Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol65.Issue2.Apr2000:

Int. J. Production Economics 65 (2000) 191}199

An extended FaK re}Lovell technical e$ciency measureq
W. Briec*
ManL tre de Confe& rences, Universite& de Rennes 1, CREREG, 11 rue Jean Mace& -BP1997, 35019 Rennes Cedex, France
Received 13 March 1998; accepted 5 February 1999

Abstract
Recently, Chambers et al. (Journal of Economic Theory 70 (1996) 407}419; Working Paper, Southern Illinois
University, Carbondale, 1996) have developed the concept of directional distance to measure technical e$ciency.
However, there is no guarantee that such a measure must intersect the e$cient subset (also called strong e$cient subset)
de"ned by Koopmans (Activity Analysis of Production and Allocation, 1951, Vol. 36, pp. 27}56), that is primarily the
Pareto e$cient subset. So, we develop a framework for measuring e$ciency in the full input/output space. Following
FaK re and Lovell (Journal of Economic Theory 19 (1978) 150}162), we introduce a graph-type extension of the Russell
measure. Moreover, we show that our new measure can be computed by using linear programming. ( 2000 Elsevier
Science B.V. All rights reserved.
Keywords: E$ciency indices; Graph measure; Technical e$ciency; Production technology; Distance functions

1. Introduction
In recent years, a number of studies on the theoretical and empirical measurement of technical e$ciency has been generated by researchers, and
two di!erent notions of technical e$ciency have

emerged in the economic literature. The "rst notion
due to Koopmans [1], de"nes a producer as technically e$cient if a decrease in any input requires
a increase of at least one other input. This de"nition
is closely related to the Pareto e$ciency concept,

* Corresponding author. Tel: (33) 02 99 84 78 08; fax: (33) 02
99 84 77 90.
E-mail address: walter.briec@univ-rennes1.fr (W. Briec)
q
The "rst version of this paper was presented to the 1996
Latin American Meeting of the Econometric Society (Rio de
Janeiro, August).

and its great intuitive appeal has led to its adoption
by several authors, in particular by FaK re and Lovell
[2]. The second notion introduced by Debreu [3]
and Farrell [4], is based on radial measures of
technical e$ciency. In the input case, the Debreu}Farrell index measures the minimum amount
that a vector can be shrunk along a ray while
holding output levels constant. This e$ciency index is constructed around a technical component

that involves equiproportionate modi"cation of
inputs, and this has received a growing interest
during the last few years. Following Charnes et al.
[5], several empirical papers have implemented
the Debreu}Farrell measure. In particular, describing the production set as a piece-wise linear
technology, it can be computed by linear programming.
Recently, several authors suggested some new
development in order to judge adjustments of both
input and output quantities simultaneously. In

0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 0 1 9 - 5

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W. Briec / Int. J. Production Economics 65 (2000) 191}199

particular, Chambers et al. [6,7] have introduced
the `directional distance functiona as an e$ciency
measure, by how much the outputs can be increased and the inputs decreased. We made independently the same thing by de"ning the so-called

`Farrell proportional distancea [8]. These new
measures are derived from a recent concept due to
Luenberger [9]: the `shortage functiona.
The aim of this paper is the introduction of a new
distance function for measuring technical e$ciency
in the full-input}output space. Why do we want
another e$ciency measure? First, the above-mentioned indices are not able to measure the technical
e$ciency (also called strong e$ciency) in the fullinput}output space, with respect to the criterion
introduced by Koopmans [1]. In particular, the
directional distance function, and the Farrell proportional distance function are constructed around
a technical component that has not the #exibility to
select a strong e$cient vector on the boundary of
the production set. These measures, in the general
case, select a vector on the weak e$cient subset,
generally smaller. Second, FaK re and Lovell [2] introduced the Russell measure that evaluates technical e$ciency, but this has not the ability to judge
adjustments of both input and output quantities
simultaneously.
From the above considerations, the objective of
this paper is to merge the ideas of the directional
distance function with the Russell measure in order

