Hindawi Publihing Corporation pplied Matheatic Volue 2013, rticle ID 658635, 7 page http://dx.doi.org/10.1155/2013/658635 Reearch rticle n Extenion of Cro Redundancy of Interval Scale Output and Input in DE Farhad Hoeinzadeh-Lotfi, 1 Ghola-Reza Jahanhahloo, 2 and Manour Mohaadpour 1 1 Departent of Matheatic, Science and Reearch ranch, Ilaic zad Univerity, Hearak, Poonak, Tehran, Iran 2 Faculty of Matheatical Science and Coputer Engineering, Univerity for Teacher Education, 599 Taleghani venue, Tehran 15618, Iran Correpondence hould be addreed to Manour Mohaadpour; anour ohaadpour@yahoo.co Received 19 Noveber 2012; ccepted 22 May 2013 cadeic Editor: Hadi Naeri Copyright 2013 Farhad Hoeinzadeh-Lotfi et al. Thi i an open acce article ditributed under the Creative Coon ttribution Licene, which perit unretricted ue, ditribution, and reproduction in any ediu, provided the original work i properly cited. It i well known that data envelopent analyi (DE) odel are enitive to election of input and output variable. the nuber of variable increae, the ability to dicriinate between the deciion aking unit (DMU) decreae. Thu, to preerve the dicriinatory power of a DE odel, the nuber of input and output hould be kept at a reaonable level. There are any cae in which an interval cale output in the aple i derived fro the ubtraction of nonnegative linear cobination of ratio cale output and nonnegative linear cobination of ratio cale input. There are alo cae in which an interval cale input i derived fro the ubtraction of nonnegative linear cobination of ratio cale input and nonnegative linear cobination of ratio cale output. Lee and Choi (2010) called uch interval cale output and input a cro redundancy. They proved that the addition or deletion of a cro-redundant output variable doe not affect the efficiency etiate yielded by the CCR or CC odel. In thi paper, we preent an extenion of cro redundancy of interval cale output and input in DE odel. We prove that the addition or deletion of a cro-redundant output and input variable doe not affect the efficiency etiate yielded by the CCR or CC odel. 1. Introduction In any DE application, uch a incoe, an interval cale outputintheapleiderivedfrotheubtractionof nonnegative linear cobination of ratio cale output and nonnegative linear cobination of ratio cale input. There are alo any cae, like cot, in which an interval cale input i derived fro the ubtraction of nonnegative linear cobination of ratio cale input and nonnegative linear cobination of ratio cale output, although the effect of uch dependencie on DE i not clear. Lee and Choi [1] called uch interval cale output and input a cro redundancy.they proved that the addition or deletion of a cro-redundant output variable doe not affect the efficiency etiate yielded by the CCR or CC odel. Francico J. López [2] generalized the contribution of Lee and Choi by introducing pecific definition and conducting oe additional analyi on the ipact of the preence of other type of linear dependencie aong the input and output of a DE odel. In