Computation of Congestion in DEA Models with Productions Trade-offs and Weight Restrictions

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1 Appled Mathematcal Scences, Vol. 5, 2011, no. 14, Computaton of Congeston n DEA Models wth Productons Trade-offs and Weght Restrctons G.R. Jahanshahloo a, M. Khodabakhsh b, F. Hossenzadeh Lotf a and M.R. Moazam Goudarz a a Scence and Research Campus Islamc Azad Unversty, Tehran, Iran Jahanshahlo@yahoo.com (G.R. Jahanshahloo) b Department of Mathematcs, Faculty of Scence Lorestan Unversty, Khorram Abad, Iran Abstract Our am n ths work s to determne congeston wth productons trade-offs or, equvalently, under weghts restrctons n data envelopment analyss (DEA). For ths purpose, we revew a two-model approach to evaluate congeston (Cooper et al, 1996), and after that we have bref vew n relaton wth computaton of effcent targets wth producton trade-off n Podnovsk s procedure (Podnovsk, 2007a). So, our method s a hybrd of two above procedure: computaton of congeston wth weght restrctons whch s supported by an expermental example (Podnovsk, 2007b). Keywords: Data envelopment analyss; Congeston; Trade-offs; Weght restrctons 1 Introducton In data envelopment analyss (DEA), the effcency of decson-makng unts (DMUs) s assessed by solvng a par of mutually dual lnear programmng problems. One of these, s referred to as the envelopment, and the other one as multpler form. These two forms provde a dfferent framework for the nterpretaton of the radal effcency of the decson-makng unt (DMU) under the assessment. The feasble set of the envelopment model s explctly related to the economc noton of the producton possblty set (PPS). The effcency measured by ths model has the meanng of the ultmate radal mprovement factor by

2 664 G.R. Jahanshahloo et al whch the nputs or outputs of the DMUo under the assessment can change, whle the resultng DMU remans wthn the PPS. In the multpler model, the effcency of DMUo s assessed by assocatng varable weghts wth ts nputs and outputs. By varyng these weghts, the model maxmzes the rato of the total weghted output to the total weghted nput of DMUo n relaton wth the smlar ratos of the other unts. The obtaned maxmum rato s the manageral effcency of DMUo, whch by dualty concdes wth the radal effcency assessed n the envelopment model. Often DEA models do not provde suffcent dscrmnaton between the effcency of DMUs. One method that tradtonally be used to overcome ths drawback, s the use of weght restrctons Allen et al. (1997) and (Thanassouls, 2001). These are usually based on the perceved mportance of dfferent nputs and outputs or ther monetary values. Ths approach has a well-known drawback, namely, the envelopment DEA forms become dstorted and the effcency measures lose ther clear technologcal meanng Allen et al. (1997). Another approach for mprovement of DEA models was suggested based on the ncorporaton of producton trade-offs n the envelopment DEA models (Podnovsk, 2004, 2005, 2007b). Such trade-offs represent smultaneous changes to the nputs and outputs that are technologcally possble. The mathematcal effect of the ncorporaton of trade-offs n the envelopment models s the same as the effect caused by weght restrctons n the multpler forms: the resultng DEA models dscrmnate better. Concept of congeston ndcates an economc state and t occurs when reducng some nputs can ncrease some outputs. (Fare and Grosskopf, 1983) frst ntroduced an approach for analyzng congeston. Later, Fare et al. (1985) dscussed data envelopment analyss related models and methods (called FGL approach) for producton effcency evaluaton. Cooper et al. (1996) ntroduced an alternatve approach for evaluatng the concept congeston. In exstng researches, so far, t s not to be attenton to computaton of congeston wth productons trade-offs or, equvalently, under weghts restrctons ssue whch s surveyed n ths paper extensvely. The paper s organzed as follows: n secton 2 we revew a two-model approach to evaluate congeston, secton 3 ntroduced a procedure for practcal applcaton of models that ncor- porate producton trade-offs between nputs and outputs or, equvalently, weght restrctons mposed on ther dual models and the proposed models for assess of congeston of such models, also a numercal example are gven n secton 4, and n secton 5 concluson s put forward.

