This note generalizes analytical relationships among activity variables of DEA models previously derived by Boussemart, Briec and Leleu (2007). We relax the asumption of constant returns to scale by showing that the key results hold under a weaker asumption of homogeneity. We use the notion of alpha-returns to scale to extend the analysis to strictly increasing and decreasing returns, covering now the whole range of returns to scale for multi-output homogenous technologies.Data envelopment analysis, Methodology, Production

Hervé Leleu

Jean-Philippe Boussemart

Walter Briec

English

Research Papers in Economics

IÉSEG Working Paper Series 2008-ECO-6 
 
1 
 
 
 
 
 
 
 
 
 
 
IÉSEG WORKING PAPER  
2008-ECO-6 
 
 
 
 
Linear Programming Solutions and Distance  
Functions Under α-Returns to Scale 
 
 
Jean-Philippe Boussemart
* 
Walter Briec
** 
Hervé Leleu
*** 
 
*University of Lille and IÉSEG School of Management, CNRS-LEM (UMR 8179) 
**LAMPS, University of Perpignan, IAE 
***IÉSEG School of Management, CNRS-LEM (UMR 8179) 
 
 
 
 
September 2008 
 
 
 
IÉSEG School of Management, Catholic University of Lille 
3, rue de la Digue, 59000 Lille, France 
www.ieseg.fr  
Tel: 33 (0)3 20 54 58 92 
Fax: 33 (0)3 20 57 48 55 
 
 
 
 
 
 
 
   IÉSEG Working Paper Series 2008-ECO-6 
 
2 
 
Linear Programming Solutions and Distance 
Functions Under α-Returns to Scale 
 
Jean-Philippe Boussemart 
University of Lille and IÉSEG School of Management,  
CNRS-LEM (UMR 8179), 
3 rue de la Digue, 59000 Lille, France. jp.boussemart@ieseg.fr 
 
Walter Briec 
LAMPS, University of Perpignan, IAE, 
52 Avenue Paul Alduy, F-66860 Perpignan Cedex, France. briec@univ-perp.fr  
 
Hervé Leleu 
IÉSEG School of Management, CNRS-LEM (UMR 8179) 
3 rue de la Digue, 59000 Lille, France. h.leleu@ieseg.fr  
 
 
 
Abstract 
This note generalizes analytical relationships among activity variables of DEA models 
previously derived by Boussemart, Briec and Leleu (2007). We relax the asumption of 
constant returns to scale by showing that the key results hold under a weaker asumption of 
homogeneity. We use the notion of α -returns to scale to extend the analysis to strictly 
increasing and decreasing returns, covering now the whole range of returns to scale for multi-
output homogenous technologies.   
 
JEL: D24  
Keywords: Data envelopment analysis, Methodology, Production.  
Running Head: DEA solutions and α -returns to scale.  IÉSEG Working Paper Series 2008-ECO-6 
 
3 
 
Introduction 
In this technical note, we generalize analytical relationships among scores and activity 
variables of DEA models previously derived by Boussemart, Briec and Leleu (2007) [BBL 
(2007) in the remainder of the text]. The results were established under a constant returns to 
scale (CRS) assumption that we relax here. More formally, we show that our previous results 
do not necessitate the CRS assumption but a weaker assumption of homogeneity. We can 
therefore consider other kinds of returns to scale. Following Boussemart et al. (2008) we refer 
to the notion of α -returns to scale to deal with strictly increasing and decreasing returns for 
multi-output homogeneous technologies. As in BBL (2007) we analyze the relationships 
among optimal solutions of the input-based, the output-based, the hyperbolic, and the 
proportional distance functions.  
 
