex ension o a iable e u ns o scale’’ p oposes he new model o anking e icien uni s. ‘‘Il- lus a ed examples’’ includes some nume ical examples. The las sec ion concludes he s udy. DEA model and anking model e iew DEA model e iew DEA is a me hodology o assessing he ela i e e iciency o decision making uni s (DMUs) whe e each DMU has mul iple inpu s used o secu e mul iple ou pu s. I is assumed in DEA ha he e a e nDMUs and o each DMU j ðj¼1;...;nÞis conside ed a column ec o o inpu s ðXjÞ o p oduce a column ec o o ou pu s ðYjÞ, whe e Xj¼ðx1j;x2j;...;xmjÞTand Yj¼ðy1j;y2j;...;ysjÞT. He e, he supe sc ip ðTÞindica es a ec o anspose. I is also assumed ha Xj0;Yj0;Xj6¼ 0;and Yj6¼ 0 o e e y j¼1;...;n. The ollowing inpu -o ien ed CCR model [see (Coope e al. 2007)] in he en elopmen o m wi h cons an Re u ns o Scale measu es he le el o DEA e iciency ðhÞ o he k h DMU ðXk;YkÞ: h¼min h s. . X n j¼1 kjxij hxik;i¼1;...;m X n j¼1 kjy j y k; ¼1;...;s kj0;j¼1;...;n ð1Þ He e, k¼ðk1;...;knÞTis a column ec o o unknown a iables used o componen s o he inpu and ou pu ec o s by a combina ion. h ep esen s he e iciency sco e o DMU k in (1), whe e he supe sc ip (*) indica es op imali y. DMU k is ela i ely e icien i and only i on op imali y, he objec i e o (1) equals o one and all he slacks a e ze o. Simila ly, he ou pu -o ien ed CCR model, co espond- ing o (1), is o mula ed as ollows: /¼max / s: :P n j¼1 kjxij xik;i¼1;...;m P n j¼1 kjy j /y k; ¼1;...;s kj0;j¼1;...;n ð2Þ He e, 1=/in ends he DEA e iciency sco e in he ou pu -o ien ed model. Also, he ollowing inpu -o ien ed BCC model [see Banke e al. (1984)] in he en elopmen o m wi h a i- able Re u ns o Scale measu es he le el o DEA e iciency ðhÞo he k h DMU ðXk;YkÞ: h¼min h s. . P n j¼1 kjxij hxik;i¼1;...;m P n j¼1 kjy j y k; ¼1;...;s P n j¼1 kj¼1; kj0;j¼1;...;n ð3Þ 238 J Ind Eng In (2016) 12:237–242 123 DMU k is ela i e e icien i and only i on op imali y, he objec i e o (3) equals o one and all he slacks a e ze o. Simila ly, he ou pu -o ien ed BCC model, co espond- ing o (3) which ob ains om (2) by adding cons ain , X n j¼1 kj¼1: Mo eo e , he ollowing addi i e model is based on inpu and ou pu slacks which accoun s he possible inpu dec eases as well as ou pu inc eases simul aneously. max X n j¼1 s iþX n j¼1 sþ s: :P n j¼1 kjxij þs ixik;i¼1;...;m P n j¼1 kjy j sþ /y k; ¼1;...;s kj;s i;sþ ;0; ð4Þ DMU k is ela i e e icien i and only i on op imali y, he objec i e o (4) equals o ze o. Ranking models In his subsec ion we e iew he some anking models in da a en elopmen analysis. The i s anking model p o- posed by Ande son and Pe e son (1993) which is he sup- pe e iciency model. In he AP model DMU unde e alua ion is excluded om e e ence se and by using o he uni s, he ank o gi en DMU is ob ained. The AP model using he CRS supe -e iciency model is as ollows: AP:min h s: :P n j¼1j6¼k kjxij hxik;i¼1;...;m P n j¼1j6¼k kjy j y k; ¼1;...;s kj0;j¼1;...;n;j6¼ k ð5Þ The main d awbacks o his model a e in easibili y and ins abili y o some DMUs. I is said ha a model is s able i a DMU unde e alua ion is e icien , i is emains e icien a e pe u ba ion on da a. The second anking model unde in es iga ion p oposed by Meh abian e al. (1999) o sol e in easibili y o AP models in some cases. The ollowing model is MAJ model: MAJ: min 1 þw s. P n j¼1j6¼k kjxij xik þw;i¼1;...;m P n j¼1j6¼k kjy j y k; ¼1;...;s kj0;j¼1;...;n;j6¼ k ð6Þ The hi d anking model p oposed by Jahanshahloo e al. (2004a), ha hei p oposed me hod o ank he ex emely e icien DMUs in DEA models wi h cons an and a iable Re u ns o Scale using he omi ed DMU unde e alua ion om p oduc ion possibili y se and applying L 1 -no m. I is shown ha he p oposed me