thi paper, we deal with cro-redundant output and input variable iultaneouly in DE odel. We prove that the addition or deletion of a cro-redundant output and input variable doe not affect the efficiency etiate yielded by the CCR or CC odel. The paper i organized a follow. In Section 2, we introduce preliinarie of DE. In Section 3, wepreent our ain reult. In Section 4, wewillillutratethatthe addition or deletion of cro-redundant output variable and input variable doe not affect the efficiency etiate yielded by the CCR or CC odel. Concluion are uarized in Section 5. 2. Preliinarie Suppoe that we have n 2peer oberved DMU, {DMU j : j,2,...,n} which produce ultiple output y rj,(r =
2 pplied Matheatic 1,...,), by utilizing ultiple input x ij,(i,...,). The input and output vector of DMU j are denoted by x j and y j, repectively, and we aue that x j and y j are eipoitive, that i, x j 0,x j =0 and y j 0,y j =0 for i,...,n.weue(x j, y j ) to decript DMU j and pecially ue (x o, y o )(oeleent of {1,2,...,n}) a the DMU under evaluation. Throughout thi paper, vector will be denoted by bold letter. The input-oriented CCR [3] ultiplierodelevaluate the efficiency of each DMU o by olving the following linear progra: θ = ax u t y o, v t x o =1.t. u t y j v t x j, j=1,...,n, u o,v o. ecaue x j and y j are eipoitive for j,...,n, θ >0. lo ince u t y o v t x o and v t x o =1,wehaveθ 1.Thu 0<θ 1. θ repreent the input-oriented CCR-efficiency value of DMU o. The output-oriented CCR ultiplier odel evaluate the efficiency of each DMU o by olving the following linear progra: φ = in v t x o, u t y o =1.t. v t x j u t y j, j=1,...,n, u 0,v 0. Since u t y o v t x o and u t y o,wehaveφ 1. 1/φ repreent the output-oriented CCR-efficiency value of DMU o.loθ =1/φ [4]. The input-oriented CC [4] ultiplierodelevaluate theefficiencyofeachdmu o by olving the following linear progra: z = ax u t y o +u o, v t x o =1.t. u t y j +u o v t x j, j=1,...,n, u 0, v 0, u o i free. Let (u, v ) be an optial feaible olution for odel (1); then (u, v,u o ),whereu o =0, will be a feaible olution of odel (3). Thu z θ ; therefore, 0<z 1. z repreent the input-oriented CC-efficiency value of DMU o. (1) (2) (3) Finally, the output-oriented CC ultiplier odel evaluate the efficiency of each DMU o by olving the following linear progra: t = in v t x o V o, u t y o =1.t. v t x j V o u t y j, j=1,...,n, u 0, v 0, V o i free. It can be eaily confired that t 1. 1/t repreent the output-oriented CC-efficiency value of DMU o. 3. Main Reult In thi ection, we prove that the addition or deletion of a cro-redundant output variable and/or input variable doe not affect the efficiency etiate yielded by the CC ultiplier odel in input- and output-oriented verion. Siilarly, it can be proved that the addition or deletion of cro-redundant variable doe not affect efficiency etiate yielded by the CCR ultiplier odel in input- and outputoriented verion. Theore 1. LeteachDMUhave+1inputand+1output, that i, x j =(x 1j,...,x (+1)j ) and y j =(y 1j,...,y (+1)j ) for j=1,2,...,n. Let x (+1)j = y (+1)j = (4) β i x ij α r y rj ; j=1,...,n, (5) a r y rj b i x ij ; j=1,...,n, (6) where β i 0, b i 0, i,...,;α r 0,a r 0,r = 1,...,. Then the optial objective function value of the following odel: ρ = ax.t +1 +1 +1 p r y ro +p o, q i x io =1, p r y ro +1 q i x io p o 0, j=1,...,n, p r 0, q i 0,,...,+1,,...,+1 i equal to the optial objective function value of the following odel (3). (7)
