- Email: [email protected]
A comment on dynamic DEA
Applied Mathematics and Computation 213 (2009) 275–276
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A comment on dynamic DEA Rolf Färe, Shawna Grosskopf * Department of Economics, Oregon State University, Corvallis, OR 97331, USA
a r t i c l e
i n f o
a b s t r a c t This brief comment provides a modification of the dynamic DEA model of Amirteimoori which appeared recently in this journal. The goal is to more directly allow the actions in the present period to have impacts on the future. Ó 2009 Elsevier Inc. All rights reserved.
Keywords: Dynamic DEA Revenue efficiency
1. Introduction In a recent paper in this journal, Amirteimoori [1] states that he wishes to ‘. . .develop a new DEA with dynamic revenue efficiency’ (p. 22). Unfortunately, the model developed is not dynamic, but rather the sum of two Shephard1 static revenue maximization models. The purpose of this comment is to show how the Amirteimoori model can be made dynamic in the sense that actions today have an impact on the future. We make use of the dynamic model discussed in Färe and Grosskopf [3]. 2. The Amirteimoori model For our purpose it suffices to study a two period, t ¼ 1; 2, version of the Amirteimoori primal model, i.e., (8), which is the dual formulation of (7). The so-called dynamic revenue for DMU p can be computed as2
max
s:t:
f1 f1 þ q1 h1
þ f2 f2 þ q2 h2
period 1 n P k1j x1j 5 x1p j¼1 n P
period 2 n P k2j x2j 5 x2p ;
ð1Þ
j¼1
k1j k0j 5 k0p
j¼1
f1 5
n P
k2j k1j 5 k1p ;
j¼1 n P
k1j k1j
f2 5
j¼1
h1 5
n P
n P
k2j k2j ;
j¼1
k1j y1j
j¼1
ktj ; ft ; ht = 0;
h2 5
n P
k2j y2j ;
j¼1
t ¼ 1; 2:
* Corresponding author. E-mail address: [email protected] (S. Grosskopf). 1 The primal and dual revenue maximization models (5) and (6) can be found in Shephard [2, p. 288]. 2 The variables are ft output prices, qt input prices. xtj ; ytj input and output quantities, respectively, ft ; ht final outputs. ktj represent input/outputs, depending upon the period. ktj are intensity variables. For details, see [1, p. 24]. 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.03.017
276
R. Färe, S. Grosskopf / Applied Mathematics and Computation 213 (2009) 275–276
With the two periods side by side one can see that the output in period t ¼ 1 does not have any impact on period t ¼ 2. Although k1j appear in both periods this is just data and hence no endogenous interaction occurs between periods, i.e., the optimization problem can be equivalently written as
max
f1 f1 þ q1 h1
s:t:
period 1;
max
f2 f2 þ q2 h2
s:t:
period 2
ð2Þ
and the solution is their sum. Hence, the Amirteimoori model is the sum of two Shephard [2] revenue models. Following Färe and Grosskopf [3] we may reformulate the model above in order to transform it into a truly dynamic modf i f i el, i.e., by allowing for intertemporal resource allocation. Let f1 ¼ k1 þ k1 where k1 is final output and k1 is investment or intermediate output. i We may now add the investment portion of output k1 to the k1p input in period t ¼ 2, so that n X
i
k2j k1j 5 k1p þ k1 :
ð3Þ
j¼1
In this specification, actions taken in period t ¼ 1 now have consequences in period t ¼ 2. By reducing consumption of f i f final output from f1 to k1 and investing the residual k1 , we have a dynamic model. It remains to enter k1 into the objective function so that for period t ¼ 1 we have
max
f
g 1 k1 þ g 1 h1 :
ð4Þ
A similar adjustment can be made for t ¼ 2. Alternatively, one may formulate a dynamic model by taking the model in Färe and Grosskopf [3, p. 164] together with intertemporal revenue maximization. References [1] A. Amirteimoori, Data envelopment analysis in dynamic framework, Appl. Math. Comput. 181 (2006) 21–26. [2] R.W. Shephard, Theory of Cost and Production Functions, Princeton University Press, Princeton, NJ, 1970. [3] R. Färe, S. Grosskopf, Intertemporal Production Frontiers: with Dynamic DEA, Kluwer Academic Publishers, Boston, MA, 1996.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A comment on dynamic DEA Rolf Färe, Shawna Grosskopf * Department of Economics, Oregon State University, Corvallis, OR 97331, USA
a r t i c l e
i n f o
a b s t r a c t This brief comment provides a modification of the dynamic DEA model of Amirteimoori which appeared recently in this journal. The goal is to more directly allow the actions in the present period to have impacts on the future. Ó 2009 Elsevier Inc. All rights reserved.
