Research Article Data Envelopment Analysis with Uncertain Inputs and Outputs

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1 Joural of Applied Mathematics, Article ID , 7 pages Research Article Data Evelopmet Aalysis with Ucertai Iputs ad Outputs Meili We, 1,2 Liha Guo, 1,2 Rui Kag, 1,2 ad Yi Yag 1,2 1 Sciece ad Techology o Reliability ad Evirometal Egieerig Laboratory, Beijig , Chia 2 School of Reliability ad Systems Egieerig, Beihag Uiversity, Beijig , Chia Correspodece should be addressed to Liha Guo; lihaguo@buaa.edu.c Received 27 Jue 2014; Accepted 1 July 2014; Published August 2014 Academic Editor: Xiag Li Copyright 2014 Meili We et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Data evelopmet aalysis (DEA), as a useful maagemet ad decisio tool, has bee widely used sice it was first iveted by Chares et al. i O the oe had, the DEA models eed accurate iputs ad outputs data. O the other had, i may situatios, iputs ad outputs are volatile ad complex so that they are difficult to measure i a accurate way. The coflict leads to the researches of ucertai DEA models. This paper will cosider DEA i ucertai eviromet, thus producig a ew model based o ucertai measure. Due to the complexity of the ew ucertai DEA model, a equivalet determiistic model is preseted. Fially, a umerical example is preseted to illustrate the effectiveess of the ucertai DEA model. 1. Itroductio Data evelopmet aalysis is a mathematical programmig techique that measures the relative efficiecy of decisio makig uits with multiple iputs ad outputs, which was iitialized by Chares et al. [1]. This was followed by variety of theory research work, icludig Baker et al. [2], Chares et al. [3], Peterse [4], ad Toe []. More DEA papers ca refer to Seiford [6] i which 00 refereces are documeted. The origial DEA models assume that iputs ad outputs are measured by exact values. However, i may situatios, such as i a maufacturig system, a productio process, or a service system, iputs ad outputs are volatile ad complex so that they are difficult to measure i a accurate way. Thus may researchers tried to model DEA with various ucertai theories. Probability theory is the earliest theory which was used to establish the stochastic DEA models. Segupta [7] geeralized the stochastic DEA model usig the expected value. Baker [8] icorporated statistical elemets ito DEA, thus developig a statistical method. May papers [9 13]have employed the chace-costraied programmig to DEA i order to accommodate stochastic variatios i data. Fuzzy theory is aother theory which was used to deal with the ucertaity i DEA. As oe of the DEA iitiators, Cooper et al. [14 16]itroducedhowtodealwithimprecisedatasuchas bouded data, ordial data, ad ratio bouded data i DEA. Kao ad Liu [17] developed a method to fid the membership fuctios of the fuzzy efficiecy scores whe some iputs or iputs are fuzzy umbers. Etai et al. [18] proposeda DEA model with a iterval efficiecy by the pessimistic ad the optimistic values. May researchers have itroduced possibility measure [19]ito DEA[20, 21]. A lot of surveys showed that huma ucertaity does otbehavelikefuzziess.forexample,wesay theiputis about 10. Geerally, we employ fuzzy variable to describe the cocept of about 10; the there exists a membership fuctio, such as a triagular oe (9, 10, 11). Based o this membership fuctio, we