hod is able o o e come he exis ing di icul ies in he AP (Ande sen and Pe e sen 1993) and MAJ (Meh abian e al. 1999) models. On he o he hand, he p oposed model is he o m o nonlinea p og amming which is di icul o be sol ed. The model o Jahanshahloo e al. (2004a) is p esen ed as ollows: L1no m:min X m i¼1 xixik jj þX s ¼1 y y k jj s: :P n j¼1j6¼k kjxij xi;i¼1;...;m P n j¼1j6¼k kjy j y ; ¼1;...;s xi0;y 0i¼1;...;m; ¼1;...;s kj0;j¼1;...;n;j6¼ k ð7Þ The ou h anking model p oposed by Rezai Bal e al. (2012) which applies o anking ex eme e icien uni s using he lea e-one-ou idea and L1-no m. The p oposed model is always easible and so, i is able o emo e he exis ing di icul ies in some me hods, such as Ande sen and Pe e sen (1993). The model o Rezai Bal e al. (2012)is o mula ed as ollows: L1no m :min mk s: :mkP n j¼1j6¼k kjxij xik;i¼1;...;m mky k P n j¼1j6¼k kjy j; ¼1;...;s kj0;j¼1;...;n;j6¼ k mk0 ð8Þ J Ind Eng In (2016) 12:237–242 239 123 The p oposed anking model o e icien DMUs In his sec ion, we suppose ha he DMUk is ex eme e icien . By excluding he DMUk om he CCR p oduc- ion possibly se , i is ob ained a new e iciency on ie . In o de o gain he anking sco e o DMUk by exploi ing he new e iciency on ie , we sugges a new model by by using he lea e-one ou idea and minimizing dis ance be ween DMU unde e alua ion and i ual DMU. The p oposed model is as ollows: min P m i¼1 aiþP s ¼1 b s: :P n j¼1j6¼k kjxij xik ai;i¼1;...;m P n j¼1j6¼k kjy j y k þb ; ¼1;...;s kj0;j¼1;...;n;j6¼ k; ai0;b 0i¼1;...;m ¼1;...;s; ð9Þ whe e a¼ða1;...;amÞ,b¼ðb1;...;bsÞand k¼ ðk1;...;kk1;kkþ1;...;knÞa e he a iables o he model (9). Theo em 1 The model (9)is easible and bounded. P oo Fo p6¼ kwe se kp¼1;kj¼0;j¼1;...;n;j6¼ k;p;ai¼min xik xipg;i¼1;...;m;b ¼min y p y kg; ¼1;...;s: Ob iously, i can be seen ha ðk;a;bÞacco ding o abo e selec ion is a easible solu ion o he model (9). Mo eo e , he objec i e unc ion o model (9) is bounded below ze o, because he a iables o model a e nonnega- i e. Also, he a ge unc ion is ze o when ai¼0 and b ¼ 0 o all i; h. Ex ension o a iable e u ns o scale In his sec ion, he p oposed model in p e ious sec ion is ex ended o a iable Re u ns o Scale model. Fo his pu pose, he model (9) is e o mula ed by adjoining he ollowing con exi y cons ain o he model: X j¼1 n j6¼k kj¼1;kj0: So, in o de o ge he anking sco e unde a iable e u ns o Scale assump ion is sol ed he ollowing model: min P m i¼1 aiþP s ¼1 b s: :P n j¼1j6¼k kjxij xik ai;i¼1;...;m P n j¼1j6¼k kjy j y k þb ; ¼1;...;s P n j¼1j6¼k kj¼1; kj0;j¼1;...;n;j6¼ k; ai0;b 0i¼1;...;m; ¼1;...;s; ð10Þ Theo em 2 The model (10)is easible and bounded. P oo The p oo o his heo em is simila o he p oo o Theo em 1. Table 1 Inpu and ou pu da a o Example 1 DMU Inpu 1 Inpu 2 Ou pu 1 Ou pu 2 1 81 87.6 5191 205 2 85 12.8 3629 0 3 56.7 55.2 3302 0 4 91 78.8 3379 8 5 216 72 5368 639 6 58 25.6 1674 0 7 112.2 8.8 2350 0 8 293.2 52 6315 414 9 186.6 0 2865 0 10 143.4 105.2 7689 66 11 108.7 127 2165 266 12 105.7 134.4 3963 315 13 235 236.8 6643 236 14 146.3 124 4611 128 15 57 203 4869 540 16 118.7 48.2 3313 16 17 58 47.4 1853 230 18 14 650.8 4578 217 19 0 91.3 0 508 Table 2 Resul s o anking by di e en models DMU 12591519 AP anking esul s 4 1 3 – 2 – MAJ anking esul s 5 3 2 6 4 1 L1-no m anking esul s 4 3 2 6 5 1 L1-no m anking esul s 5 2 3 6 4 1 P oposed model anking esul s 53426 1 240 J Ind Eng In (2016) 12:237–242 123 Illus a ed examples In his sec ion, we employ he abo e DEA model (6) and (7) on he wo da a se s which hey a e in oduced he e, wi h he assump ion o cons an e u ns o scale. Example 1 As can be seen om Table 1, he da a se consis s o 19 DMUs wi h 2 inpu s and 2 ou pu s. The da a o