Keywords: Dynamic DEA Revenue efficiency
1. Introduction In a recent paper in this journal, Amirteimoori [1] states that he wishes to ‘. . .develop a new DEA with dynamic revenue efficiency’ (p. 22). Unfortunately, the model developed is not dynamic, but rather the sum of two Shephard1 static revenue maximization models. The purpose of this comment is to show how the Amirteimoori model can be made dynamic in the sense that actions today have an impact on the future. We make use of the dynamic model discussed in Färe and Grosskopf [3]. 2. The Amirteimoori model For our purpose it suffices to study a two period, t ¼ 1; 2, version of the Amirteimoori primal model, i.e., (8), which is the dual formulation of (7). The so-called dynamic revenue for DMU p can be computed as2
max
s:t:
f1 f1 þ q1 h1
þ f2 f2 þ q2 h2
period 1 n P k1j x1j 5 x1p j¼1 n P
period 2 n P k2j x2j 5 x2p ;
ð1Þ
j¼1
k1j k0j 5 k0p
j¼1
f1 5
n P
k2j k1j 5 k1p ;
j¼1 n P
k1j k1j
f2 5
j¼1
h1 5
n P
n P
k2j k2j ;
j¼1
k1j y1j
j¼1
ktj ; ft ; ht = 0;
h2 5
n P
k2j y2j ;
j¼1
t ¼ 1; 2:
* Corresponding author. E-mail address: [email protected] (S. Grosskopf). 1 The primal and dual revenue maximization models (5) and (6) can be found in Shephard [2, p. 288]. 2 The variables are ft output prices, qt input prices. xtj ; ytj input and output quantities, respectively, ft ; ht final outputs. ktj represent input/outputs, depending upon the period. ktj are intensity variables. For details, see [1, p. 24]. 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.03.017
276
R. Färe, S. Grosskopf / Applied Mathematics and Computation 213 (2009) 275–276
With the two periods side by side one can see that the output in period t ¼ 1 does not have any impact on period t ¼ 2. Although k1j appear in both periods this is just data and hence no endogenous interaction occurs between periods, i.e., the optimization problem can be equivalently written as
max
f1 f1 þ q1 h1
s:t:
period 1;
max
f2 f2 þ q2 h2
s:t:
period 2
ð2Þ
and the solution is their sum. Hence, the Amirteimoori model is the sum of two Shephard [2] revenue models. Following Färe and Grosskopf [3] we may reformulate the model above in order to transform it into a truly dynamic modf i f i el, i.e., by allowing for intertemporal resource allocation. Let f1 ¼ k1 þ k1 where k1 is final output and k1 is investment or intermediate output. i We may now add the investment portion of output k1 to the k1p input in period t ¼ 2, so that n X
i
k2j k1j 5 k1p þ k1 :
ð3Þ
j¼1
In this specification, actions taken in period t ¼ 1 now have consequences in period t ¼ 2. By reducing consumption of f i f final output from f1 to k1 and investing the residual k1 , we have a dynamic model. It remains to enter k1 into the objective function so that for period t ¼ 1 we have
max
f
g 1 k1 þ g 1 h1 :
ð4Þ
A similar adjustment can be made for t ¼ 2. Alternatively, one may formulate a dynamic model by taking the model in Färe and Grosskopf [3, p. 164] together with intertemporal revenue maximization. References [1] A. Amirteimoori, Data envelopment analysis in dynamic framework, Appl. Math. Comput. 181 (2006) 21–26. [2] R.W. Shephard, Theory of Cost and Production Functions, Princeton University Press, Princeton, NJ, 1970. [3] R. Färe, S. Grosskopf, Intertemporal Production Frontiers: with Dynamic DEA, Kluwer Academic Publishers, Boston, MA, 1996.