pplied Matheatic 3 Proof. Let (p 1,...,p +1,q 1,...,q +1,p o ) be an optial olution for odel (7); then we have +1 ρ = +1 +1 p r y ro p r y ro p o, (8) q i x io =1, (9) +1 y (6)and(9), it follow that ρ = (p r +p +1 ) y ro q i x io p o 0. (10) lo, by (5) and(9) it conclude that Now, let p +1 b ix io +p o. (11) (q i +q +1 β i )x io q +1 α r y ro =1. (12) V i = (q i +q +1 β i )ρ u r = (p r +p +1 a r)ρ where = Then, by (7), we have + p +1 b i, for,...,, + (q +1 α r )ρ, for,...,, u o = p o ρ, (13) (p r +p +1 a r)y ro +p o =1+ q +1 α r y ro. (14) In addition, θ u r y ro u o = ρ (p r +p +1 a r)y ro + z 2 (q +1 α r ) p o ρ = ρ ( (p r +p +1 a r)y ro p ( 1) o )+ρ = z 2 () + ρ ( 1) =ρ. lo V i x ij u r y rj + u o ρ (q i +q +1 β i )x ij + 1 p +1 b ix ij 1 ρ (p r +p +1 a r)y rj 1 ρ (q +1 α r )y rj + 1 p o ρ [ρ ( q i x ij p r y rj +p o ) +ρ ( + So that by (10)wehave q +1 β i x ij p +1 a ry rj ) p +1 b ix ij ρ (q +1 α r )y rj]. (17) (18) V i 0,,...,, u r 0,,...,. (15) lo, by (12) and(13), we obtain V i x ij u r y rj + u o V i x io = ρ (q i +q +1 β i)x io + 1 (p +1 b i)x io = ρ () + 1 ( ρ )=1. (16) 1 [ρ ( (p +1 a r +q +1 α r )y rj (q +1 β i +p +1 b i)x ij )
4 pplied Matheatic +ρ ( + q +1 β i x ij p +1 a ry rj ) p +1 b ix ij ρ (q +1 α r )y rj ] [ρ q +1 α r y rj ρ p +1 b ix ij + p +1 b ix ij ρ (q +1 α r )y rj] [( ρ ) p +1 b ix ij +ρ (1 ) q +1 α ry rj ]. Therefore, V i x ij u r y rj + u o 1 [( ρ ) p +1 b ix ij +ρ (1 ) q +1 α r y rj] 0. (19) (20) Conequently, (u, v, u o ),whereu =(u 1,...,u ) and v = (V 1,...,V ), i a feaible olution for odel (1), which for. θ u r y ro u o = ρ (p r +p +1 a r)y ro + z 2 (q +1 α r ) p o ρ = ρ ( (p r +p +1 a r)y ro p ( 1) o )+ρ = z 2 () + ρ ( 1) =ρ. (21) Now, let (u, v,u o ) be an optial olution for odel (1); then (p, q, p o ),wherep = (p 1,...,p +1 ) and q = (q 1,...,q +1 ),withp r = u r, r,...,; p +1 = 0; q i = V i,,...,; q +1 =0; p o =u o, i a feaible olution for odel (2), which for θ = u r y ro u o =+1 p r y ro p o ρ.thuθ =ρ. Theore 2. Let each DMU have + 1 input and + 1 output with condition (5) and (6). Then the optial objective function value of the following odel: w = in.t +1 +1 +1 q i x io q o, p r y ro =1 +1 q i x io p r y ro q o 0, j=1,...,n, p r 0, q i 0,,...,+1,,...,+1 (22) i equal to the optial objective function value of the following odel (4). Proof. Let (p 1,...,p +1,q 1,...,q +1,q o ) be an optial olution for odel (22); then we have +1 ω = +1 +1 q i x io q o (23) p r y ro =1 (24) q i x ij p r y rj q o 0. (25) y (6)and(15), it follow that ω = (q i +q +1 β i )x io q o q +1 α r y ro. (26) lo,by (5) and(16) it conclude that Now, let (p r +p +1 a r )y ro V i = (q i +q +1 β i )w u r = (p r +p +1 a r)w p +1 b i x io =1. (27) + w p +1 b i, for,...,, + (q +1 α r ), for,...,, V o = q o w, (28) where = (q i +q +1 β i )x io q o, =1+ q +1 α r y ro. (29)