iginally a e used by Rezai Bal e al. (2012). Table 2 epo s he esul s o anking o 6 ex emely e icien DMUs ðD1;D2;D5;D9;D15;D19Þin model (7) wi h con- s an Re u ns o Scale and he p oposed me hod is com- pa ed wi h Ap, MAJ, L1and L1. The esul s imply ha he model p oposed in his pape p o ides a easy ool o anking ex emely e icien DMUs. The alue o inpu s and ou pu s. Example 2 (Empi ical example). We employ DEA model (10) on he empi ical example used in Zhu (1998), wi h he assump ion o a iable Re u ns o Scale. The da a se in Table 3p o ides 13 open coas al Chinese ci ies and i e Chinese special economic zones in 1989. Two inpu s and h ee ou pu s we e chosen o cha ac e ize he echnology o hose ci ies/zones. Two inpu s include In es men in ixed asse s by s a e-owned en e p ises, Fo eign unds ac ually used. Th ee ou pu s include To al indus ial ou pu alue, To al alue o e ail sales and Handling capaci y o coas al po s. Table 4 epo s he esul s o anking o 10 ex e- mely e icien DMUs ðD1;D2;D5;D6;D7;D9;D10;D11; D13;D16Þin model (10) wi h a iable e u ns o scale and he p oposed me hod a e compa ed wi h o he me hods. Conclusion Many DEA esea ches a e p oposed on anking o e icien decision making uni s, bu hey ha e a p oblem, e.g. he AP model may be in easible in some cases. In he p esen pape , we p oposed a model o anking ex eme e icien DMUs in DEA by exploi ing he lea e-one ou and mini- mizing dis ance be ween DMU unde e alua ion and i - ual DMU. The p oposed model is linea o m and always easible and bounded. The e o e, i is able o ank all ex eme e icien DMUs in he DEA me hods wi h con- s ain and a iable Re u ns o Scale and so, elimina e he Table 3 The alue o inpu s and ou pu s DMU# ci ies/zones Inpu 1 Inpu 2 Ou pu 1 Ou pu 2 Ou pu 3 Dalian 2874.8 16,738 160.89 80,800 5092 Qinhuangdao 946.3 691 21.14 18,172 6563 Tianjin 6854.0 43,024 375.25 44,530 2437 Qingdao 2305.1 10,815 176.68 70,318 3145 Yan ai 1010.3 2099 102.12 55,419 1225 Weihai 282.3 757 59.17 27,422 246 Shanghai 17,478.6 116,900 1029.09 351,390 14,604 Lianyungang 661.8 2024 30.07 23,550 1126 Ningbo 1544.2 3218 160.58 59,406 2230 Wenzhou 428.4 574 53.69 47,504 430 Guangzhou 6228.1 29,842 258.09 151,356 4649 Zhanjiang 697.7 3394 38.02 45,336 1555 Beihai 106.4 367 7.07 8236 121 Shenzhen 4539.3 45,809 116.46 56,135 956 Zhuhai 957.8 16,947 29.20 17,554 231 Shan ou 1209.2 15,741 65.36 62,341 618 Xiamen 972.4 23,822 54.52 25,203 513 Hainan 2192.0 10,943 25.24 40,267 895 Table 4 Resul s o se e al models anking DMU 12567910111316 AP 9184632 7 5 10 MAJ 183497106 2 5 L1-no m 483691107 2 5 L1-no m 384691107 2 5 P oposed model 173289105 6 4 J Ind Eng In (2016) 12:237–242 241 123 exis ing di icul ies in some me hods. In addi ion, i can be easily used when he numbe o inpu s and ou pu s is much la ge han he numbe o DMUs. Illus a i e examples a e included o show good anking esul s by he p oposed me hod. Open Access This a icle is dis ibu ed unde he e ms o he C ea i e Commons A ibu ion 4.0 In e na ional License (h p://c ea i ecommons.o g/licenses/by/4.0/), which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided you gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons license, and indica e i changes we e made. Re e ences Ami eimoo i A, Jahanshahloo GR, Ko d os ami S (2005) Ranking o decision making uni s in da a en elopmen analysis: a dis ance- based app oach. Appl Ma h Compu 171:122–135 Ande sen P, Pe e sen NC (1993) A p ocedu e o anking e icien uni s in da a en elopmen analysis. Manag Sci 39:1261–1264 Bal H, Ho kcu H, Celebioglu S (2008) A new me hod based on he dispe sion o weigh s in da a en elopmen analysis. 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