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Semi-disposability of undesirable outputs in data envelopment analysis for environmental assessments
Accepted Manuscript
Semi-disposability of undesirable outputs in data envelopment analysis for environmental assessments Lei Chen , Ying-Ming Wang , Fujun Lai PII: DOI: Reference:
S0377-2217(16)31076-1 10.1016/j.ejor.2016.12.042 EOR 14173
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
30 May 2016 29 November 2016 27 December 2016
Please cite this article as: Lei Chen , Ying-Ming Wang , Fujun Lai , Semi-disposability of undesirable outputs in data envelopment analysis for environmental assessments, European Journal of Operational Research (2016), doi: 10.1016/j.ejor.2016.12.042
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Highlights Semi-disposability assumption of undesirable outputs is proposed for environmental assessment
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Measure models of semi-disposability are proposed under constant and variable returns to scale
Reference point comparison is proposed to gain the non-disposal
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degree of undesirable outputs
Semi-disposability is extended to uncertain circumstances by the interval non-disposal degree
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Two empirical examples are provided to illustrate the effectiveness of
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our method
1
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Semi-disposability of undesirable outputs in data envelopment analysis for environmental assessments Lei Chena,b, Ying-Ming Wanga*,
Fujun Laib
350116, PR China
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a. Decision Science Institute , School of Economics & Management, Fuzhou University, Fuzhou
b. College of Business, University of Southern Mississippi, Hattiesburg, MS 39560, United States
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Abstract: The assumptions of strong and weak disposability for undesirable outputs have long dominated studies of data envelopment analysis for environmental assessments. Unfortunately, these assumptions cannot describe the diverse technical features of different undesirable outputs during the actual production process. Thus, we introduce a non-disposal degree to develop a new semi-disposability assumption,
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which can replace the assumptions of strong and weak disposability in environmental
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assessments. This assumption ensures that decision makers can address undesirable outputs freely within the scope of current production technology; otherwise, they have
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to reduce desirable outputs in the same proportion to decrease undesirable outputs. A reference point comparison method is proposed for determining the non-disposal
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degree from an objective perspective. The assumption of semi-disposability is extended to uncertain circumstances by using the interval non-disposal degree. Finally,
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two empirical examples are provided to illustrate the effectiveness of the semi-disposability assumption. Keywords: Data envelopment analysis; Efficiency; Reference point comparison; Return to scale; Semi-disposability.
*Corresponding author. Tel.: +86 0591 22866677; E-mail: [email protected] (L. Chen), [email protected] (Y.M. Wang), [email protected] (F.J. Lai) 2
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1 Introduction Due to rapid economic development, the increasingly severe effects of environmental pollution have attracted widespread attention among the international community. To achieve a “win-win” situation in the economy and environment, many studies have explored sustainable development models (e.g., Gorobets, 2014; Mariano
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et al., 2015). In particular, the environmental assessment of economic development is one of the main research areas, which is an important for the formulation of economic transition policies. For instance, Woo et al. (2015) examined the environmental efficiency from static and dynamic perspectives in 31 OECD countries using the data
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envelopment analysis (DEA) approach and the Malmquist productivity index, where their results provided important contributions that could be used to improve the environmental efficiency by policymakers. Bian et al. (2015) evaluated the environmental efficiency of Chinese regional industrial systems using a two-stage
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slacks-based measure approach and found the main sources of inefficiency. Therefore, scientific environmental assessment is highly significant for sustainable economic
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development.
As one of the most popular methods for efficiency assessment, DEA is a
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nonparametric approach for evaluating the relative efficiency of homogeneous decision-making units (DMUs) with multiple inputs and outputs. However, unlike
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traditional efficiency assessments (e.g., Martinez-Núñez & Pérez-Aguiar, 2014; Wanke et al., 2016), environmental assessments need to consider the desirable outputs
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produced during the production process as well as undesirable outputs, such as waste and pollution. This is because the undesirable outputs are not what decision makers expect but they usually appear with desirable outputs during the actual production process. A DEA model with undesirable outputs was proposed by Färe et al. (1989), and then the DEA approach has been employed widely to deal with the problem of environmental assessment throughout the world (e.g., Baležentis et al., 2016; Wu et al., 2016). In general, these studies can be classified into several categories. The first category mainly focuses on the treatment approach for undesirable outputs in the DEA 3
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model (e.g., Dyckhoff & Allen, 2011; Seiford & Zhu, 2002). The second category deals with undesirable outputs by using different efficiency measures in the DEA approach (e.g., Arabi et al., 2015; Chen et al., 2015; Sahoo et al. 2011). The third category focuses on the disposability of undesirable outputs (e.g., Färe et al., 1989; Sueyoshi et al. 2012b). Färe et al. (1989) proposed the assumptions of strong and weak disposability for
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undesirable outputs, and these assumptions have dominated most previous studies of DEA for environmental assessments (e.g., Hang et al., 2015; Song et al., 2013). According to the assumption of strong disposability, the undesirable outputs may be reduced freely or with no cost at the will of decision makers. For instance, Korhonen
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et al. (2004) constructed a DEA model to measure the eco-efficiency of power plants in a European country based on the assumption of strong disposability. Liu et al. (2015)
constructed
a
two-stage
DEA
model
with
undesirable
input-intermediate-outputs based on the strong disposability perspective, and used it
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to evaluate the efficiency of listed commercial banks in China. According to the assumption of weak disposability, undesirable outputs need to be reduced in
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proportion to the desirable outputs because the joint production of the desirable and undesirable outputs occurs in this case. For instance, Leleu (2013) discussed a current
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error in the modeling of weak disposability under the variable returns to scale technique, and introduced a hybrid model to ensure the economically meaningful
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jointness of desirable and undesirable outputs while constraining the shadow prices of undesirable outputs to their expected sign. Vlontzos et al. (2014) utilized a non-radial
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DEA model based on the assumption of weak disposability to evaluate the energy and environmental efficiency of the primary sectors in EU member state countries. Zha et al. (2016) considered the weak disposability of undesirable outputs and proposed radial and non-radial stochastic DEA models to evaluate the pure efficiency of energy use and CO2 emissions. The strong and weak disposability assumptions of undesirable outputs have gained acceptance among the majority of researchers, but they still have some shortcomings. Kuosmanen (2005) noted that the common specification of weak 4
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disposability implicitly assumes that all DMUs in the sample apply a uniform abatement factor, although there is no theoretical reason to explain why, so he then proposed a new model of weak disposability that allows for non-uniform abatement factors. Kuosmanen et al. (2011) argued that the assumption of strong disposability for undesirable outputs implies that a finite amount of input can produce an infinite amount of undesirable output, which is physically impossible. Yang and Pollitt (2010)
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emphasized the necessity of distinguishing weak and strong disposability among undesirable outputs in environmental assessments, and proposed a model to distinguish them based on the technical features of undesirable outputs. Sueyoshi and Goto (2014) argued that under the weak disposability assumption for undesirable
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outputs, a decrease in undesirable outputs occurs with a decrease in desirable outputs, which cannot reflect the effects of managerial effort and technology innovation on environmental
efficiency.
However,
different
disposability assumptions
for
undesirable outputs imply different production possibility sets of production activity,
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which may yield diverse evaluation results (Adler & Volta, 2016). Therefore, developing a scientific disposability assumption for undesirable outputs is necessary
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for DEA environmental assessment, which is the aim of the present study. Overall, the greatest shortcoming of the strong and weak disposability
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assumptions is that these assumptions cannot describe the diverse technical features of different undesirable outputs during the actual production process. Therefore, this
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study introduces the concept of a non-disposal degree to develop a new assumption, i.e., semi-disposability, to address undesirable outputs under constant returns to scale
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(CRS) and variable returns to scale (VRS). We then discuss this disposability according to its conceptual, methodological, and economic implications. In addition, a reference point comparison (RPC) method is developed to determine the non-disposal degree from an objective perspective. The assumption of semi-disposability is then extended to uncertain circumstances by using an interval non-disposal degree. Finally, the semi-disposability assumption of undesirable outputs is applied to two empirical examples to illustrate its effectiveness. The remainder of this paper is organized as follows. Section 2 reviews the 5
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concepts of strong and weak disposability. Section 3 describes the assumption of semi-disposability under CRS and the RPC method. Section 4 extends the semi-disposability assumption to the case of VRS. In Section 5, we describe the application of the proposed method to two empirical examples. Finally, we give our conclusions in Section 6.
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2 Weak and strong disposability
As mentioned in the previous section, Färe et al. (1989) proposed the strong and weak disposability assumptions for undesirable outputs. Following their study, we
, xm Rm as an inputs vector, Y y1 , y2 ,
, ys Rs as a
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denote X x1 , x2 ,
desirable outputs vector, and Z z1 , z2 ,
, zh Rh as an undesirable outputs
vector for DMU j . The production technology comprising all feasible (Y , Z , X ) denoted by T {(Y , Z , X ) X can produce Y , Z } . Equivalently, the corresponding
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output set can be denoted by P( X ) Y , Z Y , Z , X T . It is assumed that there
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are n DMUs, and then the weak disposability under CRS can be specified by the following vector notation: n
n
n
j 1
j 1
, n ,
jY j Y , j Z j Z , j X j X , j 0 j 1,
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P w ( X ) Y , Z
j 1
(1)
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where the subscript j represents the jth DMU and j represents the jth intensity
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variable. According to the assumption of weak disposability, inputs and desirable n
outputs are strongly disposable by the inequality constraints n
Y j 1
j
j
X j 1
j
j
X
and
Y , which allow a vertical extension, whereas undesirable outputs are not.
Then, the output-oriented efficiency measure for a specific DMU k can be obtained as follows:
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max kw n
s.t.
x j 1
j ij
xik
n
y j 1
j
rj
kw yrk
fj
z fk
n
z j 1
j
i 1,
j 0
,m
r 1, f 1,
j 1,
(2)
,s h
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, n.
where kw represents the efficiency of DMU k based on the assumption of weak disposability under CRS.