1. Background 
1.1. Production Technology: Definition and Assumptions 
The production technology transforms inputs  1 ()
n
n x xx R + = ,, ∈ L  into outputs 
1 ()
p
p yy y R + =, , ∈ L  under the technology T :  
  { } ( ) can produce
np Tx y Rx y
+
+ =, ∈ :  (1) 
 
We suppose that the technology obeys the following axioms:  
•  T1: (0 0) T ,∈, (0 ) 0 yT y ,∈ ⇒=  i.e., no free lunch;  
•  T2: the set  { } () ( ) A xu y T u x =, ∈ : ≤ of dominating observations is bounded 
n x R+ ∀∈ , i.e., infinite outputs cannot be obtained from a finite input vector;  
•  T3: T  is closed;  
•  T4: For all () x yT , ∈ , and all ( )
np uv R
+
+ ,∈ , we have 
() () ( ) x yu v u v T ,− ≤ ,− ⇒ , ∈  (free disposability of inputs and outputs).  
 IÉSEG Working Paper Series 2008-ECO-6 
 
4 
 
1.2. α -Returns to Scale Technologies and Distance Functions 
We first define the distance functions. The input and output Farrell measures are defined 
respectively by  { } () 0 ( ) inf I
II I E xy xy T θ θθ ,= ≥ : ,∈and by 
{ } () 0 ( ) sup O
OO O E xy x y T θ θθ ,= ≥ : , ∈.  It is also possible to define graph technical efficiency 
measures. The hyperbolic measure is defined by  { }
1 () 0 ( ) inf H H
HH H Ex y x yT θ θ θθ ,= ≥ : , ∈ 
and the proportional measure is given by:  ( ) { } () 0 ( 1 ) ( 1 ) sup P
PP P P Ex y x y T θ θθ θ , =≥ : − , + ∈ .   
 
In BBL (2007) these distance functions were considered under a constant returns to scale 
technology. We extend the class of technologies to homogeneous multi-ouput technologies. A 
production technology T  is said to be homogeneous of degree α  if for all  0 β > : 
  () ( ) x yT x yT
α ββ , ∈⇒ , ∈ .  (2) 
Lau (1978) termed these technologies "almost homogeneous technologies of degree 1 and α " 
for all  0 β > . A complete characterization is given by Färe and Mitchell (1993). Obviously, 
CRS corresponds to  1 α =  while strictly increasing returns corresponds to  1 α >  and strictly 
decreasing returns corresponds to  1 α < . Boussemart et. al. (2008) termed this property of the 
technology  α -returns to scale. Under such an assumption they established the following 
equalities linking both input, output and graph measures:  
  () ()
OI Ex y Ex y
α −
⎡ ⎤ , =, , ⎣ ⎦  (3) 
 
 
1
1 1 () () a n d () ()
HI HO E xy E xy E xy E xy
α
α α + + −
⎡⎤ ⎡⎤ ,= , ,= , ⎣⎦ ⎣⎦  (4) 
 
We first extend these relationships to the proportional distance function.  
 
Lemma 1.1. Suppose the technology T satisfies T1-T4. Under an assumption of α -returns to 
scale: 
  1
1( )
()
1( )
P
I
P
Ex y
Ex y
Ex y
α /
−,
,=
⎡ ⎤ +, ⎣ ⎦
 (5) 
 
Proof: Since α -returns to scale hold,  ( ) x yT ∀ ,∈ , 0 ( ) x yT
α ββ β ≥⇒ , ∈. This implies that IÉSEG Working Paper Series 2008-ECO-6 
 
5 
 
(( ) )
I Ex y x y T
α ββ ,, ∈ . Since the projected vector ( ( ) )
I Ex y x y , .,  is a frontier point in T  that 
achieves the Debreu-Farrell efficiency measure and since:  
  () [1 ( )] [1 ( )]
PP Ex y x Ex y y −, , +,  (6) 
achieves the proportional distance function, we need to find some  0 β >  that satisfies the 
relationship:  
  ( ) (( ) ) [ 1( ) ] [ 1 ( ) ]
IP P E x y x y Ex y x Ex y y
α ββ ,, =− , , + ,  (7) 
Then, we deduce both the following equalities:  
  () ( 1 () )
IP Ex y Ex y β , =− ,  (8) 
and  
  (1 ( ))
P Ex y
α β = +,  (9) 
Dividing equation 8 by equation 9 yields:  1
1( )
1( )
()
P
P
Ex y I
Ex y
Ex y α /
−,
⎡⎤ +, ⎣⎦
,= .  
 