3 Computaton of congeston n DEA models A two-model approach to evaluate congeston We start wth a two-model approach that used to evaluate congeston. Assume we have n observed DMUs, DMU j (X j,y j ), j =1, 2,..., n, and every DMU j produces the same s outputs n (possbly) dfferent amounts, y rj,r = 1, 2,..., s, usng the same nputs, x j,=1, 2,..., m, also n (possbly) dfferent amounts. All nputs and outputs are assumed to be nonnegatve, but at least one nput and one output are postve,.e., X j =(x 1j,..., x mj ) 0, X j 0 and Y j =(y 1j,..., y sj ) 0, Y j 0. Now, to mantan contact wth Cooper et al. (2002), we begn wth the followng verson of a BCC model: max φ + ε(σ m =1 s +Σ s r=1 s+ r ) s.t. λ j x j + s = x o, =1,..., m, (1) λ j y rj s + r = φy ro, r =1,..., s, λ j =1, λ j,s,s+ r 0 j =1,..., n, =1,..., m, r =1,..., s. Here ε>0s a non-archmedean element defned to be smaller than any postve real number. Ths means that ε s not a real number. The standard procedure s to avod any needs for explctly assgnng a value to ε by usng the followng two-stage process. Stage one: maxmze φ whle gnorng the slacks, s,s+ r, n the objectve. Stage two: replace φ wth φ = max φ n (1) and maxmze the sum of the slacks, then determne whether DMUo s effcent or neffcent n accordance wth the followng defnton. Defnton 2.1. (BCC Effcency ) DMUo s effcent f and only f the followng two condtons are both satsfed: () φ =1. () All slack varables are zero n optmal solutons. Defnton 2.2. (BCC-Projecton ) For a BCC-neffcent DMU o, we defne ts BCC-projecton, based on an optmal soluton for Model (1), as follows: ( x o = x o s, ŷ ro = φ y ro + s + r ),

4 666 G.R. Jahanshahloo et al the mproved actvty ( X o, Ŷo) s BCC-effcent. As descrbed n Cooper et al. (2001), the x o and ŷ ro values together wth the x j and y rj,as defned n (1), are used to erect the followng new problem: max s.t. =1 δ λ j x j δ = x o,, =1,..., m, (2) λ j y rj = ŷ ro, r =1,..., s, λ j =1, δ s, λ j 0, j =1,..., n, δ 0, =1,..., m. Fnally, to dentfy the congestng nputs and to estmate ther amounts, we utlze the =1,..., m nput constrants δ s n (2) to obtan s c = s δ, =1,2,...,m, (3) where δ s obtaned from (2). s c slack assocated wth s s then the congestng amount n the total n nput =1,..., m, as obtaned from (1) and δ s the (maxmum) amount of ths total slack that can be assgned to purely techncal (non-congestng) neffcency, as obtaned from (2). 3 Computaton of congeston wth productons trade-offs We follow the notaton n (Podnovsk, 2007) and let (P,Q), trade-offs between the nputs and/or outputs, represent the possble smultaneous change to the nputs and outputs n the entre technology. Varous examples of producton trade-offs are dscussed n (Podnovsk, 2004). Suppose that we can specfy k trade-offs: (P t,q t ),,2,...,k. (4)

5 Computaton of congeston n DEA models 667 The use of trade-offs (4) n the standard VRS technology leads to the expanded technology T VRS TO, where the abbrevaton TO stands for trade-offs, as n the followng form T VRS TO = {(X, Y ) X 0, Y 0,X n λ jx j + k π tp t, Y n λ jy j + k π tq t, Σ n λ j =1,λ j 0; π t 0; j =1,..., n; t =1,..., k}. (5) The output radal effcency of DMU o n T VRS TO s defned as max{φ (φx o,y o ) T VRS TO }. Ths leads to the followng LP formulaton: max s.t. φ λ j X j + π t P t + d = X o, (A1) λ j Y j + λ j X j + λ j Y j + λ j =1, π t Q t e = φy o, π t P t + d 0, π t Q t e 0, (A2) (A3) (A4) (A5) λ, π, e, d 0. (A6) We now observe that (A3) s redundant because t follows from (A1). Snce, (φ =1,π t =0;t =1, 2,..., k, d =0,e =0,λ o =1,λ j =1;j =1, 2,..., n, j o) s a feasble soluton for above model and the objectve functon maxmzng φ, so, φ 1 therefore, A4 can be deleted. Ths s mples that the model s

6 668 G.R. Jahanshahloo et al equvalent to the followng LP formulaton: max s.t. φ λ j X j + π t P t X o, (6) λ j Y j + λ j =1, λ, π 0. π t Q t φy o, The output radal effcency of DMUo s equal to the optmal value φ of the objectve functon n Model (6). Due to the constrants of the model, the target DMU (X o,φ Y o ) s a vald member of the PPS T VRS TO and located on ts boundary. Therefore, the radal effcency φ of DMUo has a meanng of technologcally feasble radal mprovement factor for the outputs of the DMUo. It s must to be attenton that the ntroducton of trade-offs (4) n the envelopment models s equvalent to the ncorporaton of weght restrctons u T Q t v T P t 0, t =1,..., k (8) n the dual multpler forms. In VRS DEA models wth producton trade-offs, the second stage of the optmzaton procedure s a testng for possble non-radal mprovements to