Considering that undesirable outputs are never wanted by decision makers, the
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strong disposability is specified as the following vector notation according to the studies of Yang and Pollitt (2010), and Sueyoshi et al. (2014): P s ( X ) Y , Z
n
n
n
j 1
j 1
jY j Y , j Z j Z , j X j X , j 0 j 1, j 1
n
Z j
j
(3)
Z allows for the strong disposability of
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where the inequality constraint
, n ,
j 1
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undesirable outputs. Then, under CRS, the output-oriented efficiency measure for a specific DMU k can be obtained as follows:
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max ks
AC
CE
s.t.
n
x j 1
xik
i 1,
,m
rj
ks yrk
r 1,
,s
fj
z fk
j ij
n
y j 1
j
n
z j 1
j
j 0
f 1,
j 1,
(4)
h
, n.
where ks represents the efficiency of DMU k based on the assumption of strong disposability. As described by Yang and Pollitt (2010), Fig. 1 illustrates the difference between the weak and strong disposability assumptions, where the contour lines BAD and 7
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CAD indicate the efficiency frontiers based on the assumptions of weak and strong disposability, respectively. Actually, the assumption of strong disposability for inputs and undesirable outputs makes the efficiency frontier BAD a convex hull under the constraint of CRS. By contrast, imposing weak disposability on undesirable outputs allows the efficiency frontier CAD to bend backwards. This is because undesirable outputs have an unconstrained shadow price, which may be obtained as a positive or
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negative value (Boussemart et al., 2015). We can also understand this as a direct response to the simultaneous contraction of desirable and undesirable outputs (Maghbouli et al., 2014). As the quantity of undesirable outputs is reduced, more inputs have to be used in the production system to produce a given amount of
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desirable outputs (Leleu, 2013), where the consequences of this phenomenon will affect the position of the efficiency frontier. Undesirable output Z/Y B
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C
D
A
Input X/Y
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O
Efficiency frontiers based on the assumptions of different forms of
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Fig.1
disposability
However, during a real production process, undesirable outputs cannot be
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reduced freely or without cost at the will of decision makers. In addition, these outputs may be decreased by reducing desirable outputs, as well as by managerial effort and technological innovation to a certain extent. Therefore, the assumption of only weak or strong disposability cannot reflect the real efficiency frontier somewhere between the strong and weak disposability extremes (Yang & Pollitt, 2010). This means that the real efficiency frontier is usually located in the shaded part indicated by BAC in Fig. 1. 8
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3 Semi-disposability 3.1 Semi-disposability of undesirable outputs
To extend the concept of disposability further, we consider the disposability of undesirable outputs from a new perspective to make environmental assessments more
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consistent with the actual production process. This new concept is defined as follows. Semi-disposability: The new disposability concept for undesirable outputs referred to as “semi-disposability” indicates that DMUs can deal with undesirable outputs freely within the scope of current production technology, whereas DMUs have
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to reduce desirable outputs in the same proportion to reduce undesirable outputs outside this scope. For example, consider two power plants A and B in the same area, which emit sulfur dioxide (SO2) by coal combustion. By employing desulphurization equipment with good management, power plant A can reduce its SO2 emissions by
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more than 90% at the cost of only reducing its electricity generation by several per cent, whereas power plant B does not employ any method to reduce its SO2 emissions.
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In this case, the synchronous change relationship between the desirable and undesirable outputs proposed by Färe et al. (2004) will be broken for power plant B
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because if power plant B introduces desulphurization technologies and managerial experience from power plant A, a large number of SO2 emissions could be reduced
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freely. By contrast, if power plant A wants to reduce its SO2 emissions further in the absence of new breakthroughs in technology and management, then reducing its
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electricity generation would be the only possible solution. In order to describe the semi-disposability of undesirable outputs, the
non-disposal degree concept is defined as follows. Non-disposal degree: For any undesirable output, a part cannot be reduced without cost and its decrease will lead to a synchronous decrease in desirable outputs. The percentage of this part among the total undesirable output is called the “non-disposal degree” of the undesirable output. For example, by introducing desulphurization technologies and managerial experience into power plant A, power 9
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plant B could freely reduce the SO2 emissions by 90%. However, 10% of the SO2 emissions cannot be reduced without reducing electricity generation. Thus, we consider that the non-disposal degree of SO2 emissions is equal to 0.1 for power plant B. We assume that the production technology comprises all feasible (Y , Z , X )
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denoted by T. P( X ) represents the corresponding output set, represents the non-disposal degree of undesirable outputs Z , and (Y ', Z ', X ') represents an arbitrary production activity. According to the concepts defined above, the production technique for semi-disposability can be formulated as follows.
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Case 1: If Z Z ' , then undesirable outputs Z have strong disposability. Thus, under the premise of Z Z ' , if (Y , Z , X ) T and X X ' , Y Y ' , Z Z ' , then (Y ', Z ', X ') T .
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Case 2: If Z Z ' , then undesirable outputs Z have weak disposability. Thus,
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under the premise of Z Z ' , if (Y , Z , X ) T and X X ' , Y Y ' , Z Z ' , then (Y ', Z ', X ') T .
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Z Z ' can be divided into Z Z ' and Z Z ' , so the output set of semi-disposability under CRS, which is the union of Case 1 and Case 2, can be
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specified by the following vector notation:
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P m ( X ) Y , Z
n
n
n
n
j 1
j 1
j 1
j 1
jY j Y , j Z j Z , j Z j Z , j X j X , j 0 j 1,
(5)
where 1 , 2 ,
f
1,
, n ,
, h Rh is a vector such that the value of each element f
, h is between 0 and 1, which represents the non-disposal degree of n
undesirable
outputs
Z
.
The
inequality
constraints
Z j 1
10
j
j
Z
and
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n
Z j 1
j
j
Z
allow for the semi-disposability of undesirable outputs. Then, the
output-oriented efficiency measure for a specific DMU k can be obtained as follows:
max km
x
j ij
j 1
xik
i 1,
,m
km yrk
r 1,
,s
n
y j
j 1
rj
(6)
n
j
j 1
fj
z fk
f 1,
fj
fk z fk
f 1,
n
z j 1
j
j 0
j 1,
, n.
h
h
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z
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n
s.t.
where km represents the efficiency of DMU k based on the assumption of semi-disposability under CRS.
f
1,
, h is a constant,
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It should be noted that the non-disposal degree fk
which can be determined using qualitative decision analysis methods such as the
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Delphi method and Expert meeting method. Therefore, model (6) is a linear programming function.
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Theorem 1. The weak and strong disposability assumptions are special cases of the semi-disposability assumption.
n
CE
Proof. In the case where fk 1
j z fj z fk and
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j 1 n
z j 1
j
fj
n
z j 1
j
fj
f
1,
, h , the inequality constraints
fk z fk in model (6) can be merged into the constraint
z fk ; thus, model (6) is equivalent to model (2), which shows that weak
disposability is a special case of semi-disposability. In n
z j 1
j
fj
the
case
fk z fk
where
fk 0
f
1,
, h ,
the
inequality
constraint
in model (6) is transformed into the inequality constraint 11
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n
z j 1
j
fj
j
0 . Thus, this constraint can be ignored because
j 1,
and
z fj
, h are non-negative numbers. Thus, model (6) is equivalent to
, n; f 1,
model (4), which shows that the strong disposability is also a special case of semi-disposability. □
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To further explain the economic meaning of semi-disposability, we develop dual formulations for the semi-disposability DEA technique as follows: s
h
r 1
f 1
min km wi xik (b f z fk bf fk z fk ) m
s
h
i 1
r 1
f 1
s
u y r
r 1
rk
1
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s.t. wi xij ur yrj (b f z fj bf z fj ) 0 j 1,
wi , ur , b f , bf 0, i 1,
km
represents
the
, m, r 1,
, s, f 1,
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where wi , ur , b f , bf ( i 1,
, m; r 1,
efficiency
of
DMU k
, s; f 1,
,n
(7)
,h
, h ) are intensity variables and
based
on
the
assumption
of
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semi-disposability. According to the duality theory of linear programming, we know that the optimal efficiency km* obtained by model (6) is equal to the optimal
dual
formulation
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The
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efficiency km* obtained by model (7). provides
some
economic
interpretations
of
semi-disposability for decision makers. First, let e f b f bf , and wi , ur , e f , m ; r 1,
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( i 1,
, s ; f 1,
, h ) represent the shadow prices of the inputs, and
desirable and undesirable outputs, respectively. Second, the inputs xik have a non-negative effect on the value of efficiency km via the shadow price wi , the desirable outputs yrk have a non-positive effect on the value of km via the shadow price ur , and the undesirable outputs zrk may have a non-positive or non-negative effect on the value of km via the shadow price e f . Thus, to be more efficient, 12
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decision makers should reduce the inputs and increase the desirable outputs as much as possible. In addition, they should reduce undesirable outputs in the case where
e f b f bf 0 , and if e f b f bf 0 , they would suffer a loss by limiting the increase in undesirable outputs. Hailu et al. (2001) noted that only non-positive shadow prices are theoretically acceptable. However, when decision makers address
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undesirable outputs outside the scope of the current production technology, undesirable outputs will be reduced in the same proportion as the reduction of desirable outputs. In addition, if the cost of the increase in undesirable outputs is not greater than the profit due to the increase in desirable outputs, then the shadow price of undesirable outputs can be considered as non-negative. It should be noted that
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model (6) is the output-oriented DEA model; thus, when the efficiency value km or
km is smaller, the efficiency is higher. When the efficiency value km or km is
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equal to 1, DMU k is considered to be efficient.