ca obtai that the iput is exactly 10 with possibility measure 1. O the other had, theoppositeevetof otexactly10 hasthesamepossibility measure. The coclusio that ot 10 ad exactly 10 have thesamepossibilitymeasureisotappropriate.thisispired Liu [22] to foud a ucertaity theory which has become a brach of axiomatic mathematics for modelig huma ucertaity. This paper will apply the ucertaity theory to DEA to deal with huma ucertaity, thus producig some ucertai DEA models. I this paper, we will assume the iputs ad outputs are ucertai variables ad propose some ucertai DEA models. The rest of this paper is orgaized as follows. Sectio 2

2 2 Joural of Applied Mathematics will itroduce some basic cocepts ad properties about ucertai variables. The a ucertai DEA model as well as its equivalet crisp model will be preseted i Sectio 3. Fially, a umerical example will be give to illustrate the ucertai DEA model i Sectio Prelimiaries Ucertaity theory was fouded by Liu [22] i 2007 ad refied by Liu [23] i As extesios of ucertaity theory, ucertai process, ad ucertai differetial equatios [24], ucertai calculus [2] were proposed. Besides, ucertai programmig was first proposed by Liu [26] i2009, which wats to deal with the optimal problems ivolvig ucertai variable. This work was followed by a ucertai multiobjective programmig, a ucertai goal programmig [27], ad a ucertai multilevel programmig [28]. Sice that, ucertaity theory was used to solve variety of real optimal problems, icludig fiace [29 31], reliability aalysis [32, 33], graph [34, 3], ad trai schedulig [36, 37]. I this sectio, we will state some basic cocepts ad results o ucertai variables. These results are crucial for the remaider of this paper. Let Γ be a oempty set, ad let Ł be a σ-algebra over Γ. EachelemetΛ ŁisassigedaumberΛ} [0, 1]. I order to esure that the umber Λ} has certai mathematical properties, Liu [22] preseted the four axioms. Axiom 1. Γ} = 1 for the uiversal set Γ. Axiom 2. Λ} + Λ c }=1for ay evet Λ. Axiom 3. For every coutable sequece of evets Λ 1,Λ 2,..., we have Λ i } Λ i }. (1) Axiom 4. Let(Γ k, Ł k,m k ) be ucertaity spaces for k = 1, 2,.... The the product ucertai measure M is a ucertai measure satisfyig Λ k }= M k Λ k }, (2) where Λ k are arbitrarily chose evets from Ł k for k = 1, 2,...,respectively. If the set fuctio M satisfies the first three axioms, it is called a ucertai measure. Defiitio 1 (see Liu [22]). Let Γ be a oempty set, let Ł be a σ-algebra over Γ, ad let M be a ucertai measure. The the triplet (Γ, Ł, M) is called a ucertaity space. Defiitio 2 (see Liu [22]). A ucertai variable ξ is a measurable fuctio from a ucertaity space (Γ, Ł, M) to the set of real umbers; that is, for ay Borel set B of real umbers, the set is a evet. ξ B} =γ Γ ξ(γ) B} (3) Defiitio 3 (see Liu [22]). The ucertaity distributio Φ of a ucertai variable ξ is defied by for ay real umber x. Φ (x) =ξ x} (4) Example 4. The liear ucertai variable ξ L(a, b) has a ucertaity distributio 0, if x a, Φ (x) = (x a) (b a), if a x b, 1, if x b. Example. A ucertai variable ξ is called zigzag if it has a zigzag ucertaity distributio 0, if x a, (x a), if a x b, Φ (x) = 2 (b a) (x+c 2b), if b x c, 2 (c b) 1, if x c deoted by Z(a, b, c),wherea, b, c are real umbers with a< b<c. Defiitio 6 (see Liu [2]). The ucertai variables ξ 1,ξ 2,...,ξ are said to be