to form an extended `FaK re}Lovell technical e$ciency measurea. This selects a strongly e$cient
vector and measures technical e$ciency in the fullinput}output space. According to FaK re and Lovell
we term it: Russell proportional distance function.
This measure calls a vector technically e$cient
if and only if it belongs to the strong e$cient
subset.
From the theoretical standpoint, our measure
can be used to compare the production vectors: it
characterises the observed e$cient subset, it is
weakly monotonic and unit invariant. The Russell
proportional distance has another important property. It is constructed around a technical component that has the #exibility to take into account the
particularity of the market by introducing a weighing scheme. In our opinion this property is interesting, because the producer may consider that some

particular inputs and outputs are more important
than the others.
The paper is organised as follows: in Section 2,
we focus our attention on the shortage function [9]
the directional distance function [6,7], and the Farrell proportional distance function [10]. Moreover,
we show that a proportionally modi"ed input}output vector does not necessarily belong to the strong
e$cient subset of the full-input}output space.

In Section 3, we introduce a graph-type extension of the Russell measure and we state its basic
properties. Moreover, among the outcomes of the
section is the comparison of the Russell proportional distance function and the Farrell proportional distance function [10]. In particular, we
show that the comparison is in accordance with the
result provided by FaK re et al. [11], alternatively in
input or output space.

2. Distance function and technical e7ciency
First we de"ne the standard notations used in
this paper. Let R` be the non-negative Euclidean
n
n-orthant, for z, z@3Rn with n'1 we denote:
z6z@ Q z )z@ ∀i3M1, 2 , nN, z)z@ Q z6z@
i
i
and zOz@, z(z@ Q z (z@ ∀i3M1,2, nN. A proi
i
duction technology transforming inputs x"
(x , x ,2, x )3Rn into outputs y"(y , y ,2, y )3
`

1 2
p
1 2
n
Rp can be characterised by the input correspond`
ence ¸ : yP¸(y)LRn and the output correspond`
ence P : xPP(x)LRp , where ¸(y) is the set of all
`
input vectors which yield at least y and P(x) is the
subset of all outputs vectors obtainable from x. The
set of all the input}output vectors which are feasible is called the graph T and de"ned as follows:
¹"M(x, y); x3¸(y), y3Rp N
`
"M(x, y); y3P(x), x3Rn N.
`

(1)

We assume that the production set T satis"es the
following axioms (see [11,12]):

T1: (0, 0)3¹, (0, y)3¹Ny"0,
T2: ¹(x)"M(u, y)3¹; u6xN is bounded ∀x3Rn ,
`
T3: ¹ is a closed set,
T4: ∀(x, y)3¹, (x, !y)6(u,!v)N(u, v)3¹.

W. Briec / Int. J. Production Economics 65 (2000) 191}199

The following assumption may also be introduced:
T5: ¹ is a convex set.
T5 postulates the convexity of the production
set. This assumption is often used in empirical
works and it can be useful to describe the production set as a convex combination of positive vectors
(see [5,11,13]). In such case, the production set
is de"ned as a piece-wise linear technology.
More precisely, under convexity and strong disposability assumptions, consider m activities
(x1, y1), (x2, y2),2 , (xm, ym). The production set
can now be de"ned as

H

G

m
m
¹" (u, v)3Rn`p; u7 + h xj, v6 + h yj, h3C .
`
j
j
j/1
j/1
(2)
The set C characterises the returns to scale chosen
by the producer. It is possible to characterise constant returns to scale (see [5]), various returns to
scale [14], non-increasing returns to scale [15,16],
non-decreasing returns to scale [17]:
(a) Constant returns to scale [5]:
C"
: C "Rm .
`

CRS
(b) Various returns to scale [14]:

In order to judge adjustments of both inputs and
output quantities simultaneously, it is possible to
model the technology with the graph [13]. A graph
measure of technical e$ciency is constructed
around a technical component that involves a variation of both input and output: inputs are reduced,
while outputs are simultaneously increased until it
reaches the boundary of the production set.
Recently, Luenberger [19] introduced a function
he terms the bene"t function. The bene"t function,
has its roots in a construction by Dupuit [20])
called `relative utilitya. In particular Luenberger
introduces the notion of a `shortage functiona in
his studies of production. Luenberger [19] has seen
the `shortage functiona as a shortage of (x, y) to
reach the boundary of ¹. More recently, Chambers
et al. [6,7] have interpreted the distance as an
e$ciency measure, by how much output can be

increased and input decreased. They term the new
function as `directional distancea. This can be considered as a development of what Luenberger calls
the `shortage functiona and can be used to evaluate
technical e$ciency. The directional distance function in the direction of g"(!gi, g0) can be de"ned
as follows:

(3)

G

H

(4)

G

H

(5)

G

H

m
C"
: C " h3Rm ; + h "1 .
`
i
VRS
i/1
(c) Non-increasing returns to scale [17]:
m
C"
: C " h3Rm ; + h "1 .
`
i
NIRS
i/1
(d) Non-decreasing returns to scale [15,16]:

m
(6)
C"
: C
" h3Rm ; + h *1 .
`
i
NDRS
i/1
Note that all the above speci"cations of the production set imply the convexity assumption. It is
however, possible to drop the convexity assumptions by using the FDH methodology by Tulkens
and Vanden-Eeckaut [18]. In such case the set
C can be characterized as follows:

G

193

H

m
C"
: C " h3M0, 1Nm; + h "1 .
i
FDH
i/1

(7)

De5nition 1. Let T be a production set satisfying
T1}T4. Let g"(!gi, g0)3(!Rn )]Rp be a vec`
`
tor, the function Do g : ¹PR de"ned by
T
`
Do g (x, y)"max Md*0; (x!dgi, y#dg0)3¹ N
T
d
is called directional distance function in the direction of g.
The above function can be considered as a
generalisation of the Farrell proportional distance
function (see [10]). This can be considered as a directional distance function de"ned in a particular
direction. Assume that the economy is competitive,
hence the particularities of the market are summarised by the input price vector p and the output
price vector q. It is possible to transform inputs
and outputs introducing coe$cients which are
price dependant. These coe$cients are called
orientation. Let (a(n), b(o))3Rn`p be a price`
dependant vector and let I(a(n)) and J(b(o)) which
are, respectively, two subsets of M1,2, nN and
M1,2, pN de"ned as I(a)"Mi3M1,2, nN; a (n)'0N
i

194

W. Briec / Int. J. Production Economics 65 (2000) 191}199

and J(b)"Mj3M1,2, pN; b (o)'0N. Similarly,
j
we denote I(x)"Mi3M1,2, nN; x '0N and J(y)"
i
Mj3M1,2, pN; y '0N. Now, consider the two
j
matrices A and B which are, respectively, n]n
and p]p non-negative diagonal matrices, such
that A"Diag(a(n)) and B"Diag(b(o)). The Farrell
proportional distance function is de"ned as follows:
De5nition 2. Let T be a production set satisfying
T1}T4. Let (a(n), b(o))3Rn`p be a price dependant
`
orientation. Assume that (I(x)WI(a))X(J(y)WJ(b))
O0, the function D(a,b) : ¹PR de"ned by
T,F
`
D(a,b)(x, y)"max Md*0; ((I!dA)x, (I#dB)y)3¹N
T,F
d
is called oriented Farrell proportional distance
function.
From the above considerations, the Farrell proportional distance function can be expressed at
point (x, y) as a particular directional distance function in the direction of g"(!Ax, By). This point
was suggested by Chambers et al. [6,7].
A di$culty with the measures introduced is that
a projected input}output vector does not necessarily belong to the e$cient subset of the production
set.
For instance, consider the Fig. 1. Clearly the
point M is weakly e$cient [11] and the proportional distance function is zero. However, M does
not satisfy the FaK re}Lovell [2] criterion for technical e$ciency: a decrease of the input x does not
require a decrease of the output y. This means that
the Farrell proportional distance function does not
characterize the e$cient points. On the contrary, it
is clear that M@ is technically e$cient. That is, the
Farrell proportional distance function projects an
input}output vector onto the weak e$cient subset
and not necessarily onto the graph e$cient subset,
generally smaller, de"ned by FaK re et al. [11]. The
e$cient subsets of T can be de"ned as follows:
De5nition 3. Let T be a production set satisfying
T1}T4. We have the following de"nitions:
(1) Let y*0, the e$cient subset of ¸(y) is de"ned
as LK(¸(y))"Mx3¸(y); x@)xNx@N¸(y)N and
we have LK(¸(0))"M0N.

Fig. 1.