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3.2 Method for determining the non-disposal degree
f
1,
, h; j 1,
, n is a constant, which can
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The non-disposal degree fj
be determined by many methods such as the Delphi method and Expert meeting
CE
method. However, these methods are subjective methods and they determine fj based on the experience of experts, so they cannot provide a unified and convincing
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result. Thus, we propose a RPC method to provide an objective criterion for determining the value of fj in environmental assessments, where its general procedure is as follows.
First, calculate the concentration of undesirable outputs c fj . The environmental pollutants z fj
f
1,
, n regarded as undesirable outputs in environmental
assessments are generally discharged by the emission of production wastes k fj 13
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f
1,
, h . For example, the industrial waste gas as the “production waste” is
the carrier of the SO2 regarded as the “environmental pollutant”. Thus, we measure the concentration of environmental pollutants c fj in the production wastes by using the following formulation.
z fj k fj
f 1,
,
h; j 1,
,n
(8)
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c fj
Second, set an optimal reference point c f * for the concentration of undesirable outputs. If we assume that the optimal reference point satisfies the condition that
c f * min c fj j1, , n
f
1,
, h , then this reference point represents the current
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level of production technology. If there are no special instructions, we usually let
c f * min c fj . j1, , n
Third, determine the non-disposal degree fj . We can obtain the non-disposal
M
degree fj by using the following formulation:
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fj
z*fj z fj
c f *k fj z fj
,
(9)
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where z *fj represents the optimal emissions of the fth environmental pollutant for DMU j within the scope of the current production technology.
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Clearly, RPC is an objective method for determining the non-disposal degree fj based on empirical data. By comparing all the DMUs, the current production
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technology level can also be determined. It is useful to make a scientific environmental assessment based on the assumption of semi-disposability.
3.3 Semi-disposability with the interval non-disposal degree
The RPC method can be applied to determine the crisp non-disposal degree to make environmental assessments based on the assumption of semi-disposability. 14
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However, in the practical decision-making process, it is difficult to obtain the crisp information to determine a certain value of the non-disposal degree. In general, uncertain information can be expressed as an interval number (Wang et al, 2005). Thus, we develop a semi-disposability DEA model with the interval non-disposal degree to address this problem, as follows:
max km
x
j ij
n
y j
j 1
rj
j 1
j
j 1
j
r 1,
z fk
fj
ˆ fk z fk
j 0 where ˆ fk
km yrk
fj
n
z
i 1,
,m ,s
(10)
n
z
xik
f 1,
h
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j 1
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n
s.t.
j 1,
f 1,
h
, n.
represents the non-disposal degree within the bounded interval
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Lfk , Ufk .
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Theorem 2. We assume that 1fk and 2fk are the non-disposal degree of z fk . If
1fk 2fk exists, then their corresponding optimal efficiencies have k1* k2* .
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Proof. We assume that DMU1k and DMU 2k have the same inputs xik
r 1,
, s , and undesirable outputs z fk
f
1,
, m ,
, h with
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desirable outputs yrk
i 1,
different non-disposal degrees 1fk and 2fk . For DMU1k , we obtain the optimal
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solution (11* ,
, n1* ,k1* ) by solving model (6). 1fk 2fk , so 1fk z fk 2fk z fk exists.
Obviously, the optimal solution (11* ,
, n1* ,k1* ) for DMU1k is the feasible solution
of DMU 2k , but not the optimal solution. By contrast, we cannot guarantee that the optimal solution of DMU 2k is the feasible solution for DMU1k . Considering that the objective function is used to obtain a maximum value, then we have k1* k2* . □ Fig. 2 shows the interval of the efficiency frontier based on the assumption of 15
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semi-disposability with interval non-disposal degree. In Fig. 2, the contour lines FAD and EAD indicate the efficiency frontiers based on the assumption of semi-disposability with the non-disposal degrees 1fk and 2fk , and 1fk 2fk exists. The shaded part in Fig. 2, which comprises FAD and EAD, represents the efficiency frontier based on the assumption of semi-disposability with interval non-disposal
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degree 2fk , 1fk . Undesirable output Z/Y E
B
F
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C
D
A O
Efficiency frontier with interval non-disposal degree
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Fig. 2
Input X/Y
In order to deal with the environmental assessment problem based on the
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assumption of semi-disposability with interval non-disposal degree Lfk , Ufk , the following pair of linear programming models are developed to generate the upper and
AC
CE
PT
lower bounds of interval efficiency, kmU and kmL , for each DMU k .
max kmU n
s.t.
x j 1
j ij
n
y j
j 1
rj
j 1
j
j 1
j
j 0
,m
kmU yrk
r 1,
,s
fj
z fk
f 1,
h
fj
Lfk z fk
f 1,
h
n
z
i 1,
(11)
n
z
xik
j 1,
and
16
, n.
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x
j ij
j 1 n
y j
j 1
rj
j
j 1
j 1
j
j 0
,m
kmL yrk
r 1,
,s
fj
z fk
f 1,
h
fj
Ufk z fk
f 1,
h
n
z
i 1,
(12)
n
z
xik
j 1,
, n.
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s.t.
Theorem 3. Assuming that ks* and kw* represent the optimal efficiencies of
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DMU k based on the strong and weak disposability assumptions, then we have
kw* kmL* kmU * ks* .
Proof. According to Theorem 1, if fk 0 f 1,
, h , then the semi-disposability
of the undesirable outputs is equivalent to the strong disposability assumption. In
, h , then the semi-disposability of the undesirable
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addition, if fk 1 f 1,
f
1,
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outputs is equivalent to the weak disposability assumption. Moreover, all fk
, h are constant between 0 and 1, so we have 0 akL* akU * 1 . According
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to Theorem 2, the efficiency value has a negative correlation with the non-disposal
CE
degree. Thus, we have kw* kmL* kmU * ks* .□ The value of the interval non-disposal degree Lfk , Ufk can be obtained by the
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RPC method and based on expert experience in the following manner.
First, determine the value of kmU . Let the maximum optimal concentration of environmental pollutants cUf* min c fj j1, , n
f
1,
, h , because this represents
the best situation for the actual production process. Thus, kmU can be obtained using model (9).
Second, determine the value of kmL . The minimum optimal concentration c Lf* 17
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can be obtained based on expert experience and c Lf* min c fj . This is because j1, , n
the process is equivalent to estimating the potential of the optimal reference point for improvement. Then, c Lf* can be used to determine the value of kmL using model (9).
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In this manner, models (11) and (12) can be solved as linear programming models.
4 Extension to VRS
The semi-disposability of the undesirable outputs under CRS was discussed in Section 3. In this section, the semi-disposability is considered under VRS.
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According to Färe et al. (2003), the strong disposability under VRS can be specified as the following vector notation:
where
n j 1
n
n
j 1
j 1
n
n
j 1
j 1
jY j Y , j Z j Z , j X j X , j 1, j 0 j 1,
, n ,
(13)
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PVs ( X ) Y , Z
j 1 represents the restriction of VRS. Färe et al. (2003) and Leleu
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(2013) noted that the restriction of VRS on the weak disposability assumption must consider the simultaneous contraction of desirable and undesirable outputs. Thus, the
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weak disposability under VRS can be specified as the following vector notation:
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n n n n PVw ( X ) Y , Z jY j Y , j Z j Z , j X j X , j 1, 0 1, j 0 j 1, j 1 j 1 j 1 j 1
, n ,
(14)
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where represents the relationship between the desirable and undesirable outputs. By a simple variable substitution, the weak disposability assumption under VRS can be re-specified as the following vector notation: n n n n PVw ( X ) Y , Z jY j Y , j Z j Z , j X j X , j , 0 1, j 0 j 1, j 1 j 1 j 1 j 1
, n ,
(15) We assume that (Y ', Z ', X ') represents an arbitrary production activity and 18
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represents the non-disposal degree of undesirable outputs Z . According to the concepts defined above, the semi-disposability under VRS can be formulated as follows: Case 1: If Z Z ' , then undesirable outputs Z have strong disposability. Thus, in this case, the semi-disposability can be specified as follows. n
n
n
n
j 1
j 1
j 1
j 1
n
jYj Y , j Z j Z , j Z j Z , j X j X , j 1, j 0 j 1,
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PVm1 ( X ) Y , Z
j 1
, n ,
(16)
Then, the efficiency measure for a specific DMU k can be obtained as follows:
n
s.t.
x
xik
i 1,
,m
rj
km1 yrk
r 1,
,s
fj
z fk
f 1,
h
fj
fk z fk
f 1,
h
j ij
j 1 n
y j
j 1 n
z n
j
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j 1
z j
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j 1 n
j 1
j
(17)
1
j 0
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max km1
j 1,
, n.
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where km1 represents the efficiency of DMU k under the VRS assumption of semi-disposability in Case 1.
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Case 2: If Z Z ' , then undesirable outputs Z have weak disposability. Thus,
in this case, the semi-disposability can be specified as follows. n PVm 2 ( X ) Y , Z jY j Y , j 1
n n Z Z , X X , j , 0 1, j 0 j 1, , n , j j j j j 1 j 1 j 1 n
(18) Then, the efficiency measure for a specific DMU k can be obtained as follows:
19
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x
j ij
j 1 n
y j
j 1
rj
n
z j
j 1 n
j 1
j
fj
xik
i 1,
,m
km 2 yrk
r 1,
,s
fk z fk
f 1,
h
0 1 j 0 j 1,
, n.
(19)
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n
s.t.
where km 2 represents the efficiency of DMU k under the VRS assumption of
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semi-disposability in Case 2.
Obviously, the output set of semi-disposability under VRS is the union of Case 1 and Case 2, which can be specified as follows.
PVm ( X ) PVm1 ( X ) PVm 2 ( X )
(20)
can be obtained as follows:
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Therefore, the efficiency of DMU k under the VRS assumption of semi-disposability
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km max km1 , km 2
(21)
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where km represents the efficiency of DMU k under the VRS assumption of
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semi-disposability.
