Directory UMM :Data Elmu:jurnal:S:Socio-Economic Planning Sciences:Vol34.Issue1.Mar2000:

Socio-Economic Planning Sciences 34 (2000) 35±49
www.elsevier.com/locate/orms

Network DEA
Rolf FaÈre a,b,*, Shawna Grosskopf b
a

Department of Agricultural and Resource Economics, Oregon State University, Corvallis, OR 97331-3612, USA
b
Department of Economics, Oregon State University, Corvallis, OR 97331-3612, USA

Abstract
Most traditional DEA models treat their reference technologies as black boxes. Our network models,
developed for the Swedish Institute for Health Economics (IHE), allow us to look into these boxes and
to evaluate organizational performance and its component performance. The very general structure of
the network model allows us to apply this model to a variety of situations: intermediate products,
allocation of budgets or ®xed factors and certain (time separable) dynamic systems. # 2000 Elsevier
Science Ltd. All rights reserved.

1. Introduction
The traditional models for Data Envelopment Analysis (hereafter DEA)-type performance

measurement are based on thinking about production as a ``black box''. Inputs are
transformed in this box into outputs. The actual transformation process is generally not
modeled explicitly; rather, one simply speci®es what enters the box and what exits. This is, in
fact, one of the advantages of DEA Ð it reveals rather than imposes the structure of the
transformation process. Nevertheless, when researchers apply DEA to speci®c industries or
situations, they have often added more structure to the model to better suit the application.
Examples abound and include, among others: two (or more) stage models, hybrid models,
cone-assurance regions, windows, etc. The reader is referred to Charnes et al. [1] (see especially
pp. 425±36).
The variations on DEA mentioned above are generally intended to customize the model to
suit the application. Here, we focus, in a somewhat more general way, on the transformation
process in the black box. The general formulation we use is a network model, which has
* Corresponding author.
0038-0121/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 3 8 - 0 1 2 1 ( 9 9 ) 0 0 0 1 2 - 9

36

R. FaÈre, S. Grosskopf / Socio-Economic Planning Sciences 34 (2000) 35±49

proved fruitful in engineering and operations research applications, among others. More
speci®cally, in building on work by Shephard and FaÈre [2,3] on dynamic production
correspondences, FaÈre and Grosskopf [4] developed a sequence of network models that can be
used to address various re®nements of the standard DEA1 models.
In this paper, we present three network models. The ®rst, used by FaÈre et al. [8] to study
allocation of farmland to various crops, allows for allocation of a (®xed) factor or input
among alternative uses. This general structure could also be used to introduce allocation of a
budget or allocation of resources across units or branches. The second model, used by FaÈre
and Whittaker [9], explicitly models intermediate products, i.e., products produced and used
inside the technology. This model is also used by FaÈre et al. [10] to study alternative
organizational structures of a multiplant ®rm. The third network formulation is a dynamic
DEA model in which some outputs at period t are inputs in the next period, t + 1. This
provides an alternative to the dynamic DEA models proposed by Sengupta [11]. The dynamic
DEA model introduced here is used by FaÈre and Grosskopf [12] to study the dynamic
eciency of APEC (Asian-Paci®c Economic Community) countries.
Our goal is to show the usefulness of network DEA by bringing together a sequence of such
models in one paper. We begin by discussing the network technology in a ``heuristic'' way,
illustrating the various models with simple diagrams. Next, we formalize these technologies by
specifying them as a series of linear inequality constraints, familiar from DEA. Then, we turn
to the speci®cation of those models as performance measures based on distance functions

which may be estimated using traditional DEA or Farrell [13] eciency measures. A summary
concludes and discusses policy implications and directions for further research.