(2) If P(x)OM0N the subset of P(x) de"ned by
LK(P(x))"My3P(x); y@*yNy@NP(x)N is called
e$cient subset of P(x), if P(x)"M0N, we have
LK(P(0))"M0N.
(3) The subset LK(¹ )"M(x, y)3¹; (!x, y))
(!x@, y@)N(!x@, y@) N ¹N is called the graph
e$cient subset of T.
Since the notion of technical e$ciency compels
a vector to belong to the strong e$cient subset,
there is reason to develop a graph measure that
projects an input}output vector onto the graph
e$cient subset.

3. An extended FaK re+Lovell technical e7ciency
measure
Now, let us de"ne the Russell measure of technical e$ciency, which was introduced by FaK re and
Lovell [2]. The Russell measure } theoretical and
empirical } evaluates e$ciency alternatively in
input or output space (note that FaK re et al. [11]
have de"ned a `hyperbolica Russell graph measure
of technical e$ciency). This measure is related with
the e$cient subset of ¸(y), and satis"es the weak
monotonicity conditions. These properties were
established by FaK re and Lovell [2] and Russell
[21], respectively.
The Russell input measure evaluates the maximum sum of proportionate reductions in individual inputs in coordinate direction. The Russell
output measure evaluates the maximum sum of

W. Briec / Int. J. Production Economics 65 (2000) 191}199

proportionate increasing in individual outputs in
coordinate direction. We denote Card(I(x)) the
number of positive x s, and Card(J(y)) the number
i
of positive y s. The Russell measure of technical
j
e$ciency is de"ned as follows (see [13]):
De5nition 4. Let ¹ be a production set satisfying
T1}T4.The function Ei : ¹PR de"ned by
R
`
Ei (x, y)
R

H

G

1
"min
+ j ; jx3¸(y), j 3[0, 1]
i
i
Card(I(x))
j
i|I(x)

is called Russell input measure of technical e$ciency.
The function E0 : ¹PR de"ned by
R
`
E0 (x, y)
R

G

H

1
+ k ; ky3P(x), k *1
"max
j
j
Card(J(y))
j|J(Y)
k

is called Russell output measure of technical e$ciency.
Note that contrary to the Russell input measure,
the Russell output measure does not completely
characterise the e$cient subset for all technologies,
in particular if some output is null. Now, we de"ne
an `orientationa of measurement of technical e$ciency. We modify the previous Farrell proportional distance, by maximising the modi"cations of
inputs and outputs in co-ordinate direction. Let
p and o be, respectively, inputs and outputs price
vectors as de"ned in Section 2. It is possible to
transform inputs and outputs introducing coe$cients which are price dependant:
f each individual input x , for i"1,2, n, is proi
portionally decreased by the factor, d a (n) with
i i
a (n)3[0, 1]. Thus, the input obtained is
i
(1!d a (n))x ;
i i
i
f each individual output y , for j"1,2, p, is proj
portionally increased by the factor e b (o) with
j j
b (o)3[0, 1]. Thus, the output obtained is
j
(1#e b (o))y .
j j
j

195

With this notation, it is clear that the oriented
Russell proportional distance of (x, y) exists if and
only if we have (I(x)WI(a))X(J(y)W(Jb))O0. We
will discuss later the relationship between Russell
proportional distance function and Russell
measure of technical e$ciency. The oriented Russell proportional distance function is de"ned as
follows:
De5nition 5. Let T be a production set satisfying
assumptions T1}T4. Let (a(n), b(o))3Rn`p be
`
a price-dependant orientation. and let us denote
r(x, a)"Card(I(x)WI(a)) and s(y, b)"Card(J(y)W
J(b)). Assume that (I(x)WI(a))X(J(y)WJ(b))O0, the
function D(a,b) : ¹PR de"ned by
T,R
`

G

1
1
+ d#
+ e;
D(a,b)(x, y)"max
i
j
T,R
s(y,
b)
r(x, a)
W
i|I(X) I(a)
j|J(Y)WJ(b)
d,e
((I!dA)x, (I#eB)y)3¹