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5 Illustrative examples In this section, we provide two empirical examples to illustrate the validity and
rationality of our methods. The first empirical example is used to demonstrate the effectiveness of semi-disposability assumption. Due to insufficient data of the first
empirical example, the second empirical example is taken as a supplementary example to demonstrate the RPC method.
5.1 Environmental assessment of U.S. fossil fuel power plants 20
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This example illustrates the empirical differences among strong disposability, weak disposability, and semi-disposability during the environmental assessment of fossil fuel power plants. In this example, the nameplate capacity (x1) and fuel consumption (x2) are selected as the inputs, the net generation (y) is selected as the desirable output, and the emissions of SO2 (z1), NOx (z2), and CO2 (z3) are the
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undesirable outputs. The empirical dataset comprising 20 U.S. fossil fuel power plants in 2009 was taken from the study by Sueyoshi and Goto (2012a), and the descriptive statistics for this dataset are summarized in Table 1.
Table 1 Descriptive statistics for 20 power plants x2 (1000
(MW)
MMBtu)
2842 85 889.85 772.217
146042 61 38354.15 41244.99
y
Max Min Mean Std.dev
z1
z2
z3
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x1 Indicators
(GWh)
(Tons)
(Tons)
(Tons)
14620 2 3786.25 4285.682
37109.31 0.02 6828.79 9206.757
14346.41 0.91 3530.399 3706.735
15588.09 3.162 3959.786 4344.574
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In order to better compare the different disposability assumptions, we assume
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that the non-disposal degrees of z1, z2, and z3 for all the DMUs are the interval number [0.5, 0.8], i.e., 1Lj 2Lj 3Lj 0.5 and 1Uj 2Uj 3Uj 0.8 ( j 1,
, 20 ). Then,
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the efficiencies of all the DMUs can be obtained based on different disposability assumptions and the results are shown in Table 2.
CRS Weak
Semi
( ks )
( kw )
( kmU )
( kmL )
( ks )
1.000 1.732 1.075 1.296 1.082 1.105 1.137 1.026 1.000
1.000 1.000 1.008 1.169 1.000 1.065 1.000 1.000 1.000
1.000 1.497 1.072 1.296 1.054 1.105 1.118 1.026 1.000
1.000 1.199 1.059 1.296 1.022 1.094 1.050 1.021 1.000
1.000 1.450 1.065 1.261 1.081 1.090 1.000 1.000 1.000
Strong
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Plant
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Table 2 Comparison of the results based on different assumptions
1 2 3 4 5 6 7 8 9
U
Semi
21
L
Strong
VRS Weak SemiU
SemiL
( kw )
( kmU )
( kmL )
1.000 1.000 1.003 1.169 1.000 1.063 1.000 1.000 1.000
1.000 1.316 1.057 1.261 1.054 1.090 1.000 1.000 1.000
1.000 1.126 1.027 1.261 1.022 1.088 1.000 1.000 1.000
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1.031 1.000 1.861 1.393 1.000 1.084 1.000 1.072 4.134 1.936 1.468
1.000 1.000 1.836 1.060 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.031 1.000 1.857 1.318 1.000 1.084 1.000 1.057 3.250 1.475 1.349
1.031 1.000 1.845 1.180 1.000 1.058 1.000 1.023 2.701 1.190 1.269
1.007 1.000 1.000 1.381 1.000 1.077 1.000 1.000 1.000 1.789 1.430
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.007 1.000 1.000 1.318 1.000 1.077 1.000 1.000 1.000 1.432 1.338
1.005 1.000 1.000 1.162 1.000 1.050 1.000 1.000 1.000 1.173 1.264
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10 11 12 13 14 15 16 17 18 19 20
According to Theorem 1, the strong and weak disposability assumptions represent the cases where all 0 and all 1 . In addition, kmU and kmL
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represent the efficiencies of DMU k in the cases where all 0.5 and all 0.8 under the CRS assumption of semi-disposability. ks , kw , kmU , and kmL represent the efficiencies under the VRS assumptions of different disposability. According to the results obtained under the CRS assumption in Table 2, we can
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see that for any DMU k , there is a decrease in efficiency with the decrease in the
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non-disposal degree, i.e., kw kL kU ks , which is consistent with Theorem 3. In particular, this influence is more obvious for inefficient plants, such as Plants 2 and 18.
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This is because according to Theorem 2, when the non-disposal degree is smaller, the potential for improving the efficiency is greater. This potential is reflected directly by
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the great differences between w and mL for these plants. In addition, these differences can be obtained within the scope of the current production technology, i.e.,
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after introducing advanced management and technology, the efficiency of these plants can be improved by reducing undesirable outputs with the given inputs and desirable outputs. Similarly, for some of the undesirable outputs that cannot be reduced freely, the impact on efficiency is reflected directly by the differences between s and mU , such as Plant 19. In this situation, reducing undesirable outputs with the given inputs and desirable outputs is impossible within the scope of the current production technology. 22
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Similar conclusions can be reached by comparing the results obtained with different
disposability
assumptions
under
VRS,
which
shows
that
the
semi-disposability assumption is still effective under VRS. According to the analysis given above, we can see that the assumption of weak disposability overestimates the efficiencies of DMUs, whereas the assumption of strong disposability underestimates these efficiencies. The assumption of semi-disposability gives a reasonable evaluation
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of the efficiencies of DMUs, and it also provides more information to facilitate improvement. Thus, the assumption of semi-disposability is a more scientific assumption in environmental assessments.
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5.2 Environmental assessment of Chinese regional industry
In order to further illustrate the effectiveness of the semi-disposability assumption with the RPC method, we describe the evaluation of the industrial
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efficiencies of 30 regions in mainland China in this section (due to a lack of data, Tibet is not considered in this study). It is worth noting that “industry” refers to the
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material production sector which is engaged in the extraction of natural resources and processing and reprocessing of minerals and agricultural products. A detailed of
“industry”
can
refer
to
China
Statistical
Yearbook
2014
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definition
(http://www.stats.gov.cn/tjsj/ndsj/2014/indexeh.htm).
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The number of industrial employed persons (IEP), total investment in fixed assets of industry (FAI), and total energy consumption by industry (ECI) are selected
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as the inputs, which are used to reflect the investment of industry in labor, capital, and energy (Li & Shi, 2014; Chen & Jia, 2016, Chang et al., 2013). The industrial added value (IAV), which can reflect the development level of an industrial economy, is selected as the desirable output. The total volume of sulfur dioxide emissions by industry (SDEI) and total volume of chemical oxygen demand discharged by industry (CODDI) are selected as the undesirable outputs. SDEI and CODDI are the main pollutants monitored by government in air and water, respectively, and they can reflect the degree of environmental pollution caused by industry to a great extent. 23
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There are some limitations that should be noted: first, the industrial composition in different regions of China may be different; second, the industrial economies in different regions of China are related to each other by trade connections; third, FAI has a dependency on business cycles. These limitations affect the validity of empirical results, but do not affect this empirical example as a supplementary example to demonstrate the RPC method.
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The data were obtained from the China Statistical Yearbook 2014, China Energy Statistical Yearbook 2014, and China Statistical Yearbook on Environment 2014. The descriptive statistics for this dataset are summarized in Table 3.
Table 3 Descriptive statistics for Chinese regional industry in 2013 FAI (100
ECI
persons)
million yuan)
Max Min Mean Std.dev
1055.37 13.62 209.87 211.33
18371.77 355.24 6024.83 4485.10
IAV (100
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IEP (10000 Indicators
SDEI
CODDI
(10000 tce)
million yuan)
(Tons)
(Tons)
35357.59 1720.33 14249.71 8389.38
26894.54 472.38 9266.50 6960.40
1445348 31652 611688 363233
234281 6055 106468 59623
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In addition, the total volume of industrial waste gas emissions (IWGE) and total
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volume of industrial waste water discharge (IWWD) are selected to determine the non-disposal degree for SDEI and CODDI. The data came from the China Statistical
in Table 4.
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Yearbook on Environment 2014. Descriptive statistics for the data set are summarized
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Table 4
Descriptive statistics for determining the non-disposal degree SDEI
CODDI
IWGE (100
IWWD
(Tons)
(Tons)
million cu.m)
(10000 tons)
1445348 31652 611688 363233
234281 6055 106468 59623
79121 3692 22308 16153
220559 6744 69933 56128
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Indicators Max Min Mean Std.dev
Then, we let the optimal reference point c f * min c fj for the concentration of j1, , n
undesirable outputs. According to the data shown in Table 4, the non-disposal degree for SDEI and CODDI can be obtained by the RPC method, and the results are shown in Table 5. 24
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SDEI
Region Anhui Beijing Chongqing Fujian Gansu Guangdong Guangxi Guizhou Hainan Hebei
Non-disposal degree obtained by the RPC method
CODDI
0.42 0.48 0.13 0.32 0.18 0.26 0.33 0.21 1.00 0.45
0.46 0.88 0.36 0.72 0.13 0.41 0.29 0.20 0.30 0.35
Region Heilongjiang Henan Hubei Hunan Inner Mongolia Jiangsu Jiangxi Jilin Liaoning Ningxia
SDEI
CODDI
0.20 0.23 0.26 0.20 0.17 0.37 0.19 0.20 0.21 0.16
0.28 0.43 0.37 0.37 0.23 0.59 0.41 0.37 0.52 0.09
Region Qinghai Shaanxi Shandong Shanghai Shanxi Sichuan Tianjin Xinjiang Yunnan Zhejiang
SDEI 0.29 0.15 0.22 0.52 0.24 0.18 0.26 0.17 0.17 0.28
CODDI
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Table 5
0.11 0.21 0.77 1.00 0.34 0.34 0.40 0.10 0.14 0.53
Table 5 shows that Hainan and Shanghai are the optimal regions in terms of
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SDEI and CODDI, respectively, and they represent the current technical level for addressing SO2 and chemical oxygen demand in China, i.e., SDEI 1 for Hainan and
CODDI 1 for Shanghai. Most of the regions have great potential for reducing their
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undesirable outputs within the current technical level. For example, SDEI =0.32 for Fujian, which means that the SDEI can be reduced by 68% after improving the
ED
production process rather than reducing desirable outputs. Then, by applying model (6), the efficiencies of all the regions can be obtained
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based on the CRS assumption of semi-disposability and the results are shown in Table
Region
CE
6.