2. The heuristics of network models
In this section, we provide a heuristic road map to the di€erent network models discussed in
the current paper. We denote inputs by x=(x1,. . . ,xN) and outputs by y=( y1, . . .,yM ). The
simple static non-network model, often referred to as the ``black box,'' is illustrated ®rst (see
Fig. 1).
Here, inputs x are employed in the production process P to produce output y. P may be
modeled, in the simplest case, by a production function or as a DEA model in more complex
cases, as we illustrate in the next section. Independent of how P is modeled, there is no
information about what is taking place within the production process P. Only the
transformation of inputs into outputs x 4 y is modeled.
This static model can also be used to measure performance over time, as in Fig. 2. The
comparative static model takes technology and inputs as ®xed and exogenous in each period,
however (disembodied) technical change can occur over time. This idea has been used to model
productivity change in a DEA framework, as in FaÈre et al. [14]. They show how to use DEA
1
The terminology DEA was coined by Charnes et al. [5]. Interestingly, the activity analysis model due to von
Neuman [6], see Karlin [7], could be considered as a DEA model.

R. FaÈre, S. Grosskopf / Socio-Economic Planning Sciences 34 (2000) 35±49

37

Fig. 1. The static technology.

to compute and decompose Malmquist productivity indexes into changes in eciency
(``catching up'') and technical change (shifts in the frontier). See also FaÈre et al. [15].
Next, we turn to a related model that introduces the ``linkage'' among those processes
characterizing network models. Let us assume that there are two production processes, P 1 and
P 2, each producing an output vector y 1 and y 2, respectively. Moreover, assume that the two
processes use the same source of inputs x. In this case, one can analyze the allocation of x to
P 1 and P 2. In particular, if x 1 is employed by P 1 and x 2 by P 2, then their sum cannot exceed
x, i.e., x 1+x 2 E x. This type of model is known in agricultural economics as a model with
®xed (x ) but allocatable inputs (Fig. 3).
As an example, assume that y 1 is the output of corn and y 2 is the output of soybeans. Then,
land and other resources x can be allocated to production of y 1 and y 2 under the constraint
that total use of inputs does not exceed the given resources x. This model may be used to
determine the optimal (output-maximizing, revenue maximizing or pro®t maximizing)

allocation of land to crops. It could also be used to simulate the e€ect of set-aside programs,
for example.
The static network model, which we will discuss next, attempts to analyze the ``inside'' of a
production process and to explicitly model the black box. Subprocesses and intermediate goods
are the two new concepts required to model a network. In Fig. 4, adopted from FaÈre and
Grosskopf [4], we describe a network with three subprocesses, 1, 2 and 3. To these, a source 0
and an outlet or ``sink'' 4 are added. The source gives the network exogenous inputs x, which

Fig. 2. The comparative static technology.

38

R. FaÈre, S. Grosskopf / Socio-Economic Planning Sciences 34 (2000) 35±49

are allocated to the subprocesses 0i x, i = 1,2,3 with the constraint 10x ‡ 20x ‡ 30xEx. The lower
``prescript'' refers to the source of the input whereas the upper ``prescript'' is the destination or
point of use of the input.
The subprocess 1 produces the output vector y 1, of which some are intermediate outputs 31y
and some are ®nal outputs 41y, i.e., y1 E31y ‡ 41y. The notation means that process 1, the lower
index, produces intermediate outputs that are inputs in process 3, i.e., 31y. In addition, some of

its outputs are ®nal outputs, here represented with the upper index 4. In practice, intermediate
outputs may also be ®nal outputs, as in the case of spare parts. Subprocess or node 3, uses
network exogenous inputs 30x as well as intermediate inputs produced in subprocess 1 and 2,
i.e., 31y and 32y, respectively, to produce ®nal outputs 43y.
The sink 4 collects all ®nal outputs from the network, i.e., y ˆ 41y ‡ 42y ‡ 43y. By appropriately
adding zeros in the vectors, the problem of dimensionality is avoided. This type of network
model has been used to study the organization of Swedish Pharmacies, where the subprocesses
may represent di€erent types of pharmacies (at hospitals vs more standard commercial outlets
or regional headquarters which distribute drugs to local pharmacies, for example). As another
example, nodes 1 and 2 might represent production units that send their ``seconds'' to node 3,
which is the factory outlet. Nodes 1 and 2 send their ®nal products directly to market (4).
We now proceed to show how the network notion can be used to model dynamic production
processes or technologies. Suppose we have two periods and two production processes with
period-speci®c inputs and outputs. In addition, we assume that some of the output in the ®rst
period is used as input in the second, i.e., some outputs are time-intermediate products (Fig. 5).
The two production processes are denoted by Pt and P t + 1. Each uses time-speci®c inputs xt
and x t + 1 to produce time-speci®c ®nal outputs fy t and fy t + 1. In addition, Pt produces
intermediate outputs that are used as inputs in P t + 1. P t also uses inputs from the earlier
period t ÿ1, namely iy t ÿ 1. The total output from the dynamic model consists of ®nal outputs
(fyt+fyt+1 ) and intermediate inputs (iy t, iy t + 1). Clearly, there is a trade-o€ between