pplied Matheatic 5 Then, by (7), we have V i 0,,...,, u r 0,,...,. lo, by (26)and(27), we obtain (30) [w q +1 α r y rj w p +1 b ix ij +w p +1 b ix ij (q +1 α r )y rj] u r y ro = w (p r +p +1 α r )y ro In addition + w V i x ij u r y rj V o (q +1 α r) y ro = w () + ( w ) =1. (31) Therefore, [w ( 1) p +1 b ix ij +(w ) q +1 α r y rj]. (33) + w w w (q i +q +1 β i)x ij p +1 b ix ij (p r +p +1 a r)y rj 1 (q +1 α r )y rj 1 q o w [w ( q i x ij p r y rj q o ) +w ( q +1 β i x ij p +1 a ry rj ) (32) V i x ij u r y rj V o 1 [w ( 1) p +1 b ix ij +(w ) q +1 α r y rj] 0. (34) Conequently, (u, v, V o ),whereu = (u 1,...,u ) and v = (V 1,...,V ), i a feaible olution for odel (4), which for z V i x io V o So that by (14), we have +w p +1 b ix ij (q +1 α r )y rj]. V i x ij u r y rj V o 1 [w ( (p +1 a r +q +1 α r )y rj +w ( (q +1 β i +p +1 b i)x ij ) q +1 β i x ij p +1 a ry rj ) +w p +1 b ix ij (q +1 α r )y rj] = w (q i +q +1 β r)x io + w (p +1 b i )x io q o w = w () + w ( 1) =w. (35) Now let (u, v, V o ) be an optial olution for odel (4), and then (p, q, p o ),wherep = (p 1,...,p +1 ) and q = (q 1,...,q +1 ),withp r = u r, r,...,;p +1 = 0; q i = V i, i,...,; q +1 =0; p o = V o, i a feaible olution for odel (22), which for ω +1 p r y ro p o = u r y ro u o =z.thuz =w. Theore 3. Let each DMUhave +1input and +1output with condition (5) and (6).
6 pplied Matheatic Then, the optial objective function value of the following odel: ρ =ax.t +1 +1 +1 p r y ro, q i x io =1 p r y rj +1 q i x ij 0, j=1,...,n p r 0, q i 0,,...,+1,,...,+1 (36) i equal to the optial objective function value of the following odel (1). Proof. Thi proof i iilar to the proof of Theore 1. Theore 4. Let each DMUhave +1input and +1output with condition (5) and (6). Then, the optial objective function value of the following odel: w =in.t +1 +1 +1 q i x io, p r y ro =1 +1 q i x ij p r y rj 0, j=1,...,n, p r 0, q i 0,,...,+1,,...,+1 (37) i equal to the optial objective function value of the following odel (2). Proof. Thi proof i iilar to the proof of Theore 2. 4. Illutrative Exaple In thiection, we uethedata recordedin Table 1 to illutrate that the addition or deletion of a cro-redundant output variable and input variable doe not affect the efficiency etiate yielded by the CCR or CC odel. Thee correpond to 20 DMU, whoe efficiency i aeed uing four input and four output where x 4j =(x 1j +x 2j +2x 3j ) Table 1: Dataet. Inp 1 Inp 2 Inp 3 Inp 4 Out 1 Out 1 Out 3 Out 4 Unit 1 7 1 4 12.75 1 2.5 3 1.25 Unit 2 3 7 4 14.75 2.5 1 3 1.25 Unit 3 6 6 3 14.25 2.5 2 3 2.25 Unit 4 3 1 3 1.75 4 5.5 7 6.75 Unit 5 6 0.5 3 5.25 5 3.5 6 5.75 Unit 6 4 0.5 3 3.5 2 6 6 5.5 Unit 7 1.5 2.5 3 1.5 6 4 7 7 Unit 8 0.5 4 4 6.25 1.5 5 6 4.25 Unit 9 2.75 1.75 4 4 8 3 6 6.5 Unit 10 1 3 3 1 8 3 7 7.5 Unit 11 2 2 3 1.25 5.5 5 7 7.25 Unit 12 2.5 1.5 3 2 7 3 6 6.5 Unit 13 4.5 1.5 6 13 4 2 4 2 Unit 14 2 4 7 16.25 1.5 2 4 0.25 