All these relationships are important because they show that, under an assumption of α -
returns to scale, most of the existing measures can be expressed in term of the Farrell input 
measure of technical efficiency.  
 
2. α-Returns to Scale Technologies and DEA Models 
We further propose a nonparametric model of production technologies for which the four 
distance functions can be calculated by solving DEA models. Let us consider a set of  J  firms 
( ) ( ) { } 11
np
JJ A xy xy R
+
+ = , ,..., , ∈ . We denote  {1 } JJ = ,, L . The production technology can be 
estimated by enveloping observed firms. Under this DEA framework, the production set for 
constant returns to scale is defined as:  
  () 0
np
CRS j j j j
jJ jJ
Tx y R xx y y λλ λ
+
+
∈∈
⎧⎫
=, ∈ : ≥ , ≤ , ≥ ⎨⎬
⎩⎭ ∑∑  (10) 
We also use a more general CES CET −  model introduced by Färe, Grosskopf and Njinkeu 
(1988) and adapted by Boussemart et. al. (2008) to α -returns to scale. It consists in two parts: 
the output part which is characterized by a Constant Elasticity of Transformation formula and 
the input part which is characterized by a Constant Elasticity of Substitution formula. 
Formally, we consider the map 
mm
r R R + + Φ: →  defined as  1 () ( )
rr
rm zz z Φ =, ,. L  For all  0 r > , IÉSEG Working Paper Series 2008-ECO-6 
 
6 
 
this function is an isomorphism from 
m R+  to itself and its reciprocal is defined on 
m R+  as: 
11 1
1 () ( )
rr
rm zz z
−/ / Φ= , , . L  To simplify the technical exposition we will denote in the remainder of 
the paper: 
  ()
r
r zz Φ =,  (11) 
for all  0 r > . Let us consider the following set: 
 
11 {( ) ( ) ( ) 0} jj jj
jJ jJ
Tx y x x y y
γγ δδ
γδ λλ λ
//
,
∈∈
=, : ≥ , ≤ , ≥ ∑ ∑  (12) 
Tγ δ ,  satisfies T1-T4. It is obvious to see that  11 CRS TT , = .  
We have by definition of the distance functions considered here: 
 
1 1
0
() m i n { ( ) ( ) }
I
II I
kk k j j k j j
jJ jJ
Exy x x y y
γ δ γδ
θλ
θθ λ λ
,≥ ∈∈
,= : ≥ , ≤ ∑∑  (13) 
 
 
1 1
0
() m a x { ( ) ( ) }
O
OO O
kk k j j k j j
jJ jJ
Exy x x y y
γ δ γδ
θλ
θλ θ λ
,≥ ∈∈
,= : ≥ , ≤ ∑∑  (14) 
 
 
1 1
0
1
() m i n { ( ) ( ) }
H
HH H
kk k j j k j j H
jJ jJ
Ex y x x y y
γ δ γδ
θλ
θθ λ λ
θ ,≥ ∈∈
,= : ≥ , ≤ ∑∑  (15) 
 
 
1
0
() m a x { ( 1 ) ( )
P
PP P
kk k j j
jJ
Exy x x
γ γ
θλ
θθ λ
,≥ ∈
,= : − ≥ , ∑  
 
1
(1 ) ( ) }
P
kj j
jJ
yy
δ δ θλ
∈
+≤ ∑  (16) 
 
 
3. Main Results 
Let us denote by, 
I P , 
O P ,
H P  and 
P P  the mathematical programs defined in equations (13), 
(14), (15), and (16) respectively. Let 
IOH
kkk Λ ,Λ ,Λ  and 
P
k Λ  the optimal values of λ  in 
programs. These sets may not be singletons when there exist multiple solutions, i.e. in case of 
degeneracy of the system of linear inequalities. From 
I P , we have for all observed production 
vector k  and all 
II
k λ∗ ∈Λ : 
 