idepedet if (ξ i B i )} = for ay Borel sets B 1,B 2,...,B. () (6) ξ i B i } (7) Defiitio 7 (see Liu [23]). A ucertaity distributio Φ of a ucertai variable ξ is said to be regular if its iverse fuctio Φ 1 (α) exists ad is uique for each α (0, 1). I this case, the iverse fuctio Φ 1 (α) is called the iverse ucertaity distributio of ξ. Example 8. The iverse ucertaity distributio of a zigzag ucertai variable Z(a,b,c)is Φ 1 (1 2α) a+2αb, if α 0., (α) = (2 2α) b+(2α 1) c, if α > 0.. Theorem 9 (see Liu [23]). Let ξ 1,ξ 2,...,ξ be idepedet ucertai variables with regular ucertaity distributios (8)

3 Joural of Applied Mathematics 3 Φ 1,Φ 2,...,Φ,respectively.Iff is a strictly icreasig fuctio, the ξ=f(ξ 1,ξ 2,...,ξ ) (9) is a ucertai variable with iverse ucertaity distributio Ψ 1 (α) =f(φ 1 1 (α),φ 1 2 (α),...,φ 1 (α)). (10) Example 10. Let ξ be a ucertai variable with regular ucertaity distributio Φ. Sice f(x) = ax + b is a strictly icreasig fuctio for ay costats a>0ad b,the iverse ucertaity distributio of aξ + b is Ψ 1 (α) =aφ 1 1 (α) +b. (11) Example 11. Let ξ 1,ξ 2,...,ξ be idepedet ucertai variables with regular ucertaity distributios Φ 1,Φ 2,...,Φ, respectively. Sice f (x 1,x 2,...,x ) =x 1 +x 2 + +x (12) is a strictly icreasig fuctio, the sum ξ=ξ 1 +ξ 2 + +ξ (13) is a ucertai variable with iverse ucertaity distributio Ψ 1 (α) =Φ 1 1 (α) +Φ 1 2 (α) + +Φ 1 (α). (14) Theorem 12 (see Liu [23]). Assume the costrait fuctio g(x, ξ 1,ξ 2,...,ξ ) is strictly icreasig with respect to ξ 1,ξ 2,...,ξ k ad strictly decreasig with respect to ξ k+1,ξ k+2,...,ξ.ifξ 1,ξ 2,...,ξ are idepedet ucertai variables with ucertaity distributios Φ 1,Φ 2,...,Φ, respectively, the the chace costrait holds if ad oly if g(x,φ M g (x, ξ 1,ξ 2,...,ξ ) 0} α (1) (α),...,φ 1 k 3. DEA Model (α),φ 1 k+1 (1 α),...,φ 1 (1 α)) (16) I may situatios, iputs ad outputs are volatile ad complexsothattheyaredifficulttomeasureiaaccurate way. This ispired may researchers to apply probability to DEA. As we kow, probability or statistics eeds a large amout of historical data. I the vast majority of real cases, thesamplesizeistoosmall(eveosample)toestimatea probabilitydistributio.thewehavetoivitesomedomai experts to evaluate their degree of belief that each evet will occur. This sectio will give some researches to empirical ucertai DEA usig the theory itroduced i Sectio 2.The ew symbols ad otatios are give as follows: DMU i :theith DMU,,2,...,; DMU 0 :thetargetdmu; x k =( x k1, x k2,..., x kp ):theucertaiiputsvectorof DMU k,,2,...,; Φ ki (x): the ucertaity distributio of x ki, k = 1,2,...,,,2,...,p; x 0 =( x 01, x 02,..., x 0p ): the iputs vector of the target DMU 0 ; Φ 0i (x): the ucertaity distributio of x 0i, i = 1,2,...,p; y k =( y k1, y k2,..., y kq ):theucertaioutputsvector of DMU k,,2,...,; Ψ kj (x): the ucertaity distributio of y kj, k = 1,2,...,, j=1,2,...,q; y 0 = ( y 01, y 02,..., y 0q ):theoutputsvectorofthe target DMU 0 ; Ψ 0j (x): the ucertaity distributio of y 0j, j = 1,2,...,q Ucertaity Distributios of Iputs ad Outputs. Liu ad Ha [38] proposed a questioaire survey for collectig expert s experimetal data. It is based o expert s experimetal data rather tha historical data. The startig poit is to ivite oe expert who is asked to complete a questioaire about the meaig of a ucertai iput (output) ξ like How may is the iput (output). We first ask the domai expert to choose a possible value x that the ucertai iput ξ may take ad the quiz him, How likely is ξ less tha or equal to x? Deote the expert s belief degree by α. A expert s experimetal data (x, α) is thus acquired from the domai expert. Repeatig the above process, we ca obtai the followig expert s experimetal data: (x 1,α 1 ),(x 2,α 2 ),...,(x,α ) (17) that meet the followig cosistece coditio (perhaps after arearragemet): x 1 <x 2 < <x, 0 α 1 α 2 α 1. (18) Based o those expert s experimetal data, Liu ad Ha [38] suggested a empirical ucertaity distributio, Φ (x) 0, if x x 1, = α i + (α i+1 α i )(x x i ), x i+1 x i if x i x x i+1,1 i<, 1, if x>x. (19) Assume there are m domai experts ad each produces a ucertaity distributio. The we may get m ucertaity distributios Φ 1 (x), Φ 2 (x),...,φ m (x). TheDelphimethod

4 4 Joural of Applied Mathematics was origially developed i the 190s by the RAND Corporatio based o the assumptio that group experiece is more valid tha idividual experiece. Wag et al. [39] recast the Delphi method as a process to determie the ucertaity distributio.themaistepsarelistedasfollows. Step 1. The m domai experts provide their expert s experimetal data, (x ij,α ij ), j=1,2,..., i,,2,...,m. (20) Step 2. Use the ith expert s experimetal data (x i1,α i1 ), (x i2,α i2 ),...,(x ii,α ii ) to geerate the ith expert s ucertaity distributio Φ i. Step 3. Compute Φ(x) = w 1 Φ 1 (x)+w 2 Φ 2 (x)+ +w m Φ m (x), where w 1,w 2,...,w m are covex combiatio coefficiets. Step 4. If α ij Φ(x ij ) arelessthaagivelevelε>0,the go to Step.Otherwise,theith expert receives the summary (Φ ad reasos) ad the provides a set of revised expert s experimetal data. Go to Step 2. Step. The last Φ is the ucertaity distributio of the iput (output) Ucertai DEA Model. Similar to traditioal DEA model [3], the objective of the ucertai DEA model is to maximize the total slacks i iputs ad outputs subject to the costraits. The the ucertai DEA model ca be give as follows: max subject to: p s i + q s + j j=1 λ k =1, λ k 0, x ki λ k x 0i s i } α, y kj λ k y 0j +s + j } α,,2,...,, s i 0,,2...,p,,2,...,p, j=1,2...,q, s + j 0, j=1,2,...,q. (21) Defiitio 13 (α-efficiecy). DMU 0 is α-efficiet if s i ad s + j are zero for i = 1,2,...,p ad j = 1,...,q,wheres i ad s + j are optimal solutios of (21). Sice the ucertai measure is ivolved, this defiitio is differet from traditioal efficiecy defiitio. For istace, as determied by the choice of α, there is a risk that DMU 0 will ot be efficiet eve whe the coditio of Defiitio 13 is satisfied. Sice j = 0 is oe of the DMU j,wecaalwaysgeta solutio with λ 0 = 1, λ j = 0 (j = 0),adallslackszero. ThusthisucertaiDEAmodelhasfeasiblesolutioadthe optimal value s i =s + j =0for all i, j Determiistic Model. Model (21) is a ucertai programmig model, which is too complex to compute directly. This sectio will give its equivalet crisp model to simplify the computatio process. Theorem 14. Assume that x 1i, x 2i,..., x i are idepedet ucertai iputs with ucertaity distributio Φ 1i,Φ 2i,...,Φ i for each i, i = 1,2,...,p,ad y 1i, y 2i,..., y i are idepedet ucertai outputs with ucertaity distributio Ψ 1j,Ψ 2j,...,Ψ j for each j, j = 1,2,...,q. The holds if ad oly if,k=0,k=0 x ki λ k x 0i s i } α,,2,...,p, y kj λ k y 0j +s + j } α, j=1,2...,q (22) ki (α) +λ 0 Φ 1 0i (1 α) Φ 