producing ®nal outputs today (t ) or tomorrow (t + 1). Note that this model can be applied to
study dynamic eciencies, as demonstrated by FaÈre and Grosskopf [12]. In their application,
country level aggregate production is allocated between consumption (the ®nal outputs) and
investment (the intermediate output). They solve the model for the optimal path of
consumption and investment, which maximizes discounted aggregate consumption over the
time horizon.

3. The formalistics of network models
The heuristic network models of the previous section are next formalized. In particular, we
model the networks as the constraint sets or reference technologies of DEA models. These
models have proven very useful for measuring eciency and productivity. As we shall see, the
network DEA model is not a single model but a family of models, with the common feature of
having linear constraints. In the next section, we will add an objective to these models in order
to transform them into DEA performance measures.
Assume that there are k = 1,. . . ,K Decision Making Units (DMU:s) or observations of
inputs and outputs (xk,yk )=(xk1, . . .,xkN, yk1, . . . ,ykM). The coecients (xkm,ykm )

R. FaÈre, S. Grosskopf / Socio-Economic Planning Sciences 34 (2000) 35±49

39

n = 1,. . . ,N,m = 1,. . .,M, k = 1,. . . ,K are required to satisfy certain properties (see Appendix
A). To formulate the DEA model from the data (x k,y k), we need to introduce intensity
variables zk, k = 1,. . . ,K, one for each observation or activity k. These nonnegative variables
tell us to what extent a particular DMU is involved in the production of outputs. The basic
model, written in terms of an output set, is:
(
K
K
X
X
zk x kn Ex n , n ˆ 1, . . . , N,
P…x† ˆ … y1 , . . . , yM †:ym E zk ykm , m ˆ 1, . . . , M,
kˆ1

kˆ1

…1†

)

zk e0, k ˆ 1, . . . , K :
The heuristics of this model are shown in Fig. 1. Although it satis®es certain properties, such
as constant returns to scale and free disposability of inputs and outputs (again, see Appendix
A), nothing particular can be said ex ante about its internal structure. For example, we cannot
determine how inputs are allocated to the production of the various outputs, or if intermediate
products are produced.
Our model ``Fixed but Allocative Inputs'' gives us some insights into how inputs may be
shared by di€erent processes, here, modeled by two output sets P 1 and P 2. For simplicity, we
assume that only the ®rst input can be allocated between the two processes or nodes, and that
the others are preassigned to a speci®c process for each DMU. Here, superscripts refer to the
process, P 1 or P 2. In this case, the heuristic model of Fig. 3 can be formalized as:
(


K
M ÿ
X
ÿ

X

1
1
1
2
2
y1m ‡ y2m : y1m E z1k y1km ,
P x^ 1 , x^ 2 , . . . , x^ N , x^ 2 , . . . , x^ N ˆ y1 ‡ y2 ˆ
mˆ1

m ˆ 1, . . . , M,

kˆ1

K
K
X
X
z1k x 1k1 Ex 11 ,
z1k x 1kn Ex^ 1n , n ˆ 2, . . . , N, z1k e0, k ˆ 1, . . . , K,

kˆ1

kˆ1

…2†

K
K
K
X
X
X
z2k x 2kn Ex^ 2n , n ˆ 2, . . . , N,
z2k x 2k1 Ex 21 ,
y2m E z2k y2km , m ˆ 1, . . . , M,
kˆ1

kˆ1

kˆ1

)

z2k e0, k ˆ 1, . . . , K, x 11 ‡ x 21 Ex^ 1 :
The main di€erence between the two models (1) and (2) is that in (2) the allocation of the
®rst input xÃ1 between the two subprocesses is not given a priori like the other inputs xà 12,. . . xà 1N,
xà 22, . . .xà 2N. Rather, we only require that the allocation across subprocesses be feasible, which is
the interpretation of the last inequality in (2). The model in (1) does not provide for such
allocation, since it can be viewed as an aggregation of (2) that obscures the subprocesses.
The full network model, illustrated in Fig. 4, includes intermediate products and allocatable
inputs. A product is intermediate to the production system if it is both produced and
consumed, i.e., it is both an output and an input, within the network. Of course, not all the