H

is called the oriented Russell proportional distance
function.
Since ¹ is closed, and T2 holds the maximum
exists and the Russell proportional distance is well
de"ned. A and B de"ne an orientation, and according to this we have input, output or graph-type
methodology.
Consider the two-dimensional case (see Fig. 1).
In such case, matrices A and B are scalar and thus,
we have A"a and B"b. If a"0 and b'0 the
Russell proportional distance function measures
the maximal proportional increase in output given
the technology and the input vector x. If a'0 and
b"0, the Russell proportional distance measures
the maximal proportional decrease in input given
the technology and the output vector y. If a'0 and
b'0 the Russell proportional distance is a graphtype methodology. It measures the maximum sum
of proportionate modi"cations in inputs and outputs in co-ordinate directions.
However, the Russell output measure does not
satisfy all the nice properties of the Russell input
measure (stated by FaK re and Lovell [2]), in particular if some outputs are null. For instance the Russell output measure does not satisfy the weak
monotonicity condition. The same holds for the

196

W. Briec / Int. J. Production Economics 65 (2000) 191}199

Russell proportional distance we have de"ned
above. Thus, in order to salvage these properties it
is necessary to consider the subset of all the production units (x, y) such that y'0. Since in the general
case the products of the observed decision-making
units are positive, this assumption is not very restrictive. We denote ¹H the subset of T de"ned as
¹H"M(x, y)3¹; y'0N.

(8)

Thus, ∀(x, y)3¹ we denote PH(x) the subset of P(x)
de"ned as PH(x)"My3P(x); y'0N. The properties
of the oriented Russell proportional distance function are summarized as follows:
Proposition. Let T be a production set satisfying assumptions T1}T4. Assume that (I(x)WI(a))X(J(y)W
J(b))O0, and let D(a,b) be the Russell proportional
T,R
distance function, then
(1) The ezcient subset LK(¹H) satisxes the following properties:
(a) (a, b)'0NM(x, y)3¹H; D(a,b)(x, y)"0N"
T,R
LK(¹H),
(b) a'0'b"0NMx3¸(y); D(a,b)(x, y)"0N"
T,R
LK(¸(y)),
(c) a"0'b'0NMy3PH(x); D(a,b)(x, y)"0N"
T,R
LK(PH(x)).
(2) If for all i3M1,2, nN and j3M1,2, pN we have
a "1 and b "0 then D(a,b)"1!E* .
R
T,R
i
j
(3) If for all i 3 M1,2, nN and j 3 M1,2, pN we
have a "0 and b "1 then D(a,b)"1#E0 .
R
T,R
i
j
(4) The Russell proportional distance satisxes the
weak monotonicity conditions over the subset ¹H. We
have
∀(x, y), (x@, y@)3¹H, (!x@, y@)6(!x, y)
ND(a,b)(x, y))D(a,b)(x@, y@).
T,R
T,R
(5) The Russell proportional distance function is
unit invariant.
(6) We have the relationship D(a,b)(x, y))
T,F
D(a,b)(x, y).
T,R
Proof. (1) First we prove (a). Let (x, y)3LK(¹), and
assume that D(a,b)(x, y)'0. Let (xH, yH)"
T,R
(x!dHAx, y#eHBy), where (dH, eH) is the solution

of the maximisation program computing the Russell proportional distance. Since T2 holds,
M(u, v)3Rn`p; u6x, v7yN is closed and bounded,
`
thus (xH, yH) belongs to the production set T and
D(a,b)(x, y) is de"ned. Moreover, there exists i3I(x)
T,R
WI(a) or j3J(y)WJ(b) such that xH(x or yH'y .
j
j
i
i
Thus, (x, y) does not belong to the e$cient subset.
Consequently, LK(¹)LM(x, y)3¹; D(a,b)(x, y)"0N,
T,R
and since ¹HL¹ we have LK(¹H)LM(x, y)3¹H;
D(a,b)(x, y)"0N. Conversely, if (x, y)3¹H, and
T,R
D(a,b)(x, y)"0, since y'0 it is obvious that
T,R
(x, y)3LK(¹ ). Thus, (x, y)3LK(¹H) and (a) is proved.
The proofs of (b) and (c) are similar.
(2) Assume that j"I!d. Assume that for all
i3M1,2, nN and j3M1,2, pN we have a "1 and
i
b "0, thus
j
D(a,b)(x, y)
T,R
1
1
+ d#
+
e;
"max
i
j
s(y, b)
r(x, a)
i|I(a)WI(x)
j|J(b)WJ(y)
d,e

G

H

((I!dA)x, (I#dB)y)3¹

H

G

1
+ j ; (jx, y)3¹ ,
"max 1!
i
Card(I(x)) 3
i I(x)
j
therefore, we have the relationship
D(a,b)(x, y)
T,R