1 1
([ ( )] ) (( ) ( ) ) ( )
II I
kk kk j j j j
jJ jJ
Exy xy x y st
γ δ γδ λλ ∗, ∗,
∈∈
,⋅ ,= , + , − ∑∑  (17) IÉSEG Working Paper Series 2008-ECO-6 
 
7 
 
where ( )
np st R
+
+ ,∈  is a nonnegative input-output slack vector.  
Let us consider α γδ =/. First we establish a relationship between the optimal solutions of 
the programs computing the Farrell input and output measures respectively. Since 
()
I
kk kk E xy xy T γ δ
⎛⎞
⎜⎟ , ⎝⎠ ,⋅ ,∈ and since Tγ δ ,  satisfies α -returns to scale, it follows from 2 that: 
 
1 ([ ( )] ( ) [ ( )] )
OI O
kk kk k kk k E xy Exy x Exy y T
α
γ δ
/
, , ⋅ ,⋅ , , ⋅∈ (18) 
Now from the relationship on distance functions under α -returns to scale: 
() ()
OI Ex y Ex y
α −
⎡⎤ ,= , ⎣⎦ , it follows that: 
  [( ) ]
O
kk k k x Exy y T γ δ
⎛⎞
⎜⎟ , ⎝⎠ ,, ⋅ ∈  (19) 
Combining equations 17 and 19 we obtain: 
 
1
( [ ( )] ) (( [ ( )] )
OO I
kk k k k kj j
jJ
x Exy y Exy x
γ δγ λ∗,
∈
,, ⋅ = , ⋅ , ∑  
 
1
([( ) ] ) ) ( )
OI
kk j j
jJ
Exy y st
δ δδ λ
′ ′
∗,
∈
, ⋅+ , − . ∑  (20) 
 
Where 
1 () ( [ () ] [ () ) ] ) 0
OO
kk kk st Exy s Exy t
α ′′ / ,− = , ⋅ ,− , ⋅ ≥ . Consequently, 
[( ) ]
OI O
kk k Exy
δ λ∗ ,⋅ ∈ Λ  and we can now deduce that: 
  [( ) ]
OI O
kk k k Exy
δ , ⋅Λ ⊂Λ  (21) 
Using a similar procedure if 
OO
k λ∗ ∈Λ  then there is an input-output slack () 0 st ,− ≥  such that:  
  (( ) )(( ) [( ) ] [( ) ] )
II I O
kk kk kk k kk kk k Exy xy Exy xExy Exy y
α , ⋅, = , ⋅, , , ⋅  (22) 
 
1 1
( ([( ) ] ) ([( ) ] ) ) ( )
IOIO
kk j j kk j j
jJ jJ
E xy x Exy y st
γ δ γγ γδ λλ ∗, ∗,
∈∈
=, ⋅ , , ⋅ + , − . ∑∑  (23) 
 
It follows from 23 that: 
  [( ) ]
IO I
kk k k Exy
γ , ⋅Λ ⊂Λ  (24) 
Since [ ( )] [ ( )]
IO
kk kk Exy Exy
γ δ − ,= , , we deduce that  [ ( )]
OO I
kk k k Exy
δ Λ ⊂, ⋅ Λ  which yields the 
following result. 
Lemma 3.1. For all kJ ∈ , we have [( ) ]
OI O
kk k k Exy
δ , ⋅Λ =Λ , with δ γα = / .  
 
Second let us establish a relationship between the optimal solutions of the programs 
computing the hyperbolic measure and the Farrell input measure. Since 
()
I
kk kk Exy xy T γ δ
⎛⎞
⎜⎟ , ⎝⎠ ,⋅ ,∈ and since Tγ δ ,  satisfies α -returns to scale, it follows from 2 that: IÉSEG Working Paper Series 2008-ECO-6 
 
8 
 
 
1 1 ([ ( )] ( ) [ ( )] )
HI H
kk k k k kk k Ex y E xyxEx y y T
α
γ δ
− −
, ,, ⋅ ⋅ , , ⋅ ∈  (25) 
Now from the relationship on distance functions under α -returns to scale: 
1 () ()
HI Ex y E x y
α
α+ ⎡⎤ ,= , ⎣⎦  and it follows that: 
 