40

R. FaÈre, S. Grosskopf / Socio-Economic Planning Sciences 34 (2000) 35±49

intermediate goods are necessarily consumed or used up within the network; they may be ®nal
output as well. Again, spare parts is a typical example of the latter.
We next illustrate the network in Fig. 4, which has three producing subprocesses, a source,
and an outlet, giving us a total of ®ve nodes (0,. . . ,4). Let us denote total available
(exogenous) inputs by x and let 0i x, i = 1,2,3 denote the amount of the vector of (exogenous)
inputs that is allocated to node i. The source node models the constraints for the allocation of
the exogenous inputs; in particular:
xe

3
X
i
iˆ1

0

x

…3†

or
1
2
3
x n e x n ‡ x n ‡ x n , n ˆ 1, . . . , N:
0
0
0
Denote the vector of outputs produced by subprocess or subtechnology i and delivered to
node j by jiy. Returning to Fig. 4, we see that the total production of node 1 is 31y ‡ 41y, where
4
3
1y is its output of intermediate products and 1y is its ®nal output. Node 1 does not use any
intermediate products as inputs. Node or subprocess 3, however, uses inputs from both node 1
…31y† and node 2 …32y† as well as exogenous inputs 30x. This node produces only ®nal outputs 43y.
The outlet or collection node 4, given that each subtechnology produces distinct output
1
2
3
vectors, 4j y $ RMj
+ , j = 1,2,3, where M=M +M +M , can be written as


4 4 4
y ˆ y, y, y :
…4†
1 2 3
If we don't
that each node produce distinct outputs, total production can be written as
P3insist
4
the sum
jˆ1 j y of the individual nodes' outputs. The appropriate number of zeros must be
added.

Fig. 3. The resource constraint technology.

R. FaÈre, S. Grosskopf / Socio-Economic Planning Sciences 34 (2000) 35±49

41

The piecewise linear or DEA technology associated with k = 1, . . .,K observations may be
written in terms of the output set as:


4 4 4
P…x† ˆ f y ˆ y, y, y :
1 2 3
…a†

…b†

Node or subprocess 3

…c†

…d†
…e†
…f †

Node or subprocess 1

…g†
…h†
…i†

Node or subprocess 2

…j†
…k†

Distribution of exogenous inputs

…l†

K
X
4
34
ym E z ykm , m ˆ 1, . . . , M3 ,
3
k3
kˆ1
K
X
33
3
z x kn E x n , n ˆ 1, . . . , N,
k0
0
kˆ1

K
X
33
3
z ykm E ym , m ˆ 1, . . . , M1 ,
k
1
1
kˆ1

K
X
3
33
z ykm E ym , m ˆ 1, . . . , M2 ,
2
k2
kˆ1

3
z e0, k ˆ 1, . . . , K,
k




K
X
3
4
1 3
4
:
ym ‡ ym E z
ykm ‡ ykm , m ˆ 1, . . . , M1 ,
1
1
k 1
1
kˆ1
K
X
11
1
z x kn E x n , n ˆ 1, . . . , N,
k0
0
kˆ1

1
z e0, k ˆ 1, . . . , K,
k




K
X
3
4
2 3
4
ym ‡ ym E z
ykm ‡ ykm , m ˆ 1, . . . , M2 ,
2
2
2
k
2
kˆ1
K
X
22
2
z x kn E x n , n ˆ 1, . . . , N,
k0
0
kˆ1

2
z e0, k ˆ 1, . . . , K,
k
1
2
3
x n ‡ x n ‡ x n Ex n , n ˆ 1, . . . , N g
0
0
0
…5†