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Anhui Beijing Chongqing Fujian Gansu Guangdong Guangxi Guizhou Hainan Hebei
Table 6
Results of the environmental assessments
m RPC
m0.8
Region
m RPC
m0.8
Region
m RPC
m0.8
1.002 1.000 1.240 1.049 1.679 1.000 1.080 1.377 1.463 1.014
1.002 1.000 1.200 1.049 1.513 1.000 1.031 1.099 1.340 1.014
Heilongjiang Henan Hubei Hunan Inner Mongolia Jiangsu Jiangxi Jilin Liaoning Ningxia
1.356 1.259 1.165 1.000 1.000 1.000 1.020 1.063 1.094 1.712
1.355 1.251 1.165 1.000 1.000 1.000 1.007 1.063 1.094 1.164
Qinghai Shaanxi Shandong Shanghai Shanxi Sichuan Tianjin Xinjiang Yunnan Zhejiang
1.424 1.100 1.147 1.000 1.625 1.141 1.000 1.617 1.476 1.009
1.233 1.040 1.147 1.000 1.482 1.124 1.000 1.345 1.242 1.004
25
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Note that
m represents the efficiency based on the CRS assumption of semi-disposability in RPC
the case where
SDEI and CODDI are determined using the RPC method. In addition, m0.8
represents the efficiency based on the CRS assumption of semi-disposability in the case where all
SDEI CODDI 0.8 .
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m Table 6 shows that there are seven efficient regions where RPC 1 according to
the industrial environmental assessments based on the CRS assumption of semi-disposability using the RPC method, where most of them are the developed areas in China, such as Beijing, Tianjin, and Jiangsu. Thus, they provide a benchmark for the surrounding inefficient regions. In addition, the inefficient regions are usually
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distributed in the undeveloped and developing areas, which have great potential for improvement, such as Shanxi, Yunnan, and Gansu. These regions can introduce advanced technology, equipment, and management skills from the efficient regions to increase the desirable outputs and reduce undesirable outputs using the given inputs.
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Moreover, to illustrate the effectiveness of the RPC method, we let all 0.8 and the evaluation results are also shown in Table 6. By comparing the results, we can
ED
m see that the values of m0.8 are always smaller than the values of RPC , such as
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Guizhou and Xinjiang because the actual non-disposal degrees for SDEI and CODDI in most of the regions are usually smaller than 0.8. Obviously, the estimation error in
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the non-disposal degree may cause some deviations in the results of environmental assessments. The RPC method proposed in this study can provide an objective
AC
criterion for determining the non-disposal degree based on empirical data, and thus environmental assessments can be performed scientifically based on the assumption of semi-disposability.
6 Conclusion The assumptions of strong and weak disposability on undesirable outputs have long dominated previous studies of data envelopment analysis (DEA) in 26
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environmental assessments. However, some studies have noted that different undesirable outputs may have quite different technical features in a production system. In addition, these diverse features are difficult to describe based only on the strong or weak disposability assumption. In fact, weak disposability usually overestimates the efficiencies of decision-making units, whereas strong disposability underestimates these efficiencies. This is because some undesirable outputs may be reduced freely
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within the scope of the current production technology, whereas others cannot.
In this study, we reviewed the concepts of strong and weak disposability according to their conceptual and methodological implications. Moreover, based on their disadvantages,
we proposed
a new disposability assumption called
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semi-disposability to replace the strong and weak disposability assumptions in DEA for environmental assessments. Semi-disposability introduces the concepts of a non-disposal degree and interval non-disposal degree, which are determined using a new reference point comparison method to describe the diverse technical features of
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different undesirable outputs, so the evaluation results are closer to reality. Finally, we provided two empirical examples to illustrate the validity and rationality of the
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semi-disposability assumption.
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Acknowledgments
This research is supported by National Natural Science Foundation of China
CE
(#71371053 and #71471125), Humanities and Social Science Foundation of the Ministry of Education (#14YJC630056), and Social Science Planning Fund project of
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Fujian Province (#FJ2016C199).
References Adler N., Volta N., 2016. Accounting for externalities and disposability: A directional economic environmental distance function. European Journal of Operational Research 250(1), 314-327. Arabi B., Munisamy S., Emrouznejad A, 2015. A new slacks-based measure of 27
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Malmquist-Luenberger index in the presence of undesirable outputs. Omega 21, 29-37. Baležentis T., Li T., Streimikiene D., et al., 2016. Is the Lithuanian economy approaching the goals of sustainable energy and climate change mitigation? Evidence from DEA-based environmental performance index. Journal of Cleaner Production 116, 23-31.
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Bian Y.W., Liang N.N., Xu H., 2015. Efficiency evaluation of Chinese regional industrial systems with undesirable factors using a two-stage slacks-based measure approach. Journal of Cleaner Production 87, 348-356.
Boussemart J.P., Leleu H., Shen Z.Y., 2015. Environmental growth convergence
AN US
among Chinese regions. China Economic Review 34, 1-18.
Chang Y.T., Zhang N., Danao D., et al., 2013. Environmental efficiency analysis of transportation system in China: A non-radial DEA approach. Energy policy 58:
M
277-283.
Chen, L., Jia, G., 2016. Environmental efficiency analysis of China's regional industry:
ED
a data envelopment analysis (DEA) based approach. Journal of Cleaner Production,
PT
http://dx.doi.org/10.1016/j.jclepro.2016.01.045. Chen P.C., Yu M.M., et al., 2015. The enhanced Russell-based directional distance
CE
measure with undesirable outputs: Numerical example considering CO2 emissions. Omega 53, 30-40.
AC
Dyckhoff H., Allen K., 2011. Measuring ecological efficiency with data envelopment analysis (DEA). European Journal of Operational Research 132(2), 312-325.
Färe, R., Grosskopf, S., 2003. Nonparametric Productivity Analysis with Undesirable Outputs: comment. American Journal of Agricultural Economics 85(4), 1070-1074. Färe R., Grosskopf S., 2004. New directions: efficiency and productivity. Kluwer Academic Publishers, Boston. Färe R., Grosskopf S., Lovell C.A.K., 1989. Multilateral productivity comparisons 28
ACCEPTED MANUSCRIPT
when some outputs are undesirable: a nonparametric approach. The Review of Economics and Statistic 71, 90-98. Gorobets A., 2014. Eco-centric policy for sustainable development. Journal of Cleaner Production. Journal of Cleaner Production 61(1), 654-655. Hailu A., Veeman T.S., 2001. Non-parametric productivity analysis with undesirable
of Agricultural Economics 83, 605-616.
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outputs: an application to the Canadian pulp and paper industry. American Journal
Hang Y., Sun J.S., et al., 2015. Measuring energy inefficiency with undesirable outputs and technology heterogeneity in Chinese cities. Economic Modelling 49, 46-52.
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Korhonen P., Luptacik M., 2004. Eco-efficiency analysis of power plants: An extension of data envelopment analysis. European Journal of Operational Research 154, 437-446.
Kuosmanen T., 2005. Weak disposability in nonparametric production analysis with
M
undesirables. American Journal of Agricultural Economics 87(4), 1077-1082. Kuosmanen T., Matin R.K., 2011. Duality of weakly disposable technology. Omega
ED
39, 504-512.
Leleu H., 2013. Shadow pricing of undesirable outputs in nonparametric analysis.
PT
European Journal of Operational Research 231, 474-480. Li H., Shi J.F., 2014. Energy efficiency analysis on Chinese industrial sectors: an
CE
improved Super-SBM model with undesirable outputs. Journal of Cleaner Production 65, 97-107.
AC
Liang L., Wu J., et al., 2008. The DEA game cross-efficiency model and its Nash equilibrium. Operations Research 56, 1278-1288.
Liu W.B., Zhou Z.B., et al., 2015. Two-stage DEA models with undesirable input-intermediate-outputs. Omega 56, 74-87. Maghbouli M., Amirteimoori A., Kordrostami S., 2014. Two-stage network structures with undesirable outputs: a DEA based approach. Measurement 48, 109-118. Mariano E.B., Sobreiro V.A., et al., 2015. Human development and data envelopment analysis: A structured literature review. Omege 54, 33-49. 29
ACCEPTED MANUSCRIPT
Martinez-Núñez M., Pérez-Aguiar W.S., 2014. Efficiency analysis of information technology and online social networks management: An integrated DEA-model assessment. Information & Management 51(6), 712-725. Sahoo B.K., Luptacik M., Mahiberg B., 2011. Alternative measures of environmental technology structure in DEA: an application. European Journal of Operational Research 215, 750-762.
European Journal of Operational Research 142, 16–20.
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Seiford L.M., Zhu J., 2002. Modeling undesirable factors in efficiency evaluation.
Song M.L., Wang S.H., Liu Q.L., 2013. Environmental efficiency evaluation considering the maximization of desirable outputs and its application.
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Mathematical and Computer Modelling 58(5-6), 1110-1116.
Sueyoshi T., Goto M., 2012a. DEA radial measurement for environmental assessment and planning: desirable procedures to evaluate fossil fuel power plants. Energy Policy 41: 422-432.
M
Sueyoshi T., Goto M., 2012b. Weak and strong disposability vs. natural and managerial disposability in DEA environmental assessment: Comparison between
34, 686-699.