1 ( [ () ] [ () ])
HH
kk k kk k Ex y xEx y y T γ δ
−
, ,⋅ , , ⋅ ∈  (26) 
Combining equations 17 and 26 we obtain: 
 
1 ([ ( )] [ ( )] )
HH
kk k kk k Ex y xEx y y
− , ⋅, , ⋅  (27) 
 
1 1
( ([( ) ] ) ([( ) ] ) ) ( )
HI HI
kk j j kk j j
jJ jJ
E xy x E xy y s t
γ δ δγ δδ λλ
−− ′ ′
∗, ∗,
∈∈
=, ⋅ , , ⋅ + , − . ∑∑  
 
where 
1 1 () ( [ () ][ () ] ) 0
HH
kk kk st Ex y s Ex y t
α − ′′ − ,− = , ⋅ , , ⋅ ≥ . Consequently, we obtain 
[( ) ]
H IH
kk k Ex y
δ λ
−
∗ ,⋅ ∈ Λ  and we can now deduce that: 
  [( ) ]
H IH
kk k k Ex y
δ − , ⋅Λ ⊂Λ  (28) 
Using a similar procedure if 
H H
k λ∗ ∈Λ  then there is an input-output slack ( ) 0 st ,− ≥  such that: 
  (( ) )
I
kk kk E xy xy , ⋅,  (29) 
 
1 1 ([ ( )] ( ) ( ) [ ( )] )
HH H H
kk kk k kk kk k Ex y Ex yx Ex y Ex y y
α − =, ⋅ , ⋅ , , ⋅ , ⋅  (30) 
 
1 1
( ([( ) ] ) ([( ) ] ) ) ( )
HHHH
kk j j kk j j
jJ jJ
E xy x E xy y st
γ δ δγ δδ λλ ∗, ∗,
∈∈
=, ⋅ , , ⋅ + , − . ∑∑  (31) 
 
It follows that:  
  [( ) ]
H HI
kk k k Ex y
δ , ⋅Λ ⊂Λ . (32) 
This implies that  [ ( )]
H HI
kk k k Ex y
δ − Λ⊂ , ⋅ Λ  yields the following result.  
Lemma 3.2. For all kJ ∈ , we have  [ ( )]
H HI
kk k k Ex y
δ − Λ =, ⋅ Λ ,  with δ γα = / .  
 
Finally, we establish the link between the solutions of the programs computing Farrell and 
proportional distance functions. Since  ( )
I
kk kk E xy xy T γ δ
⎛⎞
⎜⎟ , ⎝⎠ ,⋅ ,∈ and since Tγ δ ,  satisfies α -
returns to scale, it follows from 2 that: 
 
1
(1 ( ) ( ) 1 ( ) )
PI P
kk k k Ex y Exy x Ex y y T
α
γ δ
/
, ⎡⎤ ⎡⎤ +, , ⋅ ⋅ , +, ⋅ ∈ ⎣⎦ ⎣⎦  (33) 
Now from Lemma 1:  1
1( )
1( )
()
P
P
Ex y I
Ex y
Ex y α /
−,
⎡⎤ +, ⎣⎦
,=  and it follows that: 
  (1 ( ) 1 ( ) )
PP
kk E xy x E xy y T γ δ , ⎡⎤ ⎡⎤ −, ⋅ , +, ⋅ ∈ ⎣⎦ ⎣⎦  (34) IÉSEG Working Paper Series 2008-ECO-6 
 
9 
 
Combining equations 17 and 34 we obtain: 
  (1 ( ) 1 ( ) )
PP
kk Ex y x Ex y y ⎡⎤ ⎡⎤ −, ⋅ , +, ⋅ ⎣⎦ ⎣⎦  
 