1 0i (1 α) s i, λ k Ψ 1 kj (1 α) +λ 0Ψ 1 0j,2,...,p, (α) Ψ 1 0j (α) +s j, j=1,2,...,q. (23) Proof. Without loss of geerality, let ad x 0 =x 1 ;the we will cosider the equatio Rewrite (24)as M x k1 λ k x 11 s i } α. (24) x k1 λ k (1 λ 1 ) x 11 s i } α. (2) Sice (1 λ 1 ) x 11 is a ucertai variable which is decreasig with respect to x 11, its iverse ucertaity distributio is Υ 1 11 (α) = (1 λ 1)Φ 1 11 (1 α), 0<α<1. (26) For each 2 k, x k1 λ k is a ucertai variable whose iverse ucertaity distributio is Υ 1 k1 (α) =λ kφ 1 k1 (α), 0 < α < 1. (27)

5 Joural of Applied Mathematics Table 1: DMUs with two ucertai iputs ad two ucertai outputs. DMU i Iput 1 Z(3., 4.0, 4.) Z(2.9, 2.9, 2.9) Z(4.4, 4.9,.4) Z(3.4, 4.1, 4.8) Z(.9, 6., 7.1) Iput 2 Z(2.9, 3.1, 3.3) Z(1.4, 1., 1.6) Z(3.2, 3.6, 4.0) Z(2.1, 2.3, 2.) Z(3.6, 4.1, 4.6) Output 1 Z(2.4, 2.6, 2.8) Z(2.2, 2.2, 2.2) Z(2.7, 3.2, 3.7) Z(2., 2.9, 3.3) Z(4.4,.1,.8) Output 2 Z(3.8, 4.1, 4.4) Z(3.3, 3., 3.7) Z(4.3,.1,.9) Z(.,.7,.9) Z(6., 7.4, 8.3) Table 2: Results of evaluatig the DMUs with α = 0.6. DMUs (λ 1,λ 2,λ 3,λ 4,λ ) p s i + q j=1 s + j The result of evaluatig DMU 1 (0, 0.2, 0, 0.7, 0) 1.89 Iefficiecy DMU 2 (0,1,0,0,0) 0 Efficiecy DMU 3 (0, 0, 0, 0.78, 0.22) 1.4 Iefficiecy DMU 4 (0,0,0,1,0) 0 Efficiecy DMU (0,0,0,0,1) 0 Efficiecy It follows from the operatioal law that the iverse ucertaity distributio of the sum x k1λ k (1 λ 1 ) x 11 is Υ 1 (α) = = Υ 1 21 (α) k1 (α) (1 λ 1)Φ 1 11 (1 α), 0 < α < 1. (28) From which we may derive the result immediately for ad x 0 =x 1.Similarly,wecagetotherresults. Followig Theorem 14, the ucertai DEA model ca be coverted to the crisp model as follows: which is a liear programmig model. Thus it ca be easily solved by may traditioal methods. 4. A Numerical Example This example wats to illustrate the ucertai DEA model. For simplicity, we will oly cosider five DMUs with two iputs ad two outputs which are all zigzag ucertai variables deoted by Z(a, b, c). Table 1 gives the iformatio of the DMUs. For illustratio, let DMU 1 bethetargetdmu;thethe ucertai DEA model (29)ca be writte as max subject to: p s i +,k=0,k=0 q s + j j=1 ki (α) +λ 0 Φ 1 0i (1 α) Φ 1 0i (1 α) s i, λ k Ψ 1 kj (1 α) +λ 0Ψ 1 0j (α) Ψ 1 0j (α) +s+ j, λ k =1, λ k 0,,2,...,, s i 0,,2...,p,,2,...,p, j=1,2,...,q, s + j 0, j=1,2,...,q (29) max s 1 +s 2 +s+ 1 +s+ 2 subject to: k1 (α) +λ 1Φ 1 11 (1 α) Φ 1 11 (1 α) s 1, k2 (α) +λ 1Φ 1 12 (1 α) Φ 1 12 (1 α) s 2, λ k Ψ 1 k1 (1 α) +λ 1Ψ 1 11 λ k Ψ 1 k2 (1 α) +λ 1Ψ 1 12 λ k =1, λ k 0,,2,...,, (α) Ψ 1 11 (α) +s+ 1, (α) Ψ 1 12 (α) +s+ 2,

6 6 Joural of Applied Mathematics Table 3: Results of evaluatig the DMUs with differet cofidece level α. α DMU 1 DMU 2 DMU 3 DMU 4 DMU 0. Iefficiecy Efficiecy Iefficiecy Efficiecy Efficiecy 0.6 Iefficiecy Efficiecy Iefficiecy Efficiecy Efficiecy 0.7 Iefficiecy Efficiecy Iefficiecy Efficiecy Efficiecy 0.8 Iefficiecy Efficiecy Efficiecy Efficiecy Efficiecy 0.9 Efficiecy Efficiecy Efficiecy Efficiecy Efficiecy s 1 0, s 2 0, s + 1 0, s (30) Table 2 showstheresultsofevaluatigdmuswithcofidece level α = 0.6. The results ca be iterpreted i the followig way: DMU 1 ad DMU 3 are iefficiet, whereas DMU 2,DMU 4,adDMU are efficiet. Moreover, DMU 3 is more efficiet tha DMU 1 from the total slacks p s i + q j=1 s+ j, sice they are both iefficiet. Ucertai efficiecies obtaied from model (30) for