7 Computaton of congeston n DEA models 669 the radal targets. A lnear program for ths purpose s developed as follows: max s.t. m d + =1 s r=1 λ j X j + λ j Y j + λ j X j + λ j =1, e r π t P t + w + d = X o, π t Q t e = φ Y o, π t P t + w>=0, (8) λ, π, e, d, w 0. Model (8) maxmzes the sum of resdual slacks subject to the explct condton that the resultng effcent target has only nonnegatve nputs. Ths stage produces a fully effcent target of DMUo. Let λ,π,e,w andd be any optmal soluton to model (8). Defne X o = n λ j x j + k π t P t + w, Ŷ o = n λ j y j + k π t Q t. (9) Clearly, DMU ( X o, Ŷo) s a member of technology T VRS TO. Theorem 1. The DMU ( X o, Ŷo) s paretto-effcent n technology T VRS TO. Accordng to Theorem (1), f φ = 1 and optmal vectors e and d are zero vectors, DMUo concde wth DMU ( X o, Ŷo) and s therefore effcent. Otherwse DMUo s neffcent and ( X o, Ŷo) can be regarded as ts effcent target. As was told n secton 2, frstly, by applyng Model (2), the BCC-projecton of a DMUo, that s, ( X o, Ŷo) accordng to defnton (2.2) s obtaned, whch s used to compute the congeston. Smlarly, at frst, under weght restrctons, we want to fnd the radus target of a DMUo usng model (6), whch wll be used to obtan the correspondence effcent target n model (8) accordng to (9). By ncorporatng the mentoned effcent target n Model (10) we are gong

8 670 G.R. Jahanshahloo et al to compute the congeston. max s.t. =1 δ λ j x j + λ j y rj + λ j =1, π t P t δ = x o,, =1,..., m, (10) π t Q t = ŷ ro, r =1,..., s, δ d, λ j 0, j =1,..., n, π t 0, t =1,..., k, δ 0, =1,..., m. Fnally, to dentfy the congestng nputs and to estmate ther amounts, we utlze the =1,..., m nput constrants δ d n (10) to obtan d c = d δ, =1,2,...,m. (11) where δ s obtaned from (10). d c slack assocated wth d c s then the congestng amount n the total n nput =1,..., m, as obtaned from (1), and δ s the (maxmum) amount of ths total slack that can be assgned to purely techncal (non-congestng) neffcency, as obtaned from (10). for numercal treatment of the above explanatons we gve an expermental example. 4 Expermental example The example used n ths paper nvolves sx hypothetcal unversty departments (Table 1), each of whch has two nputs and three outputs as follows: Inputs T/S: Teachng staff R/S: Research staff Outputs U/S: Undergraduate students M/S: Master students

9 Computaton of congeston n DEA models 671 P: Publcatons Table 1 The data set Department U/S M/S P T/S R/S A B C D E F Table 2 optmal slack solutons and effcency n the standard VRS model Department s 1 s 2 s + 1 s + 2 s + 3 Effcency A B C D E F The man mpled reason for usng ths example that already used n (Podnovsk, 2007b) s that the functonng of unversty departments s an area that s almost ntutvely understood by everyone. Table (2) shows the effcency of all departments and the correspondng optmal slack solutons accordng to Model (2). As s noted, only the second department, B, s neffcent. Therefore we compute the BCC-projecton of ths unt usng Defnton 2.2. In the followng we gve the results: ( x 1 = , x 2 = , ŷ 1 = , ŷ 2 = , ŷ 3 = ). If we ncorporate the BCC-projecton n Model (2), the congestng nputs and ther amounts obtan as follows: s c 1 = , s c 2 = ,

10 672 G.R. Jahanshahloo et al Table 3 optmal weghts and effcency n the standard VRS model Department u 1 u 2 v1 v2 v3 Effcency A B C D E F In ths part, we want to know the condtons that a vector would be called trade-off. Specally n current example, we would lke to know the structure of these vectors. On the left-hand sde (LHS) of model (6) we see the composte DMU made by a convex combnaton of observed DMUs wth coeffcents λ j. Ths, technologcally, mples possblty n our VRS technology. The added term on the LHS s the nonnegatve combnaton of vectors (P t,q t ) wth coeffcents π t 0 as followng: ( π t P t, π t Q t ) (12) In the standard envelopment model, wthout ths added term (12), the composte unt on the LHS outperforms, n the weak sense of non-strct nequaltes, the unt on the rght-hand sde (RHS). Ths explans why the scalng factor φ s regarded as a technologcally possble output ncreasng factor. The term (12), modfes the composte DMU on the LHS of (6). We want to make sure that ths modfcaton s meanngful and the resultng unt on the LHS s technologcally possble. If ths s acheved, ths unt on the LHS wll domnate n the weak sense the unt on the RHS, and the meanng of effcency φ wll reman ntact. To acheve ths goal, we have to make sure that happen smultaneous changes n outputs and nputs, respectvely, by forcng Q t and P t wll be possble. If ths s true, we shall call vector (P t,q t )asproducton trade-off. The above development gves us a method to the constructon of weght restrctons that preserves the technologcal meanng of effcency. We refer to ths method as the trade-off approach. Accordng to ths approach, at frst, a technologcally possble producton trade-off should be constructed and then t must be rewrtten wth mposng weght restrctons. Our am s to show how producton trade-offs can be assessed n real technologes and specally, n ths example.

