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106
reduction in the output of electricity, holding the input vector constant (see
Färe and Primont, 1995).
Suppose we have
n independent decision making units, denoted by
DMU
j
( 1, 2, ,jn= K ). Each DMU consumes m inputs x
ij
, (i = 1, 2, ..., m) to
produce
s desirable outputs y
rj
, (r = 1, 2, ..., s) and emits
k
undesirable
outputs b
tj
, (t = 1, 2, ..., k).
When the undesirable outputs are strong disposable, the production
possibility set can be expressed as
11 1
{( , , ) | , , ,
0, 1, 2,..., }.
nn n
s
jj j j
jj j
j
jj
Txy xx yy b
jn
bb
ηη η
η
== =
=≤≥≥
≥=
∑∑ ∑
When the undesirable outputs are weakly disposable, as suggested by
Shephard (1970), the production possibility set may be written as
11 1
{( , , ) | , , ,
0, 1, 2,..., }.
nn n
w
jj j j
jj j
j
jj
Txy xxyy b
jn
bb
ηη η
η
== =
=≤≥=
≥=
∑∑ ∑
However, the concept of strong and weak disposability is relative and
internal to each DMU’s technology while the concept of regulations
(reduction limits of undesirable outputs) is imposed externally upon DMUs.
For more detailed discussion on strong and weak disposability of
undesirable outputs, the reader is referred to Färe and Primont (1995), and
Färe and Grosskopf (2004a).
3. THE HYPERBOLIC OUTPUT
This section presents the hyperbolic output efficiency measure developed
in Färe et al. (1989). Hyperbolic efficiency explicitly incorporates the fact
that the performance of production processes should be compared to an
environmentally friendly standard, that is, reference facets for efficiency
measurement should not be those defined by observed activities that produce
both larger amounts of desirable production and waste but larger amounts of
the former and smaller amounts of the latter (Zofìo and Prieto, 2001)
Based on the assumption of strong disposability of undesirable outputs,
DMUs’ eco-efficiencies can be computed by solving the following
programming problem:
EFFICIENCY MEASURE
Chapter 6
107
1
1
1
1
0
0
00
0
0
, 1, 2,..., ,
, 1, 2,..., ,
1
, 1, 2,..., ,
1,
, , , 0, , , , .
n
jij i
j
n
jr
j
n
jt
j
n
j
j
ji r t
i
rj
r
tj
t
Max
subject to
xs x i m
y
syr s
bs b t k
s s s for all j i r t
η
η
η
η
η
ρ
ρ
ρ
−
=
+
=
+
=
=
−++
+= =
−= =
−= =
=
≥
∑
∑
∑
∑
(1)
Obviously, model (1) is a non-linear programming problem, and
increases desirable outputs and decreases undesirable outputs
simultaneously. Similarly, based on the assumption of weak disposability of
undesirable outputs, we have:
0
M
ax
ρ
subject to
1
0
, 1, 2,..., ,
n
jij i
j
i
x
sxi m
η
−
=
+= =
∑
1
00
, 1,2,..., ,
n
jr
j
rj
r
y
syr s
η ρ
+
=
−= =
∑
1
0
0
1
, 1, 2,..., ,
n
j
j
tj
t
bbt k
η
ρ
=
==
∑
1
1,
n
j
j
η
=
=
∑
, , 0, , , .
ji r
s
sforalljir
η
−+
≥
(2)
To solve models (1) and (2), Färe et al. (1989) suggested that the
nonlinear constraints
1
0
0
1
n
j
j
tj t
bb
η
ρ
=
≥
∑
approximated by linear expressions
Hua & Bian, DEA with Undesirable Factors
108
00
1
0
2
n
tt j
j
tj
bb b
η
ρ
=
−≤
∑
; similarly, the constraints
1
0
0
1
n
j
j
tj t
bb
η
ρ
=
=
∑
can be
approximated by linear expressions
00
1
0
2
n
tt j
j
tj
bb b
η
ρ
=
−=
∑
.