In the network model (5), we can identify the three subtechnologies. The third,
®rst, P1 …10x†, is given by (f)±(h), and the last,
activity analysis model (1) with the network
model (5) shows that the former has one set of intensity variables while the latter has three sets
of such variables. Moreover, P(x ) has a distribution node that allows us to study optimal
P3 …30x, 31y, 32y), consists of expressions (a)±(e). The
P2 …20x†, by (i)±(k). A comparison of the standard

42

R. FaÈre, S. Grosskopf / Socio-Economic Planning Sciences 34 (2000) 35±49

Fig. 4. The network technology.

distribution of the exogenous inputs among subtechnologies, whereas the standard model does
not. Furthermore, the network model allows us to explicitly model intermediate inputs,
whereas the standard model does not.
FaÈre and Grosskopf [3] have shown that if the subtechnologies Pj, j = 1,2,3, satisfy
properties such as free disposability of inputs and outputs and constant returns to scale then,
so too, will the network technology (5).
Turning to the dynamic network model, illustrated by Fig. 5, we note that it consists of two
distinct subtechnologies, Pt and Pt + 1, one for each period. These subtechnologies are
interactive in a way similar to the nodes in the network model above. The output from Pt
consists of ®nal output fy t and intermediate output iyt. The latter is used as input at t + 1.
Thus, iyt is the ``investment'' from period t. The other inputs are the exogenous inputs at each
period xt and xt + 1. Although not considered here, some of these may be storable i.e., they
can be used in a later period (see FaÈre and Grosskopf [4]). The technology illustrated in Fig. 5
can be expressed as a dynamic activity analysis or DEA model as follows:





f t f t‡1 i t‡1
t
t‡1 i tÿ1
:
P x,x , y
ˆf y, y ‡ y

f

ytm

‡

i

ytm




K
X
i t
t f t
E zk ykm ‡ ym , m ˆ 1, . . . , M,
kˆ1

Fig. 5. The dynamic technology.

R. FaÈre, S. Grosskopf / Socio-Economic Planning Sciences 34 (2000) 35±49

43

K
X
ztk x tkn Ex tn , n ˆ 1, . . . , N,
kˆ1

K
X
i tÿ1 i tÿ1
E ym , m ˆ 1, . . . , M,
ztk ykm
kˆ1

ztk e0, k ˆ 1, . . . , K,


f

t‡1
ym

‡

i

t‡1
ym



E

K
X

zkt‡1

kˆ1


f

t‡1
ykm

‡

i

t‡1
ykm



, m ˆ 1, . . . , M,

K
X
t‡1
zkt‡1 x kn
Ex nt‡1 , n ˆ 1, . . . , N,
kˆ1

K
X
t
t
zkt‡1 i ykm Ei ym , m ˆ 1, . . . , M,
kˆ1

zkt‡1 e0, k ˆ 1, . . . , K g:

…6†

The dynamic DEA model (6) is related to the network model (5) in the sense that both have
multiple subtechnologies or nodes, and both have intermediate products, making the
subtechnologies interdependent. FaÈre and Grosskopf [5] have shown that if the subtechnologies
Pt and Pt+1 satisfy properties such as free disposability of inputs and outputs and constant
returns to scale, then so, too, does the dynamic model (6), i.e., the dynamic model inherits
properties from the subtechnologies.

4. Performance measures
The performance of a particular ®rm, observation, or DMU k ', can, in principle, be
evaluated under two di€erent situations: ®rst, when no price information is available, and
second, when prices are known. Of course, intermediate cases, when some prices are available,
are also possible. Here, we address the no-price situation, and use distance functions as our
yardsticks. Standard DEA or Farrell [13] technical eciency measures are closely related to
distance functions, as we show below. The most general of these functions is the technology
directional distance function, which is de®ned on the technology P(x ) by


~
…7†
D…x,
y;gx , gy † ˆ max b:… y ‡ bgy † 2 P…x ÿ bgx † ,

44

R. FaÈre, S. Grosskopf / Socio-Economic Planning Sciences 34 (2000) 35±49

where (ÿgx, gy ) is the direction in which the distance is measured.2 Less general, but better
known, are Shephard's [13] input and output distance functions, which are de®ned as