11 Computaton of congeston n DEA models 673 Table 3 shows the effcency of all departments and the correspondng optmal weghts (usng multpler form of BCC model). The output weghts are shown as u 1 for undergraduate students, u 2 for masters, u 3 for publcatons. The nput weghts are shown as v 1 for teachng staff and v 2 for research staff. Snce fve out of sx departments are effcent, the second optmzaton stage performed n ths case confrms that none of these fve departments exhbts mx neffcency (Cooper et al, 2000), the above model s not suffcently dscrmnatng. The low dscrmnaton of our analyss can be partly attrbuted to the relatvely large number of zero weghts n the optmal solutons to the VRS model. A way to resolve ths problem and mprovng the dscrmnaton of analyss s to mpose addtonal producton trade-offs n the envelopment model. (Podnovsk; 2007b) In present example, dfferent producton trade-offs are applcable that we brefly presented here some of them. (For more nformaton aboat them, one can refer to (Podnovsk; 2007b)) 1. We shall frst assume that teachng master students requres no more than twce the amount of resources used to teach undergraduates. Ths means that no department can clam extra resources (teachng and research staff) f the number of undergraduate students s reduced by 2 and the number of master students s ncreased by 1. We can also expect that research output should not be affected by such a change. Ths judgement generates, n the notaton used n model (3), the followng trade-off: P 1 =(0, 0) T, Q 1 =( 2, 1, 0) T, (13) Ths can be read as the followng: (two undergraduate students out, one master n) s feasble provded nothng else changes. 2. No extra resources (teachng and research staff) are needed f the number of undergraduate students s ncreased by 1 and the number of master students s reduced by 1. The correspondng trade-off s represented as P 2 =(0, 0) T, Q 2 =(1, 1, 0) T, (14) 3. If the number of publcatons decreases by one paper n a year, the released tme should be suffcent to teach two extra undergraduate students: P 3 =(0, 0) T, Q 3 =(2, 0, 1) T, (15) 4. The number of undergraduate students could go up by at least fve for each teachng post ntroduced, whch s by 25 n total. We can, therefore, formulate the followng trade-off, whch appears to be entrely plausble: P 4 =(5, 1) T, Q 4 = (25, 0, 0) T, (16)

12 674 G.R. Jahanshahloo et al 5. At last, consder another stuaton by assumng that one teachng post s replaced by one research poston. Snce research staff do not generally teach, some detrmental effect on students should be factored n ths trade-off. Even n the worst-case scenaro no department should lose more than, say, 20 undergraduate students. At the same tme, we may expect the ncrease of publcatons by at least 0.3 papers a year because research staff should publsh more than teachng staff. (One may argue for the use of a much larger ncrease than 0.3 publcatons, but ths s not our pont here.) These estmates result n the followng trade-off whch nvolves two nputs and two outputs at once: P 5 =( 1, 1) T, Q 5 =( 20, 0, 0.3) T, (17) Table 4 optmal slack solutons and effcency n the model wth producton tradeoffs (12) Department s 1 s 2 s + 1 s + 2 s + 3 Effcency A B C D E F By ncorporatng the trade-offs (13), (14)(15), (16) and (17) n the model (6), t can be shown that only department B s non radal effcent. To check full effcency of DMUs, we maxmze the sum of slacks by solvng model (8). The results s shown n Table (4). Accordng to (9), f we compute ( X o, Ŷo) and ncorporate t n Model (10), the congestng nputs and ther amounts wll be obtan as the followng: d c 1 =0, dc 2 = As t was seen, when producton trade-offs are applyng the values of the congestons n the nputs are varyng. Here, the frst nputs of the DMU B has no congeston and the value of the congeston n the second nput s varyng. 5 Concluson In ths paper, we computed congeston n DEA models wth productons trade-offs and weght restrctons. So far, many approaches have been presented to evaluate congeston, but the evaluaton of congeston under weghts

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