G

H

1
+ j ; (jx, y)3¹ .
"1!min
i
Card(I(x)) 3
j
i I(x)
Consequently D(a,b)(x, y)"1!E* (x, y).
R
T,R
(3) The proof is similar to (2) making the transformation h"I#d.
(4) First, we prove the input monotonicity. Let
(x@, y)3¹, with x6x@ and assume that X(x, y)"
M(d, e)3Rn`p; ((I!dA)x, (I#eB)y)3¹N, thus we
`
have X(x, y)LX(x@, y) and the input monotonicity
is proved. Now, assume that y7y@'0, since
y@'0 we have necessarily X(x, y@)M(x, y). Thus,
the output monotonicity holds. We deduce immediately (4).
(5) Let L be a (n#p)](n#p) positive diagonal
matrices de"ned over Rn`p such that

A

¸"

¸

0

1

0

B

¸
2

,

W. Briec / Int. J. Production Economics 65 (2000) 191}199

where ¸ and ¸ are, respectively, n]n and p]p
1
2
positive diagonal matrices. We need to prove that
D(a,b) (¸ x, ¸ y)"D(a,b)(x, y). From De"nition 5
T,R
LT,R 1
2
D(a,b) (¸ x, ¸ y)
LT,R 1
2

G

1
1
+
+
d#
e;
"max
i
j
s(y,
b)
r(x, a)
W
W
i|I(a) I(X)
j|J(b) J(Y)
d,e

H

((I!dA)¸ x, (I#eB)¸ y)3¸¹
1
2

is the Russell proportional distance de"ned over
¸(¹). (¸ x, ¸ y) is the input}output vector ob1
2
tained with respect to a change in the units of
measurement. Thus we have
D(a,b) (¸ x, ¸ y)
LT,R 1
2

G

1
1
d#
e;
"max
+
+
i
j
s(y,
b)
r(x, a)
d,e
i|I(a)WI(X)
j|J(b)WJ(Y)

H

¸((I!dA)x, (I#eA)y)3¸¹ .
Since L is an isomorphism over Rn`p, thus
D(a,b) (¸ x, ¸ y)
LT,R 1
2

G

1
1
"max
+
d#
+
e;
i
j
r(x, a)
s(y,
b)
d,e
i|I(a)WI(X)
j|J(b)WJ(Y)

H

((I!dA)x, (I#eA)y)3¹

197

But we have
M(d, e)3Rn`p; ((I!dA)x, (I#eB)y)3¹, d "e "jN
`
i
j
LM(d, e)3Rn`p; ((I!dA)x, (I#eB)y)3¹N.
`
Thus, we deduce immediately that D(a,b)(x, y))
T,F
D(a,b)(x, y). h
T,R
The fourth result proves that the generalised
Russell}FaK re}Lovell measure characterises the
observed decision-making units that are strongly
e$cient. In the same idea, it is possible to relate the
input and output e$cient subsets and the Russell
proportional distance. Property (4) shows that the
Russell proportional distance function satis"es the
weak monotonicity condition. A similar proof is for
instance given in Ref. [21]. (5) proves that the
proportional distance is invariant with respect to
a change in the units of measurement.
The property (6) states a comparison between the
Farrell and Russell proportional distances. A similar result is obtained by FaK re et al. [11] alternatively in the input or output space. From properties
(2) and (3), it is obvious that (6) is in accordance
with the result obtained by FaK re et al.
Now, we focus on the particular relationship
between the proportional distance and the Russell
input and output measurements of technical e$ciency. Assume that b"0. The input set dimension
is two. Fig. 2 illustrates the di!erent cases.
Assume that a "a "1, the Russell propor1
2
tional distance function coincides with the Russell
measurement of technical e$ciency and we have
the relationship D(a,b)"1!Ei . The factors of
R
R

"D(a,b)(x, y).
T,R
Consequently, the Russell proportional distance
function is invariant with respect to a change in the
units of measurement.
(6) It is possible to show that
D(a,b)(x, y)
T,F

G

1
1
e;
+ d#
+
"max
i
j
s(y,
b)
r(x, a)
W
W
i|I(a) I(x)
j|J(b) J(y)
d,e

H

((I!dA)x, (I#eB)y)3¹, d "e"j .
i

Fig. 2.