…8†
Di … y, x† ˆ max l:y 2 P…x=l†
and



D0 …x, y† ˆ min y:… y=y† 2 P…x† ,

…9†

respectively. The input and output distance functions are special cases of the directional
distance function (7). In particular, if we choose the direction (ÿgx,gy ) to be (x,0), then (7)
takes the form
~
D…x,
y; x, 0† ˆ 1 ÿ 1=Di … y, x†

…10†

i.e., the directional distance function becomes the input distance function. Similarly, if we take
(ÿgx,gy )=(0,y ), then
~
D…x,
y;0, y† ˆ …1=D0 …x, y†† ÿ 1

…11†

and the output distance function is obtained. The relation between the input and output
distance function is given by
D0 …x, y† ˆ 1=Di … y, x†

…12†

provided the technology exhibits constant returns to scale, i.e., P(lx )=l P(x ), l > 0.
We may illustrate the three distance functions (7), (8) and (9) as shown in Fig. 6. To do this,
we introduce the graph of technology T. It is de®ned in terms of output sets as
o
n
…13†
T ˆ …x, y†:y 2 P…x†, x 2 RN
‡ :

Clearly, P(x ) can be retrieved from T as


P…x† ˆ y:…x, y† 2 T :

…14†

The reference technology is given by T, the area between the ray from the origin and the xaxis. When the observation (x,y ) is evaluated by the directional distance function, the optimal
~
value is attained at A where b=D(x,
y; gx,gy ). The output distance function brings (x,y ) to B

where D0(x,y )=y while the input distance function takes (x,y ) to C where Di( y,x )=l.
To make our example a lot more concrete, suppose that the observed (x,y ) is (2,1), and we
choose the direction vector to be (ÿ2,1). If the frontier of technology is described by y=x,
then point B would have coordinates (2,2) and point C coordinates (1,1). The associated
ecient point using the directional distance function (point A) would have a value of (4/3,4/3).
The corresponding value of beta would be 1/3 for the directional distance function. The output

2
Luenberger [16,17] introduced the directional distance function, which he called the shortage function. Here, we
follow Chambers et al. [18].

R. FaÈre, S. Grosskopf / Socio-Economic Planning Sciences 34 (2000) 35±49

45

Fig. 6. Distance functions.

distance function in this case would have a value of 1/2 while the input distance function
would have a value of 2.
For our ®rst illustration of performance measurement, we choose to evaluate ®rm or DMU
k' relative to the static technology (1) by means of the Farrell output eciency measure [13].
This measure is the reciprocal of the output distance function,3 namely:
ÿ ÿ 0
ÿ 0




ÿ1
0
0
D0 x k , yk
ˆ F0 x k , yk ˆ max y
s:t:

yyk 0 m E

K
X
zk ykm , m ˆ 1, . . . , M,
kˆ1

…15†

K
X
zk x kn Ex k 0 n , n ˆ 1, . . . , N,
kˆ1

zk e0, k ˆ 1, . . . , K:

F0(x k ',yk ' ) is the solution to the linear programming problem (15). A value equal to one
signals eciency while a value larger than one signals ineciency. Output could be
proportionally increased by (F0(xk 'yk ' )-1) if the DMU k ' is inecient.
A more sophisticated measure than the Farrell measure is the Malmquist productivity index
[14]. As a measure of productivity change, it tells us how much the ratio of aggregate output
to aggregate input (an index of average product) has changed between any two time periods.
This index can be input or output oriented, dependent on whether input or output distance
functions are used. Here, we discuss the output oriented case and start with an illustration
adopted from FaÈre and Grosskopf [4] (see Fig. 7).
The two input±output observations (x t,y t) and (x t + 1,y t + 1) belong to technologies T t and
3
This is the interpretation of FaÈre, et al. [18]. Farrell's original output measure is the output distance function
[13].

