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An enhanced procedure for estimating returns-to-scale in DEA

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An enhanced procedure for estimating returns-to-scale in DEA
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  An enhanced procedure forestimating returns-to-scale in DEA G.R. Jahanshahloo  a , M. Soleimani-damaneh  a,* ,M. Rostamy-malkhalifeh  b a Faculty of Mathematical Sciences and Computer Engineering, Teacher Training University,599 Taleghani avenue, Tehran 15618, Iran b Department of Mathematics, Azad University, Science & Research Branch, Tehran, Iran Abstract In this paper, by improving the Golany and Yu  s method [Estimating returns to scalein DEA, European Journal of Operational Research 103 (1) (1997) 28–37], a new algo-rithm to estimate returns-to-scale (RTS) in data envelopment analysis (DEA) models isproposed. We show that the proposed algorithm overcomes existing disadvantages inthe Golany and Yu  s method, introduced for this purpose. The advantages of thenew algorithm are illustrated.   2005 Elsevier Inc. All rights reserved. Keywords:  DEA; Returns-to-scale; Efficiency 1. Introduction Data envelopment analysis (DEA) is a mathematical programming method-ology for assessing the relative efficiencies of a number of decision making units 0096-3003/$ - see front matter    2005 Elsevier Inc. All rights reserved.doi:10.1016/j.amc.2005.01.136 * Corresponding author. E-mail address:  soleimani_d@yahoo.com (M. Soleimani-damaneh). Applied Mathematics and Computation 171 (2005) 1226–1238 www.elsevier.com/locate/amc  (DMUs), in which each DMU uses multiple inputs to produce outputs. Esti-mating returns-to-scale (RTS) is a most important topic in DEA. There area number of approaches to do this. At first, Banker [1] showed how theCCR formulation can be employed to estimate RTS. Also, Banker et al. [3]introduced BCC model, under a variable returns-to-scale (VRS) technology.Banker and Thrall [5] proved the equivalence of the CCR method of RTS esti-mation in Banker [1] and the BCC method of RTS estimation in Banker et al.[3]. Both of these methods had problems in the presence of alternative optimalsolutions; so, they were improved by Banker et al. [2] and Jahanshahloo andSoleimani-damaneh [8]. Fa¨re and Grosskopf  [6], Golany and Yu [7], Kerstens and Vanden Eeckaut [9], Jahanshahloo and Soleimani-damaneh [8], and Banker et al. [4] provided alternative methods for the estimation of RTS inDEA.Golany and Yu [7] introduced an approach for determining left and rightRTS by testing the existence of solutions in four regions defined in the neigh-borhood of the analyzed unit. In this approach they use two LP-models to esti-mate RTS for each DMU, but they fail to do so when at least one of thesemodels is infeasible. This method is reviewed in Section 2 of this paper. In Sec-tion 3 we introduce some basic theorems and definitions and an enhanced pro-cedure which can be used to estimate RTS in problematic cases in whichGolany and Yu  s method (GY method hereafter) is not applicable. Our pro-posed procedure is compared with GY method, in Section 4 of this paper, usingtwo illustrative examples. Finally, some conclusions are provided in Section 5. 2. Golany and Yu  s method Suppose that we have  n  DMUs, where each DMU  j  ,  j   = 1, . . . , n , produces thesame  s  outputs in (possibly) different amounts  y rj  ,  r  = 1, . . . , s , using the same  m inputs  x ij  ,  i   = 1, . . . , m , also in (possibly) different amounts. The efficiency scoreof a specific DMU 0  can be evaluated by the CCR or BCC model of DEA, inthe input and output orientation as follows:(Input-oriented):  z  in  ¼  min  h   X mi ¼ 1  s  i  þ X  sr  ¼ 1  s þ r  !  ð 1 Þ s : t : X n j ¼ 1 k  j  x ij  þ  s  i  ¼  h  x i 0 ;  i  ¼  1 ;  . . .  ;  m ;  ð 1a Þ X n j ¼ 1 k  j  y  rj    s þ r   ¼  y  r  0 ;  r   ¼  1 ;  . . .  ;  s ;  ð 1b Þ  s  i  ;  s þ r   P 0 ;  8 i ;  r  ; k  ¼ ð k 1 ;  . . .  ; k n Þ 2  K ; G.R. Jahanshahloo et al. / Appl. Math. Comput. 171 (2005) 1226–1238  1227  and(Output-oriented):  z  out  ¼  max  u þ  X mi ¼ 1 t   i  þ X  sr  ¼ 1 t  þ r  !  ð 2 Þ s : t : X n j ¼ 1 l  j  x ij  þ t   i  ¼  x i 0 ;  i  ¼  1 ;  . . .  ;  m ;  ð 2a Þ X n j ¼ 1 l  j  y  rj   t  þ r   ¼  u  y  r  0 ;  r   ¼  1 ;  . . .  ;  s ;  ð 2b Þ t   i  ;  t  þ r   P 0 ;  8 i ;  r  ; l  ¼ ð l 1 ;  . . .  ; l n Þ 2  ; where K CRS ¼  k  :  k  j P 0 ;  j  ¼  1 ;  . . .  ;  n  ;  ð 3 Þ K VRS ¼  k  :  k  2  k CRS ; X n j ¼ 1 k  j  ¼  1 () ;  ð 4 Þ   CRS ¼  l  :  l  j P 0 ;  j  ¼  1 ;  . . .  ;  n  ;  ð 5 Þ and   VRS ¼  l  :  l  2    CRS ; X n j ¼ 1 l  j  ¼  1 () .  ð 6 Þ In each of the above models DMU 0  is considered an efficient DMU if   z  = 1and  slacks  = 0 in all of the optimal solutions of the respective model.Golany and Yu introduced an approach to estimate left RTS and right RTSfor each unit by testing the existing of solutions in different regions. They usedthe following models to determine the RTS to the right and the left of DMU 0 ,respectively:min  b   X mi ¼ 1  s  i  þ X  sr  ¼ 1  s þ r  !  ð 7 Þ s : t : X n j ¼ 1 k  j  x ij  þ  s  i  ¼  b  x i 0 ;  i  ¼  1 ;  . . .  ;  m ; X n j ¼ 1 k  j  y  rj    s þ r   ¼ ð 1 þ d Þ  y  r  0 ;  r   ¼  1 ;  . . .  ;  s ; X n j ¼ 1 k  j  ¼  1 ; k  j ;  s  i  ;  s þ r   P 0 ;  8  j ;  i ;  r  ; 1228  G.R. Jahanshahloo et al. / Appl. Math. Comput. 171 (2005) 1226–1238  andmax  a þ  X mi ¼ 1  s  i  þ X  sr  ¼ 1  s þ r  !  ð 8 Þ s : t : X n j ¼ 1 k  j  x ij  þ  s  i  ¼ ð 1  g Þ  x i 0 ;  i  ¼  1 ;  . . .  ;  m ; X n j ¼ 1 k  j  y  rj    s þ r   ¼  a  y  r  0 ;  r   ¼  1 ;  . . .  ;  s ; X n j ¼ 1 k  j  ¼  1 ; k  j ;  s  i  ;  s þ r   P 0 ;  8  j ;  i ;  r  ; where,  d  and  g  are arbitrary small positive numbers. Then they proposed thefollowing procedure to estimate RTS of DMU 0 . Hereafter, the superscript‘‘*’’ indicates the optimal value of the related variable. (GY procedure): Step 1. Solve (7) for the estimation of the RTS to the right of DMU 0 :1(i). (1 +  d ) >  b * > 1  )  increasing RTS.1(ii). 1 P b * )  DMU 0  is BCC-inefficient.1(iii). (1 +  d ) =  b * )  constant RTS.1(iv). (1 +  d ) <  b * )  decreasing RTS.1(v). No feasible solution  )  failure of method.Step 2. Solve (8) for the estimation of the RTS to the left of DMU 0 :2(i). 1 >  a * > (1  g )  )  decreasing RTS.2(ii).  a * P 1  )  DMU 0  is BCC-inefficient.2(iii). (1  g ) =  a * )  constant RTS.2(iv).  a * < (1  g )  )  increasing RTS.2(v). No feasible solution  )  failure of method. 3. Enhanced procedure 3.1. Basic theorems and definitions We start our discussion in this section with CCR method for determiningRTS as follows: G.R. Jahanshahloo et al. / Appl. Math. Comput. 171 (2005) 1226–1238  1229  (CCR method to estimate RTS): Assuming that ( x 0 , y 0 ) is a BCC-efficient point, the following conditionsidentify the situation of RTS at this point:(i) Increasing RTS (IRS) prevail  () P n j ¼ 1 k   j  <  1 for all optimal solutionsof the input-oriented CCR model.(ii) Decreasing RTS (DRS) prevail  () P n j ¼ 1 k   j  >  1 for all optimal solu-tions of the input-oriented CCR model.(iii) Constant RTS (CRS) prevail  () P n j ¼ 1 k   j  ¼  1 in any optimal solution of the input-oriented CCR model.See [5] for validity of this method. Now, in the following theorem we extendthis method to the output-oriented case. Theorem 1.  Assuming that ( x 0 , y 0 ) is a BCC-efficient point, the following conditions about the situation of RTS at this point are correct: (i) P n j ¼ 1 l   j  <  1  for all optimal solutions of the output-oriented CCR model   ) IRS. (ii)  DRS   ) P n j ¼ 1 l   j  >  1  for all optimal solutions of the output-oriented CCRmodel. (iii)  CRS   () P n j ¼ 1 l   j  ¼  1  in any optimal solution of the output-oriented CCRmodel. Proof.  To prove part (i), by contradiction we have P n j ¼ 1 k   j  P 1 in any optimalsolution of the input-oriented CCR model. For this optimal solution, by (1a)and (1b) in the CRS case we have, X n j ¼ 1 k   j h   x ij  þ  s  i h   ¼  x i 0 ;  i  ¼  1 ;  . . .  ;  m ; X n j ¼ 1 k   j h   y  rj    s þ r  h   ¼  1 h   y  r  0  ¼  u   y  r  0 ;  r   ¼  1 ;  . . .  ;  s .Using these statements and by noting    as a non-Archimedean infinitesimal,( l  ¼  k  h  ,  t  ¼  s  h   ,  t þ ¼  s þ h   ,  u  ¼  1 h  ) is an optimal solution for the output-orientedCCR model and P n j ¼ 1 l  j  ¼  1 h  P n j ¼ 1 k   j  P 1, while this contradicts the assump-tion and proof of part (i) is completed. Part (ii) can be proved by contradictionand using (2a) and (2b) in the same way. Proof of part (iii) is not difficultregarding that CRS prevail if and only if DMU 0  is CCR efficient.  h Now we focus on parts 1(v) and 2(v) in the GY procedure and on the valuesof   d  and  g  in models (7) and (8). Feasibility or infeasibility of these models isdependent on  d  and  g  values. 1230  G.R. Jahanshahloo et al. / Appl. Math. Comput. 171 (2005) 1226–1238
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