198

W. Briec / Int. J. Production Economics 65 (2000) 191}199

production are equiproportionately decreased. If
a (a the second input is reduced more than the
1
2
"rst one, and the Russell proportional distance
function is oriented more to the direction of the
x -axis. If a 'a , the "rst input is reduced more
1
1
2
than the second one, and the Russell proportional
distance is oriented more in the direction of the
x -axis. Of course, a similar analysis can be pro2
vided in the output case. We will not discuss this
point.
We are now able to, from the above results,
provide some linear program in order to compute
the Russell proportional distance function when
the production set is de"ned as a piece-wise linear
technology. From relationship (2), we get

e$ciency. Moreover, it has the #exibility to take
into account some managerial criterion, and can be
computed by linear programming.

1
1
+ d#
+ e ,
max
i s(yk, b)
j
r(xk, a)
i|I(a)
j|J(b)
d,e
m
(I!dA)xk7 + h xi,
i
i/1
m
(I#eB)yk6 + h yi,
i
i/1
d, e60, h3C,

[1] T.C. Koopmans, Analysis of production as an e$cient
combination of activities, in: T.C. Koopmans (Ed.), Activity Analysis of Production and Allocation, vol. 36, 1951,
pp. 27}56.
[2] R. FaK re, C.A.K. Lovell, Measuring the technical e$ciency
of production, Journal of Economic Theory 19 (1978)
150}162.
[3] G. Debreu, The coe$cient of resource utilization, Econometrica 19 (1951) 273}292.
[4] M.J. Farrell, The measurement of productive e$ciency,
Journal of the Royal Statistical Society 120 (1957)
253}281.
[5] A. Charnes, W.W. Cooper, E. Rhodes, Measuring the
e$ciency of decision-making units, European Journal of
Operational Research 3 (1978) 429}444.
[6] R. Chambers, Y. Chung, R. FaK re, Bene"t and distance
functions, Journal of Economic Theory 70 (1996) 407}419.
[7] R. Chambers, Y. Chung, R. FaK re, Pro"t, directional
distance functions, and Nerlovian e$ciency, Working Paper Southern Illinois University, Carbondale; Journal of
Optimization Theory and Application (1996) fothcoming.
[8] W. Briec, An extended FaK re}Lovell technical e$ciency
measure, Presented at XIV Latin American Meeting of the
Econometric Society, 5}12 August 1996, Rio de Janeiro.
[9] D.G. Luenberger, Microeconomic Theory, McGraw Hill,
Boston, 1995.
[10] W. Briec, A graph-type extension of Farrell technical
e$ciency measure, Journal of Productivity Analysis
8 (1997) 1.
[11] R. FaK re, S. Grosskopf, C.A.K. Lovell, The Measurement of
E$ciency of E$ciency of Production, Kluwer Nijho!
Publishing, Dordrecht, 1985.
[12] R.W. Shephard, Theory of Cost and Production Functions, Princeton University Press, Princeton, NJ, 1970.
[13] R. FaK re, S. Grosskopf, C.A.K. Lovell, Production Frontier,
Cambridge University Press, Cambridge, 1994.
[14] R.D. Banker, A. Charnes, W.W. Cooper, Some models for
estimating technical and scale e$ciency in data envelopment analysis, Management Science 30 (1984) 1078}1092.

(P)

where C is a polyhedral subset that characterises
the returns to scale. Recalling the relationship
between the proportional Distance and Russell
measurement of technical e$ciency, for some particular values of the parameters, the above linear
program is identical to the linear program computing Russell input and output measures (see [11]).
Hence, the new methodology presented in this
paper generalises the Russell measure introduced
by FaK re and Lovell [2].

4. Conclusion
The foregoing discussion is entirely in the spirit
of the recent literature on measures of technical
e$ciency, discussing radial and non-radial
measures. The measure we have introduced above
is a generalisation, in the full-input space, of the
Russell measure of technical e$ciency, introduce
by FaK re}Lovell [2]. From the practical standpoint
this measure has the advantage to select a strong
e$cient vector onto the frontier of technical

Acknowledgements
I am grateful to C. Blackorby for helpful suggestions and I would like to thank two anoymous
referees for comments that greatly improved the
exposition of the paper.

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