The above approaches of treating undesirable outputs can also be
extended to treat undesirable inputs of a recycling process. Suppose DEA
inputs data are
(,)
D
U
jjj
X
XX= , where
D
j
X
and
U
j
X
represent desirable
inputs to be decreased and undesirable inputs to be increased, respectively.
Based on the assumption of strong disposability of undesirable inputs, we
have
00
1
0
1
0
1
0
0
,
1
,
,
n
DD
jj
j
n
UU
jj
j
n
j
j
j
Min
subject to
x
sx
x
sx
ys y
η
η
η
θ
θ
θ
−
=
−
=
+
=
+=
+=
−=
∑
∑
∑
1
1,
0, 1, 2,..., .
n
j
j
j
j
n
η
η
=
=
≥=
∑
(3)
Obviously, model (3) decreases desirable inputs and increases
undesirable inputs simultaneously. Similarly, based on the assumption of
weak disposability of undesirable inputs, we have:
00
1
0
1
0
1
1
0
0
,
1
,
,
1,
0, 1, 2,..., .
n
D
D
jj
j
n
UU
jj
j
n
j
j
n
j
j
j
j
Min
subject to
x
sx
xx
ys y
j
n
η
η
η
η
η
θ
θ
θ
−
=
=
+
=
=
+=
=
−=
=
≥=
∑
∑
∑
∑
(4)
Chapter 6
109
Models (3) and (4) are nonlinear programming problems, and can also be
solved by applying similar linear approximations to the nonlinear constraints
as those used in models (1) and (2).
4. A LINEAR TRANSFORMATION
This section presents the work of Seiford and Zhu (2002). Under the
context of the BCC model (Banker et al., 1984), Seiford and Zhu (2002)
developed an alternative method in dealing with desirable and undesirable
factors in DEA. They introduced a linear transformation approach to treat
undesirable factors and then incorporated transformed undesirable factors
into standard BCC DEA models.
Consider the treatment of undesirable outputs. In order to maintain
translation invariance, there are at least two translation approaches to treat
undesirable outputs: a
linear monotone decreasing transformation or a
nonlinear monotone decreasing transformation (e.g.,
1 b ). As indicated by
Lewis and Sexton (1999), the non-linear transformation will demolish the
convexity relations. For the purpose of preserving convexity relations,
Seiford and Zhu (2002) suggested a linear monotone decreasing
transformation,
0
jj
bbv=− + ≥ , where v is a proper translation vector that
makes
0
j
b > . That is, we multiply each undesirable output by ( 1)
−
and
find a proper translation vector
v to convert negative data to non-negative
data.
Based upon the above linear transformation, the standard BCC DEA
model can be modified as the following linear program:
0
1
0
1
0
1
1
, 1, 2,..., ,
, 1,2,..., ,
, 1, 2,..., ,
1,
, , , 0, , , , .
n
jij i i
j
n
jrj r r
j
n
jtj t t
j
n
j
j
ji r t
Max h
subject to
x
sxi m
yshyr s
bs hbt k
s
ss foralljirt
η
η
η
η
η
−
=
+
=
+
=
=
−++
+= =
−= =
−= =
=
≥
∑
∑
∑
∑
(5)
FOR UNDESIRABLE FACTORS
Hua & Bian, DEA with Undesirable Factors
110
Model (5) implicates that a DMU has the ability to expand desirable
outputs and to decrease undesirable outputs simultaneously. A
0
DMU is
efficient if
*
0
1h =
and all
0
irt
sss
−++
===
. If
*
0
1h >
and (or) some
i
s
−
,
r
s
+
or
t
s
+
is non-zero, then the
0
DMU is inefficient.