ED
Japanese electric power industry and manufacturing industries. Energy Economics
PT
Sueyoshi T., Goto M., 2014. Environmental assessment for corporate sustainability by resource utilization and technology innovation: DEA radial measurement on
CE
Japanese industrial sectors. Energy Economics 46, 295-307. Sueyoshi T., Goto M., Snell M.A., 2013. DEA environmental assessment:
AC
Measurement of damages to scale with unified efficiency under managerial disposability or environmental efficiency. Applied Mathematical Modelling 37, 7300-7314.
Vlontzos G., Niavis S., Manos B., 2014. A DEA approach for estimating the agricultural energy and environmental efficiency of EU countries. Renewable and Sustainable Energy Reviews 40, 91-96. Wang Y.M., Greatbanks R., Yang J.B., 2005. Interval efficiency assessment using data envelopment analysis. Fuzzy Sets and Systems 153, 347-370. 30
ACCEPTED MANUSCRIPT
Wanke P., Barros C.P., Emrouznejad A., 2016. Assessing productive efficiency of banks using integrated Fuzzy-DEA and bootstrapping: A case of Mozambican banks. European Journal of Operational Research 249, 378–389. Woo C.W., Chung Y.H., et al., 2015. The static and dynamic environmental efficiency of renewable energy: a Malmquist index analysis of OECD countries. Renewable and Sustainable Energy Reviews 47, 367-376.
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Wu J., An Q.X., et al., 2013. Congestion measurement for regional industries in China: A data envelopment analysis approach with undesirable outputs. Energy policy 57, 7-13.
Wu J., Yin P.Z., Sun J.S., et al., 2016. Evaluating the environmental efficiency of a
AN US
two-stage system with undesired outputs by a DEA approach: An interest preference perspective. European Journal of Operational Research 254(3), 1047-1062.
Yang H.L., Pollitt M., 2010. The necessity of distinguishing weak and strong
M
disposability among undesirable outputs in DEA: environmental performance of Chinese coal-fired power plants. Energy Policy 28, 4440-4444.
ED
Zha Y., Zhao L., Bian Y., 2016. Measuring regional efficiency of energy and carbon dioxide emissions in China: A chance constrained DEA approach. Computers &
AC
CE
PT
Operations Research 66, 351-361.
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Semi-disposability of undesirable outputs in data envelopment analysis for environmental assessments Lei Chen , Ying-Ming Wang , Fujun Lai PII: DOI: Reference:
S0377-2217(16)31076-1 10.1016/j.ejor.2016.12.042 EOR 14173
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
30 May 2016 29 November 2016 27 December 2016
Please cite this article as: Lei Chen , Ying-Ming Wang , Fujun Lai , Semi-disposability of undesirable outputs in data envelopment analysis for environmental assessments, European Journal of Operational Research (2016), doi: 10.1016/j.ejor.2016.12.042
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Highlights Semi-disposability assumption of undesirable outputs is proposed for environmental assessment
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Measure models of semi-disposability are proposed under constant and variable returns to scale
Reference point comparison is proposed to gain the non-disposal
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degree of undesirable outputs
Semi-disposability is extended to uncertain circumstances by the interval non-disposal degree
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Two empirical examples are provided to illustrate the effectiveness of
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our method
1
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Semi-disposability of undesirable outputs in data envelopment analysis for environmental assessments Lei Chena,b, Ying-Ming Wanga*,
Fujun Laib
350116, PR China
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a. Decision Science Institute , School of Economics & Management, Fuzhou University, Fuzhou
b. College of Business, University of Southern Mississippi, Hattiesburg, MS 39560, United States
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Abstract: The assumptions of strong and weak disposability for undesirable outputs have long dominated studies of data envelopment analysis for environmental assessments. Unfortunately, these assumptions cannot describe the diverse technical features of different undesirable outputs during the actual production process. Thus, we introduce a non-disposal degree to develop a new semi-disposability assumption,
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which can replace the assumptions of strong and weak disposability in environmental
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assessments. This assumption ensures that decision makers can address undesirable outputs freely within the scope of current production technology; otherwise, they have
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to reduce desirable outputs in the same proportion to decrease undesirable outputs. A reference point comparison method is proposed for determining the non-disposal
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degree from an objective perspective. The assumption of semi-disposability is extended to uncertain circumstances by using the interval non-disposal degree. Finally,
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two empirical examples are provided to illustrate the effectiveness of the semi-disposability assumption. Keywords: Data envelopment analysis; Efficiency; Reference point comparison; Return to scale; Semi-disposability.
*Corresponding author. Tel.: +86 0591 22866677; E-mail: [email protected] (L. Chen), [email protected] (Y.M. Wang), [email protected] (F.J. Lai) 2
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1 Introduction Due to rapid economic development, the increasingly severe effects of environmental pollution have attracted widespread attention among the international community. To achieve a “win-win” situation in the economy and environment, many studies have explored sustainable development models (e.g., Gorobets, 2014; Mariano
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et al., 2015). In particular, the environmental assessment of economic development is one of the main research areas, which is an important for the formulation of economic transition policies. For instance, Woo et al. (2015) examined the environmental efficiency from static and dynamic perspectives in 31 OECD countries using the data
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envelopment analysis (DEA) approach and the Malmquist productivity index, where their results provided important contributions that could be used to improve the environmental efficiency by policymakers. Bian et al. (2015) evaluated the environmental efficiency of Chinese regional industrial systems using a two-stage
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slacks-based measure approach and found the main sources of inefficiency. Therefore, scientific environmental assessment is highly significant for sustainable economic
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development.
As one of the most popular methods for efficiency assessment, DEA is a
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nonparametric approach for evaluating the relative efficiency of homogeneous decision-making units (DMUs) with multiple inputs and outputs. However, unlike
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traditional efficiency assessments (e.g., Martinez-Núñez & Pérez-Aguiar, 2014; Wanke et al., 2016), environmental assessments need to consider the desirable outputs
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produced during the production process as well as undesirable outputs, such as waste and pollution. This is because the undesirable outputs are not what decision makers expect but they usually appear with desirable outputs during the actual production process. A DEA model with undesirable outputs was proposed by Färe et al. (1989), and then the DEA approach has been employed widely to deal with the problem of environmental assessment throughout the world (e.g., Baležentis et al., 2016; Wu et al., 2016). In general, these studies can be classified into several categories. The first category mainly focuses on the treatment approach for undesirable outputs in the DEA 3
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model (e.g., Dyckhoff & Allen, 2011; Seiford & Zhu, 2002). The second category deals with undesirable outputs by using different efficiency measures in the DEA approach (e.g., Arabi et al., 2015; Chen et al., 2015; Sahoo et al. 2011). The third category focuses on the disposability of undesirable outputs (e.g., Färe et al., 1989; Sueyoshi et al. 2012b). Färe et al. (1989) proposed the assumptions of strong and weak disposability for
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undesirable outputs, and these assumptions have dominated most previous studies of DEA for environmental assessments (e.g., Hang et al., 2015; Song et al., 2013). According to the assumption of strong disposability, the undesirable outputs may be reduced freely or with no cost at the will of decision makers. For instance, Korhonen
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et al. (2004) constructed a DEA model to measure the eco-efficiency of power plants in a European country based on the assumption of strong disposability. Liu et al. (2015)
constructed
a
two-stage
DEA
model
with
undesirable
input-intermediate-outputs based on the strong disposability perspective, and used it
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to evaluate the efficiency of listed commercial banks in China. According to the assumption of weak disposability, undesirable outputs need to be reduced in
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proportion to the desirable outputs because the joint production of the desirable and undesirable outputs occurs in this case. For instance, Leleu (2013) discussed a current
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error in the modeling of weak disposability under the variable returns to scale technique, and introduced a hybrid model to ensure the economically meaningful
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jointness of desirable and undesirable outputs while constraining the shadow prices of undesirable outputs to their expected sign. Vlontzos et al. (2014) utilized a non-radial
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DEA model based on the assumption of weak disposability to evaluate the energy and environmental efficiency of the primary sectors in EU member state countries. Zha et al. (2016) considered the weak disposability of undesirable outputs and proposed radial and non-radial stochastic DEA models to evaluate the pure efficiency of energy use and CO2 emissions. The strong and weak disposability assumptions of undesirable outputs have gained acceptance among the majority of researchers, but they still have some shortcomings. Kuosmanen (2005) noted that the common specification of weak 4
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disposability implicitly assumes that all DMUs in the sample apply a uniform abatement factor, although there is no theoretical reason to explain why, so he then proposed a new model of weak disposability that allows for non-uniform abatement factors. Kuosmanen et al. (2011) argued that the assumption of strong disposability for undesirable outputs implies that a finite amount of input can produce an infinite amount of undesirable output, which is physically impossible. Yang and Pollitt (2010)
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emphasized the necessity of distinguishing weak and strong disposability among undesirable outputs in environmental assessments, and proposed a model to distinguish them based on the technical features of undesirable outputs. Sueyoshi and Goto (2014) argued that under the weak disposability assumption for undesirable
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outputs, a decrease in undesirable outputs occurs with a decrease in desirable outputs, which cannot reflect the effects of managerial effort and technology innovation on environmental
efficiency.
However,
different
disposability assumptions
for
undesirable outputs imply different production possibility sets of production activity,
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which may yield diverse evaluation results (Adler & Volta, 2016). Therefore, developing a scientific disposability assumption for undesirable outputs is necessary
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for DEA environmental assessment, which is the aim of the present study. Overall, the greatest shortcoming of the strong and weak disposability
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assumptions is that these assumptions cannot describe the diverse technical features of different undesirable outputs during the actual production process. Therefore, this
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study introduces the concept of a non-disposal degree to develop a new assumption, i.e., semi-disposability, to address undesirable outputs under constant returns to scale
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(CRS) and variable returns to scale (VRS). We then discuss this disposability according to its conceptual, methodological, and economic implications. In addition, a reference point comparison (RPC) method is developed to determine the non-disposal degree from an objective perspective. The assumption of semi-disposability is then extended to uncertain circumstances by using an interval non-disposal degree. Finally, the semi-disposability assumption of undesirable outputs is applied to two empirical examples to illustrate its effectiveness. The remainder of this paper is organized as follows. Section 2 reviews the 5
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concepts of strong and weak disposability. Section 3 describes the assumption of semi-disposability under CRS and the RPC method. Section 4 extends the semi-disposability assumption to the case of VRS. In Section 5, we describe the application of the proposed method to two empirical examples. Finally, we give our conclusions in Section 6.