46

R. FaÈre, S. Grosskopf / Socio-Economic Planning Sciences 34 (2000) 35±49

T t+1, respectively. The eciency change between periods is represented by:
ÿ




D0t‡1 x t‡1 , yt‡1
Oc Oe
ˆ
:
EFFCH ˆ
Dt0 …x t , yt †
Oa Of

…16†

Eq. (16) captures the change in distance/di€erence between observed production and the
frontier Ð sometimes referred to as a measure of ``catching up''. Countries or ®rms that are
engaged in successful learning either by doing or by imitation would exhibit improvements
(values greater than one) in this component of productivity change.
Technical change is measured by:
ÿ


ÿ

!1=2
1=2

Oa Ob
Dt0 x t‡1 , yt‡1 Dt0 x t , yt
ÿ


ˆ
:
…17†
TECH ˆ
Od Oe
Dt0 x t‡1 , yt‡1 D0t‡1 …x t , yt †

Note that Eq. (17) captures, for example, shifts in the frontier due to innovation.
The product of eciency and technical change is the Malmquist productivity measure, given
by Eq. (18).
"
ÿ


ÿ

#1=2
D0t‡1 x t‡1 , yt‡1 Dt0 x t‡1 , yt‡1
M0 …t, t ‡ 1† ˆ EFFCH  TECH ˆ
:
…18†
Dt0 …x t , yt †
D0t‡1 …x t , yt †
The Malmquist index thus consists of distance functions evaluating (x t,y t) and (x t + 1,y t + 1)
relative to the two reference technologies, Tt and T t + 1, respectively. These are indicated by
the superscripts on the distance functions, Dt0 and D t0 + 1, respectively.
We next turn to the dynamic network model (6), which was used by FaÈre and Grosskopf
[12] to compute dynamic eciency. Here, we need to resolve the question of whether a single
overall scaling vs a year-by-year scaling should be used for the model. In the ®rst case, the

Fig. 7. The Malmquist index.

R. FaÈre, S. Grosskopf / Socio-Economic Planning Sciences 34 (2000) 35±49

47

®nal output for each year, here fy t and fy t + 1, would be scaled with the same scalar, y. In the
second case, two scalars, yt and y t + 1 would be used and the objective function would be their
sum. FaÈre and Grosskopf [12] used the latter formulation as they were interested in comparing
the dynamic model with a sequence of static models. In either case, eciency can be estimated
using the LP techniques familiar from DEA.

5. Summary
This paper has shown the ¯exibility of the DEA modeling framework by focusing on
network DEA. These DEA models allow the researcher to study the ``inside'' of the usual
black box technology both in static and dynamic settings. They accomplish this by providing a
very general framework for specifying (endogenizing) the inner workings of the black box. The
basic idea of the network model is thus to ``connect'' processes Ð providing a single model
framework for multi-stage production (with intermediate products, for example) or multiperiod production. This situation has typically been handled in the DEA literature as a rather
ad hoc series of DEA problems, or through the use of multiple stages.
The ``links'' between processes (or nodes) in the network model may also be used to analyze
the allocation of resources across various units or processes. Examples include the allocation of
a ®xed resource, such as land at farm level at a given point in time, over various crop uses;
and the allocation of a school budget across various budgetary categories. The links can also
be used to solve for the ``optimal'' linkages and for the structure of units Ð in cases where a
®rm has many plants or locations, etc.
All of these scenarios Ð intermediate goods, allocatable resources, and multi-period
production Ð may be represented as network models. This then allows the researcher to
endogenize allocations across space, time, etc. What we have also shown is that such models
are readily estimatable using standard LP formulations familiar from the DEA literature. Our
hope is that this rather simple framework will prove useful to those who use DEA, while
moving researchers in other arenas to investigate the power of DEA.

Acknowledgements
The authors are grateful to Dr B. Parker for his ``red-pen adjustments.''