The above transformation approach can also be applied to situations
when some inputs are undesirable and need to be increased to improve
DMUs’ performance. In this case, Seiford and Zhu (2002) rewrite the DEA
input data as
(,)
D
U
jjj
X
XX= , where
D
j
X
and
U
j
X
represent inputs to be
decreased and increased, respectively. We multiply
U
j
X
by “-1” and then
find a proper translation vector
w to let all negative data to non-negative
data, i.e.,
0
UU
jj
xxw=− + ≥ , then we have
0
1
0
1
0
1
,
,
,
n
D
D
jj
j
n
UU
jj
j
n
jj
j
Min
subject to
x
sx
x
sx
ys y
τ
ητ
ητ
η
−
=
−
=
+
=
+=
+=
−=
∑
∑
∑
1
1,
n
j
j
η
=
=
∑
0, 1, 2,...,
j
jn
η
≥=
.
(6)
In model (6),
D
j
X
is decreased and
U
j
X
is increased for a DMU to
improve its performance.
The linear transformation for undesirable factors can preserve the
convexity and linearity of BCC DEA model. However, such approach can
only be applied to the BCC model.
5. A DIRECTIONAL DISTANCE FUNCTION
Based on weak disposability of undesirable outputs, Färe and Grosskopf
(2004a) provided an alternative approach to explicitly model a joint
environmental technology. The key feature of the proposed approach is that
it distinguishes between weak and strong disposability of undesirable outputs
and uses a directional distance function.
Based on weak disposability of undesirable outputs, the production
outputs set can be written as
Chapter 6
111
11 1
() {(, , )| , , ,
0, 1, 2,..., }.
nn n
w
jj j j
jj j
j
jj
Px xy x x y y b
jn
bb
ηη η
η
== =
=≤≥=
≥=
∑∑ ∑
Null-jointness is imposed via the following restriction on the undesirable
outputs:
0
tj
b >
, (
1, 2,...,tk=
,
1, 2,...,
j
n=
).
Since the traditional DEA model is a special case of directional output
distance function, Färe and Grosskopf (2004a) adopted a more general
directional output distance function to model the null-jointness production
technology.
Let
(, )
y
b
ggg
=−
be a direction vector. We have:
000
0
1
(, ,;)
, 1,2,..., ,
n
jij
i
j
Dx y b g Max
subject to
x
xi m
β
η
=
=
≤=
∑
r
0
1
, 1, 2,..., ,
r
n
jrj y
r
j
yy gr s
ηβ
=
≥+ =
∑
0
1
, 1, 2,..., ,
t
n
jtj
t
b
j
bb gt k
ηβ
=
=− =
∑
0, 1, 2,... .
j
j
n
η
≥=
(7)
Model (7) can also be rewritten as
000
00
,
(, ,;)
( ) ( ).
w
yb
b
Dx y b g Max
subject to
yg gPx
β
ββ
+−
=
∈
r
(8)
A DMU is efficient when
000
(, ,;) 0Dx y b g
=
r
and inefficient when
000
(, ,;) 0Dx y b g >
r
.
The key difference between this and the traditional DEA model is the
inclusion of the direction vector
(, )
y
b
ggg
=−
. If
1
y
g
=
and 1
b
g
−
=− ,
then the solution represents the net performance improvement in terms of
feasible increases in desirable outputs and decreases in undesirable outputs.
Therefore, we have for desirable outputs
0000 0
((,,;)1)
rr
yDxybg y
+
=⋅−
r
000
(, ,;)Dx y b g
r
and for undesirable outputs
0000 0
((,,;)1)
tt
bDxybg b−
⋅
−
r
000
(, ,;)Dx y b g=
r
.
Based on weak disposability of undesirable outputs and directional
distance function approach, Färe and Grosskopf (2004b) also presented an
index number approach to address eco-efficiency evaluation problem.