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2 Weak and strong disposability
As mentioned in the previous section, Färe et al. (1989) proposed the strong and weak disposability assumptions for undesirable outputs. Following their study, we
, xm Rm as an inputs vector, Y y1 , y2 ,
, ys Rs as a
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denote X x1 , x2 ,
desirable outputs vector, and Z z1 , z2 ,
, zh Rh as an undesirable outputs
vector for DMU j . The production technology comprising all feasible (Y , Z , X ) denoted by T {(Y , Z , X ) X can produce Y , Z } . Equivalently, the corresponding
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output set can be denoted by P( X ) Y , Z Y , Z , X T . It is assumed that there
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are n DMUs, and then the weak disposability under CRS can be specified by the following vector notation: n
n
n
j 1
j 1
, n ,
jY j Y , j Z j Z , j X j X , j 0 j 1,
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P w ( X ) Y , Z
j 1
(1)
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where the subscript j represents the jth DMU and j represents the jth intensity
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variable. According to the assumption of weak disposability, inputs and desirable n
outputs are strongly disposable by the inequality constraints n
Y j 1
j
j
X j 1
j
j
X
and
Y , which allow a vertical extension, whereas undesirable outputs are not.
Then, the output-oriented efficiency measure for a specific DMU k can be obtained as follows:
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max kw n
s.t.
x j 1
j ij
xik
n
y j 1
j
rj
kw yrk
fj
z fk
n
z j 1
j
i 1,
j 0
,m
r 1, f 1,
j 1,
(2)
,s h
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, n.
where kw represents the efficiency of DMU k based on the assumption of weak disposability under CRS.
Considering that undesirable outputs are never wanted by decision makers, the
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strong disposability is specified as the following vector notation according to the studies of Yang and Pollitt (2010), and Sueyoshi et al. (2014): P s ( X ) Y , Z
n
n
n
j 1
j 1
jY j Y , j Z j Z , j X j X , j 0 j 1, j 1
n
Z j
j
(3)
Z allows for the strong disposability of
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where the inequality constraint
, n ,
j 1
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undesirable outputs. Then, under CRS, the output-oriented efficiency measure for a specific DMU k can be obtained as follows:
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max ks
AC
CE
s.t.
n
x j 1
xik
i 1,
,m
rj
ks yrk
r 1,
,s
fj
z fk
j ij
n
y j 1
j
n
z j 1
j
j 0
f 1,
j 1,
(4)
h
, n.
where ks represents the efficiency of DMU k based on the assumption of strong disposability. As described by Yang and Pollitt (2010), Fig. 1 illustrates the difference between the weak and strong disposability assumptions, where the contour lines BAD and 7
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CAD indicate the efficiency frontiers based on the assumptions of weak and strong disposability, respectively. Actually, the assumption of strong disposability for inputs and undesirable outputs makes the efficiency frontier BAD a convex hull under the constraint of CRS. By contrast, imposing weak disposability on undesirable outputs allows the efficiency frontier CAD to bend backwards. This is because undesirable outputs have an unconstrained shadow price, which may be obtained as a positive or
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negative value (Boussemart et al., 2015). We can also understand this as a direct response to the simultaneous contraction of desirable and undesirable outputs (Maghbouli et al., 2014). As the quantity of undesirable outputs is reduced, more inputs have to be used in the production system to produce a given amount of
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desirable outputs (Leleu, 2013), where the consequences of this phenomenon will affect the position of the efficiency frontier. Undesirable output Z/Y B
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C
D
A
Input X/Y
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O
Efficiency frontiers based on the assumptions of different forms of
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Fig.1
disposability
However, during a real production process, undesirable outputs cannot be
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reduced freely or without cost at the will of decision makers. In addition, these outputs may be decreased by reducing desirable outputs, as well as by managerial effort and technological innovation to a certain extent. Therefore, the assumption of only weak or strong disposability cannot reflect the real efficiency frontier somewhere between the strong and weak disposability extremes (Yang & Pollitt, 2010). This means that the real efficiency frontier is usually located in the shaded part indicated by BAC in Fig. 1. 8
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3 Semi-disposability 3.1 Semi-disposability of undesirable outputs
To extend the concept of disposability further, we consider the disposability of undesirable outputs from a new perspective to make environmental assessments more
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consistent with the actual production process. This new concept is defined as follows. Semi-disposability: The new disposability concept for undesirable outputs referred to as “semi-disposability” indicates that DMUs can deal with undesirable outputs freely within the scope of current production technology, whereas DMUs have
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to reduce desirable outputs in the same proportion to reduce undesirable outputs outside this scope. For example, consider two power plants A and B in the same area, which emit sulfur dioxide (SO2) by coal combustion. By employing desulphurization equipment with good management, power plant A can reduce its SO2 emissions by
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more than 90% at the cost of only reducing its electricity generation by several per cent, whereas power plant B does not employ any method to reduce its SO2 emissions.
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In this case, the synchronous change relationship between the desirable and undesirable outputs proposed by Färe et al. (2004) will be broken for power plant B
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because if power plant B introduces desulphurization technologies and managerial experience from power plant A, a large number of SO2 emissions could be reduced
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freely. By contrast, if power plant A wants to reduce its SO2 emissions further in the absence of new breakthroughs in technology and management, then reducing its
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electricity generation would be the only possible solution. In order to describe the semi-disposability of undesirable outputs, the
non-disposal degree concept is defined as follows. Non-disposal degree: For any undesirable output, a part cannot be reduced without cost and its decrease will lead to a synchronous decrease in desirable outputs. The percentage of this part among the total undesirable output is called the “non-disposal degree” of the undesirable output. For example, by introducing desulphurization technologies and managerial experience into power plant A, power 9
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plant B could freely reduce the SO2 emissions by 90%. However, 10% of the SO2 emissions cannot be reduced without reducing electricity generation. Thus, we consider that the non-disposal degree of SO2 emissions is equal to 0.1 for power plant B. We assume that the production technology comprises all feasible (Y , Z , X )
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denoted by T. P( X ) represents the corresponding output set, represents the non-disposal degree of undesirable outputs Z , and (Y ', Z ', X ') represents an arbitrary production activity. According to the concepts defined above, the production technique for semi-disposability can be formulated as follows.
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Case 1: If Z Z ' , then undesirable outputs Z have strong disposability. Thus, under the premise of Z Z ' , if (Y , Z , X ) T and X X ' , Y Y ' , Z Z ' , then (Y ', Z ', X ') T .
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Case 2: If Z Z ' , then undesirable outputs Z have weak disposability. Thus,
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under the premise of Z Z ' , if (Y , Z , X ) T and X X ' , Y Y ' , Z Z ' , then (Y ', Z ', X ') T .
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Z Z ' can be divided into Z Z ' and Z Z ' , so the output set of semi-disposability under CRS, which is the union of Case 1 and Case 2, can be
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specified by the following vector notation:
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P m ( X ) Y , Z
n
n
n
n
j 1
j 1
j 1
j 1
jY j Y , j Z j Z , j Z j Z , j X j X , j 0 j 1,
(5)
where 1 , 2 ,
f
1,
, n ,
, h Rh is a vector such that the value of each element f
, h is between 0 and 1, which represents the non-disposal degree of n
undesirable
outputs
Z
.
The
inequality
constraints
Z j 1
10
j
j
Z
and
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n
Z j 1
j
j
Z
allow for the semi-disposability of undesirable outputs. Then, the
output-oriented efficiency measure for a specific DMU k can be obtained as follows:
max km
x
j ij
j 1
xik
i 1,
,m
km yrk
r 1,
,s
n
y j
j 1
rj
(6)
n
j
j 1
fj
z fk
f 1,
fj
fk z fk
f 1,
n
z j 1
j
j 0
j 1,
, n.
h
h
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z
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n
s.t.
where km represents the efficiency of DMU k based on the assumption of semi-disposability under CRS.
f
1,
, h is a constant,
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It should be noted that the non-disposal degree fk
which can be determined using qualitative decision analysis methods such as the
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Delphi method and Expert meeting method. Therefore, model (6) is a linear programming function.