differet cofidece levels α are show i Table 3.DMU 1 is iefficiet at all cofidece levels, whereas DMU 2,DMU 4, ad DMU arealwaysefficietatalllevels.itcabeseethat theumberoftheefficietdmusisaffectedbythecofidece level α. The higher the cofidece level α is, the bigger theumberofefficietdmusis.thispheomeaidicate that ucertai DEA is more complex tha the traditioal DEA because of the iheret ucertaity cotaied i iputs ad outputs.. Coclusio Due to its widely practical used backgroud, data evelopmet aalysis (DEA) has become a pop area of research. Sicethedatacaotbepreciselymeasuredisomepractical cases, may papers have bee published whe the iputs ad outputs are ucertai. This paper has give some researches to ucertai DEA model. A ew DEA model as well as its equivalet determiistic model was preseted. For illustratio, a umerical example was desiged. Coflict of Iterests The authors declare that they have o coflict of iterests regardig the publicatio of this paper. Ackowledgmet This work was supported by Natioal Natural Sciece Foudatio of Chia (os ad ). Refereces [1] A. Chares, W. W. Cooper, ad E. Rhodes, Measurig the efficiecy of decisio makig uits, Europea Joural of Operatioal Research,vol.2,o.6,pp ,1978. [2] R. D. Baker, A. Chares, ad W. W. Cooper, Some models for estimatig techical ad scale efficiecies i data evelopmet aalysis, Maagemet Sciece, vol. 30, o. 9, pp , [3] A. Chares, W. W. Cooper, B. Golay, L. Seiford, ad J. Stutz, Foudatios of data evelopmet aalysis for Pareto- Koopmas efficiet empirical productio fuctios, Joural of Ecoometrics, vol. 30, o. 1-2, pp , 198. [4] N. C. Peterse, Data evelopmet aalysis o a relaxed set of assumptios, Maagemet Sciece, vol. 36, o. 3, pp , [] K. Toe, A slacks-based measure of efficiecy i data evelopmet aalysis, Europea Joural of Operatioal Research, vol. 130, o. 3, pp , [6] L. M. Seiford, A DEA bibliography ( ), i Data Evelopmet Aalysis: Theory, Methodology ad Applicatios,A. Chares,W.W.Cooper,A.Lewi,adL.Seiford,Eds.,Kluwer Academic Publishers, Bosto, Mass, USA, [7] J. K. Segupta, Efficiecy measuremet i stochastic iputoutput systems, Iteratioal Joural of Systems Sciece, vol. 13, o.3,pp ,1982. [8] R. D. Baker, Maximum likelihood, cosistecy ad data evelopmet aalysis. A statistical foudatio, Maagemet Sciece, vol. 39, o. 10, pp , [9] O. Olese ad N. C. Peterse, Chace costraied efficiecy evaluatio, Maagemet Sciece,vol.141,pp ,199. [10] R. D. Baker, Stochastic Data Evelopmet Aalysis, Caregie- Mello Uiversity, Pittsburgh, Pesylvaia, [11] S. Grosskopf, Statistical iferece ad oparametric efficiecy: a selective survey, Joural of Productivity Aalysis,vol. 7, o. 2-3, pp , [12] W. W. Cooper, Z. Huag, ad S. X. Li, Satisficig DEA models uder chace costraits, Aals of Operatios Research, vol. 66, pp , [13] W. W. Cooper, Z. M. Huag, V. Lelas, S. X. Li, ad O. B. Olese, Chace costraied programmig formulatios for stochastic characterizatios of efficiecy ad domiace i DEA, Joural of Productivity Aalysis,vol.9,o.1,pp.3 79,1998. [14]W.W.Cooper,K.S.Park,adG.Yu, IdeaadAR-IDEA: models for dealig with imprecise data i DEA, Maagemet Sciece,vol.4,o.4,pp ,1999. [1] W.W.Cooper,K.S.Park,adG.Yu, Aillustrativeapplicatio of idea (imprecise data evelopmet aalysis) to a Korea mobile telecommuicatio compay, Operatios Research,vol. 49,o.6,pp ,2001.

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