Hua & Bian, DEA with Undesirable Factors
112
As pointed out by Seiford and Zhu (2005), model (7) can be linked to the
additive DEA model (Charnes et al., 1985). Note in model (7), that
y
g
and
b
g can be treated as user specified weights, and 0
β
=
is a feasible solution
to model (7). Seiford and Zhu (2005) replace
y
g
and
b
g by
r
g
+
and
t
g
−
,
respectively and obtain
11
()
sk
rt
rt
g
gMax
subject to
β
+−
==
+
∑∑
0
1
, 1,2,..., ,
n
jij
i
j
x
xi m
η
=
≤=
∑
0
1
, 1,2,..., ,
r
n
jrj
r
j
ygyr s
ηβ
+
=
− ≥=
∑
0
1
0.
, 1,2,..., ,
0, 1,2,... ,
t
n
jtj
t
j
j
bgbt k
jn
β
ηβ
η
−
=
≥
+= =
≥=
∑
(9)
If we use
r
β
+
and
t
β
−
rather than a single
β
in model (9), then we have
(Seiford and Zhu, 2005)
11
0
1
0
1
0
1
, 0
, 1,2,..., ,
, 1, 2,..., ,
, 1, 2,..., ,
0, 1,2,... ,
sk
rt
rt
r
t
rt
n
jij
i
j
n
jrj r
r
j
n
jtj t
t
j
j
Max
subject to
xxi m
yy gr s
bb gt k
jn
ββ
η
ηβ
ηβ
η
ββ
+−
==
+
−
+−
=
+
=
−
=
+
+
≥
≤=
==
=− =
≥=
∑∑
∑
∑
∑
.
(10)
As noted in Section 2, weak disposability is applied when inequalities are
changed into equalities related to the constraints of undesirable outputs. In
this case, weak disposability is automatically applied in model (10), because
in optimality the equality always holds for the undesirable outputs (see Zhu
(1996) and Seiford and Zhu (2005)).
Chapter 6
113
Let
rrr
sg
β
+++
= and
ttt
sg
β
−−−
= , then model (10) can be written as the
following additive DEA model:
11
0
1
0
1
0
1
, 0
, 1,2,..., ,
, 1, 2,..., ,
, 1, 2,..., ,
0, 1,2,... ,
sk
t
r
rt
rt
r
t
rt
n
jij
i
j
n
jrj
r
j
n
jtj
t
j
j
s
s
s
s
ss
Max
gg
subject to
xxi m
yyr s
bbt k
jn
η
η
η
η
−
+
+−
==
+
−
+−
=
=
=
+
−
+
≥
≤=
==
==
≥=
∑∑
∑
∑
∑
.
(11)
Model (11) treats undesirable outputs as inputs, and decreases
undesirable outputs and increases desirable outputs simultaneously.
6. NON-DISCRETIONARY INPUTS
We here consider a performance evaluation problem provided in Hua et
al. (2005) where the eco-efficiency evaluation of paper mills along the Huai
River in China is discussed To deal with the increasingly serious
environmental problem of the Huai River, State Environmental Protection
Administration of China (EPAC) had passed an Act to limit the total
quantities of main pollutants emission (e.g., biochemical/chemical oxygen
demand (BOD/COD), nitrogen (N), etc) from industrial enterprises (e.g.,
paper mills) along the River. According to the EPAC’s Act, an annual
emission quota of BOD is allocated to each paper mill, and each paper mill
should not emit more BOD than the allocated quota.
For these paper mills, emission of BOD is an undesirable output, and the
corresponding emission quota is a non-discretionary input allocated by
EPAC. The non-discretionary input, on the one hand, restricts pollutants
emission. On the other hand, it also affects the production of desirable
outputs (paper products). When we evaluate efficiencies of paper mills along
the Huai River, undesirable outputs and non-discretionary inputs should be
modeled in a single model.