Unit 15 4 3 6 12.25 6.5 3.5 3.5 3.75 Unit 16 2 5 4 8.75 5 3.5 4 4.25 Unit 17 1.5 6 4 8.5 4.5 4.5 5 5 Unit 18 0.5 4 3 3 3.5 5.5 6 6 Unit 19 3.5 0.75 3 2.5 7.5 2.5 6 6.5 Unit 20 6 3.5 4 11 3.5 3.5 6 4.5 Table 2: Exaple reult. θ ρ z ρ Unit 1 0.3461538 0.3461538 0.7500000 0.7500000 Unit 2 0.3214286 0.3214286 0.7500000 0.7500000 Unit 3 0.4285714 0.4285714 1.0000000 1.0000000 Unit 4 1.0000000 1.0000000 1.0000000 1.0000000 Unit 5 1.0000000 1.0000000 1.0000000 1.0000000 Unit 6 1.0000000 1.0000000 1.0000000 1.0000000 Unit 7 1.0000000 1.0000000 1.0000000 1.0000000 Unit 8 1.0000000 1.0000000 1.0000000 1.0000000 Unit 9 0.9973190 0.9973190 1.0000000 1.0000000 Unit 10 1.0000000 1.0000000 1.0000000 1.0000000 Unit 11 1.0000000 1.0000000 1.0000000 1.0000000 Unit 12 1.0000000 1.0000000 1.0000000 1.0000000 Unit 13 0.4444444 0.4444444 0.6666667 0.6666667 Unit 14 0.3809524 0.3809524 0.6666667 0.6666667 Unit 15 0.5607702 0.5607702 0.5714286 0.5594240 Unit 16 0.6052279 0.6052279 0.7500000 0.7500000 Unit 17 0.6847156 0.6847156 0.7500000 0.7500000 Unit 18 1.0000000 1.0000000 1.0000000 1.0000000 Unit 19 1.0000000 1.0000000 1.0000000 1.0000000 Unit 20 0.6428571 0.6428571 0.7500000 0.7500000 (0.5y 1j + 0.5y 2j + 0.5y 3j ); j=1,...,n, y 4j = (0.5y 1j + 0.5y 2j + 0.5y 3j ) 0.5x 3j ; j=1,...,n. (38) In other word, the forth input and the forth output are croredundant variable. In Table 2, θ, z, ρ,and ρ,repectively, record the efficiency eaure provided by odel (1), odel (3), odel (7), and odel (36). It i evident fro Table 2 that the addition or deletion of cro-redundant output variable
pplied Matheatic 7 and/or input variable doe not affect the efficiency etiate yielded by the input-oriented CCR or CC ultiplier odel. 5. Concluion In thi paper, we have tudied the effect of the cro redundancy between interval cale input and output variable on the efficiency etiate yielded by the CCR ultiplier odel in input- and output-oriented verion and the CC ultiplier odel in input- and output-oriented verion. We proved that the addition or deletion of a cro-redundant output variable and input variable doe not affect the efficiency etiate yielded by the input-oriented CC ultiplier odel and the output-oriented CC ultiplier odel. Siilarly, it can be proved that the addition or deletion of cro-redundant variable doe not affect efficiency etiate yielded by the CCR ultiplier odel in input- and outputoriented verion. Reference [1] K. Lee and K. Choi, Cro redundancy and enitivity in DE odel, Productivity nalyi,vol.34,no.2,pp.151 165, 2010. [2] F. J. López, Generalizing cro redundancy in data envelopent analyi, European Operational Reearch, vol. 214, no. 3, pp. 716 721, 2011. [3].Charne,W.W.Cooper,andE.Rhode, Meauringthe efficiency of deciion aking unit, European Operational Reearch, vol. 2, no. 6, pp. 429 444, 1978. [4] R. D. anker,. Charne, and W. W. Cooper, Soe odel for etiating technical and cale inefficiencie in data envelopent analyi, Manageent Science, vol. 30, no. 9, pp. 1078 1092, 1984.
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