Appendix A
Assume there are k = 1,. . .,K DMU:s. These can be di€erent ®rms or a ®rm at di€erent
times.
Each
DMU
is
characterized
by
its
input
and
output
vector
(x k,y k)=(xk1,. . . ,xkN,yk1,. . . ,ykM). The coecients (xkn,ykm), m = 1,. . . ,M, n = 1,. . . ,N,
k = 1,. . . ,K are required to satisfy certain conditions. These are:

48

R. FaÈre, S. Grosskopf / Socio-Economic Planning Sciences 34 (2000) 35±49

…i†

…ii†

x kn e0, ykm e0, k ˆ 1, . . . , K, n ˆ 1, . . . , N, m ˆ 1, . . . , M,
K
X
x kn > 0, n ˆ 1, . . . , N:
kˆ1

…iii†

N
X

x kn > 0, k ˆ 1, . . . , K:

nˆ1

…iv†

K
X
ykm > 0, m ˆ 1, . . . , M:
kˆ1

…v†

M
X

ykm > 0, k ˆ 1, . . . , K:

mˆ1

The conditions in (i) merely state that inputs and outputs are non-negative numbers, but need
not all be positive. The requirement (ii) means that each input is used in at least one activity.
The third condition says that each activity uses at least one input. The ®rst of the two output
conditions (iv) requires that each output is produced by some activity while (v) states that each
activity produces some output.
The model of (1) satis®es the properties:
1.
2.
3.
4.

P(0)={O}
y ey 1 $ P(x )= > y $ P(x )
x 1 e x= > P(x 1) uP(x )
P(lx )=lP(x ), l > 0

no ``free lunch''
free (strong) disposability of outputs
free (strong) disposability of inputs
constant returns to scale.

References
[1] Charnes A, Cooper WW, Lewin AY, Seiford LM. The DEA process, usages and interpretations. In: Data
envelopment analysis: theory, methodology and applications, ch. 21. Boston: Kluwer Academic, 1993.
[2] Shephard RW, FaÈre R. A dynamic theory of production correspondences. ORC UC Berkeley 75-13, Berkeley,
CA 1975.
[3] Shephard RW. Theory of cost and production. Princeton: Princeton University Press, 1970.
[4] FaÈre R, Grosskopf S. Intertemporal production frontiers: with dynamic DEA Boston: Kluwer Academic
Publishers, 1996.
[5] Charnes A, Cooper WW, Rhodes E. Measuring the eciency of decision making units. European journal of
operational research 1978;2(6):429±44.
[6] von Neumann J. A model of general economic equilibrium. Review of Economic Studies 1945;13(1):1±9.
[7] Karlin S. Mathematical methods and theory in games, programming and economics. Reading: Addison-Wesley,
1959.

R. FaÈre, S. Grosskopf / Socio-Economic Planning Sciences 34 (2000) 35±49

49

[8] FaÈre R, Grabowski R, Grosskopf S, Kraft S. Eciency of a ®xed but allocatable input: a non-parametric
approach. In: Economics Letters, vol. 56, 1996. p. 187±93.
[9] FaÈre R, Whittaker G. An intermediate input model of dairy production using complex survey data. Journal of
agricultural economics 1995;46(2):201±23.
[10] FaÈre R, Grosskopf S, Roos P. Network and production models of swedish pharmacies, mimeo 1996.
[11] Sengupta J. Dynamics of DEA. Dordrecht: Kluwer Academic Publishers, 1995.
[12] FaÈre R, Grosskopf S. Eciency and productivity in rich and poor countries. In: Jensen B, Wong KY, editors.
Dynamics, economic growth and international trade. Ann Arbour: University of Michigan Press, 1997.
[13] Farrell MJ. The measurement of productive eciency. Journal of the royal statistical society Series A
1957;120:253±90.
[14] FaÈre R, Grosskopf S, Lindgren B, Roos P. Productivity developments in Swedish hospitals: A Malmquist output index approach. In: Charnes A, Cooper WW, Lewin A, Seiford L, editors. Data envelopment analysis: theory, methodology and applications. Boston: Kluwer Academic, 1994.
[15] FaÈre, Grosskopf RS, Lovell CAK. Production frontiers. Cambridge: Cambridge University Press, 1994.
[16] Luenberger D. Bene®t functions and quality. Journal of Mathematical Economics 1992;21:461±81.
[17] Luenberger D. Microeconomic theory. New York: Mcgraw Hill, 1995.
[18] Chambers RG, Chung Y, FaÈre R. Pro®t, directional distance functions, and nerlovian eciency. Journal of optimization theory and applications 1998;98(2):351±64.