1 1
(( 1 ( ) ) ( 1 ( ) ) ) ( )
PI PI
jj jj
jJ jJ
E xy x E xy y s t
γ δ
δδ γδ λλ
′′
∗, ∗,
∈∈
⎡⎤ ⎡⎤ = + ,⋅ , + ,⋅ + , − . ⎣⎦ ⎣⎦ ∑∑  (35) 
 
where 
1
() ( 1( ) 1( ) ) 0
PP st Ex y s Ex y t
α / ′′ ⎡⎤ ⎡⎤ ,− = + , ⋅ , + , ⋅ ≥ ⎣⎦ ⎣⎦ . Consequently, we obtain 
[1 ( )]
P IP
kk k Exy
δ λ∗ +, ⋅ ∈ Λ  and we can now deduce that: 
  1( )
P IP
kk Ex y
δ
⎡⎤ + ,⋅ Λ ⊂ Λ ⎣⎦ (36) 
Using a similar procedure if 
P P
k λ∗ ∈Λ  then there is an input-output slack ( ) 0 st ,− ≥  such that: 
  (( ) )
I
kk kk E xy xy , ⋅,  (37) 
 
1 1 ([1 ( )] [1 ( )] [1 ( )] [1 ( )] )
PP P P
kk kk k kk kk k Exy Exy x Exy Exy y
α − − = +,⋅ +, ⋅ , +,⋅ +, ⋅  (38) 
 
1 1
(( [1 ( )] ) ( [1 ( )] ) ) ( )
PP PP
kk j j kk j j
jJ jJ
Exy x Exy y st
γ δ δγ δδ λλ
−−
∗, ∗,
∈∈
= +,⋅ ,+,⋅ + , − . ∑∑  (39) 
 
It follows that:  
  [1 ( )]
P PI
kk k k Exy
δ − + ,⋅ Λ ⊂ Λ .  (40) 
This implies that  [1 ( )]
P PI
kk k k Exy
δ Λ⊂ + , ⋅ Λ yields the following result.  
Lemma 3.3. For all kJ ∈ , we have  [1 ( )]
P PI
kk k k Exy
δ Λ =+ , ⋅ Λ ,  with δ γα = / .  
 
By letting δ = γ = 1, Lemmas 3.1 to 3.3 reduce to the results found in BBL (2007) for constant 
returns to scale technologies. 
 
Conclusion 
This note formally establishes the relationships among the input-based, the output-based, the 
hyperbolic, and the proportional distance functions as well as their relative DEA optimal 
solutions for multi-output homogeneous technologies. In BBL (2007), the assumption of 
constant returns to scale was considered as the key assumption to allow some links among the 
various DEA models. For practitioners, it could have been seen like a weakness since CRS is 
often judged as a restrictive assumption for empirical works. The main contribution of this 
generalization is to show that previous results were not specific to constant returns but rely on IÉSEG Working Paper Series 2008-ECO-6 
 
10 
 
a weaker assumption of homogeneity.  
 
References 
Boussemart J-P., Briec W., Leleu H. (2007). Linear programming solutions and distance 
functions under a constant returns to scale technology. Journal of the Operational Research 
Society, doi:10.1057/palgrave.jors.2602519.  
 
Boussemart J-P., Briec W., Peypoch N., Tavera C. (2008). α −Returns to scale and multi-
output production technologies. European Journal of Operational Research, 
doi:10.1016/j.ejor.2008.05.28.  
 
Färe R., Grosskopf S., Njinkeu D. (1988). On Piecewise reference technologies, Management 
Science, 34: 1507-1511.  
 
Färe R., Mitchell T.(1993). Multiple outputs and homotheticity, Southern Economic Journal, 
60: 287-296.  
 
Lau L.J. (1978). Application of profit functions, in Production Economics: A Dual Approach 
to Theory and Applications, edited by Fuss and McFadden, North-Holland, Amsterdam.  
 

Application of profit functions, in Production Economics: A Dual Approach to Theory and Applications, edited by Fuss and McFadden,

Linear programming solutions and distance functions under a constant returns to scale technology.

On Piecewise reference technologies,

T.(1993). Multiple outputs and homotheticity,

α −Returns to scale and multioutput production technologies.

Linear programming solutions and distance functions under a constant returns to scale technology

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Linear programming solutions and distance functions under a constant returns to scale technology

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