AND UNDESIRABLE OUTPUTS IN DEA
Hua & Bian, DEA with Undesirable Factors
114
Denote by
12
( , ,..., )
T
jjjpj
Z
zz z= the given
p
non-discretionary inputs,
12
( , ,..., )
T
jjjmj
Xxx x= the m discretionary inputs,
12
(, ,...,)
T
jjjsj
Yyy y=
the
s desirable outputs and
12
(, ,...,)
T
jjjkj
B
bb b= the
k
undesirable outputs
(
kp≥
). We next follow Seiford and Zhu (2002) (or section 4) to make a
linear monotone decreasing transformation,
0
jj
bbv
=
−+≥, where v is a
proper translation vector that makes
0
j
b > . We then use the non-radial
DEA model to generate the following model
0
11
11
sk
rt
rt
Max
sk sk
φ
αβ
==
=+
++
∑∑
subject to
0
1
, 1, 2,..., ,
n
jij i i
j
x
sxi m
η
−
=
+= =
∑
0
1
, 1, 2,..., ,
n
jrj rr
j
y
yr s
ηα
=
==
∑
0
1
, 1,2,..., ,
n
jtj tt
j
bbt k
ηβ
=
==
∑
1
1,
n
j
j
η
=
=
∑
1, 1, , ,
rt
f
or all r t
α
β
≥≥
, 0, , .
ji
s for all j i
η
−
≥
(12)
Model (12) is an output-oriented DEA model. Values of
r
α
and
t
β
are
used to determine the amount of possible augmentation and reduction in
each desirable and undesirable output, respectively. If some transformed
undesirable outputs
0t
b become zero, then we set 1
t
β
=
for those t with
0
0
t
b = .
To incorporate the non-discretionary inputs into model (12), we let
'
0
jj
zQz=−≥( 1, 2,..., )jn= , where Q is a proper translation vector that
makes
0
j
z
′
> .
Banker and Morey (1986) used the following to model non-discretionary
inputs
''
0
1
n
jlj l
j
zz
η
=
≥
∑
( 1, 2,...,lp= ).
(13)
In the presence of undesirable outputs, Eq.(13) is inappropriate because
it considers only positive impacts of non-discretionary inputs on desirable
outputs while ignoring impact of non-discretionary inputs on undesirable
outputs.
Chapter 6
115
Hua et al. (2005) modified (13) into
''
0
1
n
jlj l
j
zz
η
=
=
∑
(
1, 2,...,lp=
).
(14)
Equation (14) requires that reference units utilize the same levels of the
transformed non-discretionary inputs as that of the assessed DMU on
average.
Based on the reference set defined in Equation (14), model (12) can be
extended to the following output-oriented DEA model:
0
11
11
sk
rt
rt
Max
sk sk
ϕ
αβ
==
=+
++
∑∑
subject to
0
1
, 1, 2,..., ,
n
jij i i
j
x
sxi m
η
−
=
+= =
∑
''
0
1
, 1, 2,..., ,
n
jlj l
j
zzl p
η
=
==
∑
0
1
, 1, 2,..., ,
n
jrj rr
j
yyr s
ηα
=
==
∑
0
1
, 1, 2,..., ,
n
jtj tt
j
bbt k
ηβ
=
==
∑
1
1,
n
j
j
η
=
=
∑
1, 1, , ,
rt
f
or all r t
α
β
≥≥
,0, ,
ji
s for all j i
η
−
≥ .
(15)
For any given non-discretionary inputs, model (15) increases each
desirable output and each transformed undesirable output simultaneously
with different proportions.
Let (
** **
, , ,
rti
s
α
βη
−
) be the optimal solution to model (15). A
0
DMU is efficient if
*
0
1
ϕ
= (i.e.,
*
1
r
α
= and
*
1
t
β
=
) and all
*
0
i
s
−
=
. If
*
0
1
ϕ
> (i.e., any of
*
r
α
or
*
t
β
is greater than one) and (or) some of
*
i
s
−
are
non-zero, then the
0
DMU is inefficient.
We now apply this approach to 32 paper mills. Table 6-1 reports
characteristics of the data set with labor and capital as the two discretionary
inputs, paper products as the desirable output, emitted BOD as the
undesirable output, and the emission quota of BOD (BOD-Q) allocated to
each paper mill as a non-discretionary input.
Hua & Bian, DEA with Undesirable Factors