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Theorem 1. The weak and strong disposability assumptions are special cases of the semi-disposability assumption.
n
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Proof. In the case where fk 1
j z fj z fk and
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j 1 n
z j 1
j
fj
n
z j 1
j
fj
f
1,
, h , the inequality constraints
fk z fk in model (6) can be merged into the constraint
z fk ; thus, model (6) is equivalent to model (2), which shows that weak
disposability is a special case of semi-disposability. In n
z j 1
j
fj
the
case
fk z fk
where
fk 0
f
1,
, h ,
the
inequality
constraint
in model (6) is transformed into the inequality constraint 11
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n
z j 1
j
fj
j
0 . Thus, this constraint can be ignored because
j 1,
and
z fj
, h are non-negative numbers. Thus, model (6) is equivalent to
, n; f 1,
model (4), which shows that the strong disposability is also a special case of semi-disposability. □
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To further explain the economic meaning of semi-disposability, we develop dual formulations for the semi-disposability DEA technique as follows: s
h
r 1
f 1
min km wi xik (b f z fk bf fk z fk ) m
s
h
i 1
r 1
f 1
s
u y r
r 1
rk
1
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s.t. wi xij ur yrj (b f z fj bf z fj ) 0 j 1,
wi , ur , b f , bf 0, i 1,
km
represents
the
, m, r 1,
, s, f 1,
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where wi , ur , b f , bf ( i 1,
, m; r 1,
efficiency
of
DMU k
, s; f 1,
,n
(7)
,h
, h ) are intensity variables and
based
on
the
assumption
of
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semi-disposability. According to the duality theory of linear programming, we know that the optimal efficiency km* obtained by model (6) is equal to the optimal
dual
formulation
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The
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efficiency km* obtained by model (7). provides
some
economic
interpretations
of
semi-disposability for decision makers. First, let e f b f bf , and wi , ur , e f , m ; r 1,
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( i 1,
, s ; f 1,
, h ) represent the shadow prices of the inputs, and
desirable and undesirable outputs, respectively. Second, the inputs xik have a non-negative effect on the value of efficiency km via the shadow price wi , the desirable outputs yrk have a non-positive effect on the value of km via the shadow price ur , and the undesirable outputs zrk may have a non-positive or non-negative effect on the value of km via the shadow price e f . Thus, to be more efficient, 12
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decision makers should reduce the inputs and increase the desirable outputs as much as possible. In addition, they should reduce undesirable outputs in the case where
e f b f bf 0 , and if e f b f bf 0 , they would suffer a loss by limiting the increase in undesirable outputs. Hailu et al. (2001) noted that only non-positive shadow prices are theoretically acceptable. However, when decision makers address
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undesirable outputs outside the scope of the current production technology, undesirable outputs will be reduced in the same proportion as the reduction of desirable outputs. In addition, if the cost of the increase in undesirable outputs is not greater than the profit due to the increase in desirable outputs, then the shadow price of undesirable outputs can be considered as non-negative. It should be noted that
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model (6) is the output-oriented DEA model; thus, when the efficiency value km or
km is smaller, the efficiency is higher. When the efficiency value km or km is
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equal to 1, DMU k is considered to be efficient.
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3.2 Method for determining the non-disposal degree
f
1,
, h; j 1,
, n is a constant, which can
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The non-disposal degree fj
be determined by many methods such as the Delphi method and Expert meeting
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method. However, these methods are subjective methods and they determine fj based on the experience of experts, so they cannot provide a unified and convincing
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result. Thus, we propose a RPC method to provide an objective criterion for determining the value of fj in environmental assessments, where its general procedure is as follows.
First, calculate the concentration of undesirable outputs c fj . The environmental pollutants z fj
f
1,
, n regarded as undesirable outputs in environmental
assessments are generally discharged by the emission of production wastes k fj 13
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f
1,
, h . For example, the industrial waste gas as the “production waste” is
the carrier of the SO2 regarded as the “environmental pollutant”. Thus, we measure the concentration of environmental pollutants c fj in the production wastes by using the following formulation.
z fj k fj
f 1,
,
h; j 1,
,n
(8)
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c fj
Second, set an optimal reference point c f * for the concentration of undesirable outputs. If we assume that the optimal reference point satisfies the condition that
c f * min c fj j1, , n
f
1,
, h , then this reference point represents the current
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level of production technology. If there are no special instructions, we usually let
c f * min c fj . j1, , n
Third, determine the non-disposal degree fj . We can obtain the non-disposal
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degree fj by using the following formulation:
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fj
z*fj z fj
c f *k fj z fj
,
(9)
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where z *fj represents the optimal emissions of the fth environmental pollutant for DMU j within the scope of the current production technology.
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Clearly, RPC is an objective method for determining the non-disposal degree fj based on empirical data. By comparing all the DMUs, the current production
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technology level can also be determined. It is useful to make a scientific environmental assessment based on the assumption of semi-disposability.
3.3 Semi-disposability with the interval non-disposal degree
The RPC method can be applied to determine the crisp non-disposal degree to make environmental assessments based on the assumption of semi-disposability. 14
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However, in the practical decision-making process, it is difficult to obtain the crisp information to determine a certain value of the non-disposal degree. In general, uncertain information can be expressed as an interval number (Wang et al, 2005). Thus, we develop a semi-disposability DEA model with the interval non-disposal degree to address this problem, as follows:
max km
x
j ij
n
y j
j 1
rj
j 1
j
j 1
j
r 1,
z fk
fj
ˆ fk z fk
j 0 where ˆ fk
km yrk
fj
n
z
i 1,
,m ,s
(10)
n
z
xik
f 1,
h
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j 1
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n
s.t.
j 1,
f 1,
h
, n.
represents the non-disposal degree within the bounded interval
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Lfk , Ufk .
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Theorem 2. We assume that 1fk and 2fk are the non-disposal degree of z fk . If
1fk 2fk exists, then their corresponding optimal efficiencies have k1* k2* .
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Proof. We assume that DMU1k and DMU 2k have the same inputs xik
r 1,
, s , and undesirable outputs z fk
f
1,
, m ,
, h with
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desirable outputs yrk
i 1,
different non-disposal degrees 1fk and 2fk . For DMU1k , we obtain the optimal
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solution (11* ,
, n1* ,k1* ) by solving model (6). 1fk 2fk , so 1fk z fk 2fk z fk exists.
Obviously, the optimal solution (11* ,
, n1* ,k1* ) for DMU1k is the feasible solution
of DMU 2k , but not the optimal solution. By contrast, we cannot guarantee that the optimal solution of DMU 2k is the feasible solution for DMU1k . Considering that the objective function is used to obtain a maximum value, then we have k1* k2* . □ Fig. 2 shows the interval of the efficiency frontier based on the assumption of 15
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semi-disposability with interval non-disposal degree. In Fig. 2, the contour lines FAD and EAD indicate the efficiency frontiers based on the assumption of semi-disposability with the non-disposal degrees 1fk and 2fk , and 1fk 2fk exists. The shaded part in Fig. 2, which comprises FAD and EAD, represents the efficiency frontier based on the assumption of semi-disposability with interval non-disposal
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degree 2fk , 1fk . Undesirable output Z/Y E
B
F
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C
D
A O
Efficiency frontier with interval non-disposal degree
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Fig. 2
Input X/Y
In order to deal with the environmental assessment problem based on the
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assumption of semi-disposability with interval non-disposal degree Lfk , Ufk , the following pair of linear programming models are developed to generate the upper and
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CE
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lower bounds of interval efficiency, kmU and kmL , for each DMU k .
max kmU n
s.t.
x j 1
j ij
n
y j
j 1
rj
j 1
j
j 1
j
j 0
,m
kmU yrk
r 1,
,s
fj
z fk
f 1,
h
fj
Lfk z fk
f 1,
h
n
z
i 1,
(11)
n
z
xik
j 1,
and
16
, n.
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x
j ij
j 1 n
y j
j 1
rj
j
j 1
j 1
j
j 0
,m
kmL yrk
r 1,
,s
fj
z fk
f 1,
h
fj
Ufk z fk
f 1,
h
n
z
i 1,
(12)
n
z
xik
j 1,
, n.
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s.t.
Theorem 3. Assuming that ks* and kw* represent the optimal efficiencies of
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DMU k based on the strong and weak disposability assumptions, then we have
kw* kmL* kmU * ks* .
Proof. According to Theorem 1, if fk 0 f 1,
, h , then the semi-disposability
of the undesirable outputs is equivalent to the strong disposability assumption. In
, h , then the semi-disposability of the undesirable
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addition, if fk 1 f 1,
f
1,
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outputs is equivalent to the weak disposability assumption. Moreover, all fk
, h are constant between 0 and 1, so we have 0 akL* akU * 1 . According
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to Theorem 2, the efficiency value has a negative correlation with the non-disposal
CE
degree. Thus, we have kw* kmL* kmU * ks* .□ The value of the interval non-disposal degree Lfk , Ufk can be obtained by the
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RPC method and based on expert experience in the following manner.
First, determine the value of kmU . Let the maximum optimal concentration of environmental pollutants cUf* min c fj j1, , n
f
1,
, h , because this represents
the best situation for the actual production process. Thus, kmU can be obtained using model (9).
Second, determine the value of kmL . The minimum optimal concentration c Lf* 17
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can be obtained based on expert experience and c Lf* min c fj . This is because j1, , n
the process is equivalent to estimating the potential of the optimal reference point for improvement. Then, c Lf* can be used to determine the value of kmL using model (9).
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In this manner, models (11) and (12) can be solved as linear programming models.
4 Extension to VRS
The semi-disposability of the undesirable outputs under CRS was discussed in Section 3. In this section, the semi-disposability is considered under VRS.
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According to Färe et al. (2003), the strong disposability under VRS can be specified as the following vector notation:
where
n j 1
n
n
j 1
j 1
n
n
j 1
j 1
jY j Y , j Z j Z , j X j X , j 1, j 0 j 1,
, n ,
(13)
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PVs ( X ) Y , Z
j 1 represents the restriction of VRS. Färe et al. (2003) and Leleu
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(2013) noted that the restriction of VRS on the weak disposability assumption must consider the simultaneous contraction of desirable and undesirable outputs. Thus, the
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weak disposability under VRS can be specified as the following vector notation:
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n n n n PVw ( X ) Y , Z jY j Y , j Z j Z , j X j X , j 1, 0 1, j 0 j 1, j 1 j 1 j 1 j 1
, n ,
(14)
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where represents the relationship between the desirable and undesirable outputs. By a simple variable substitution, the weak disposability assumption under VRS can be re-specified as the following vector notation: n n n n PVw ( X ) Y , Z jY j Y , j Z j Z , j X j X , j , 0 1, j 0 j 1, j 1 j 1 j 1 j 1
, n ,
(15) We assume that (Y ', Z ', X ') represents an arbitrary production activity and 18
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represents the non-disposal degree of undesirable outputs Z . According to the concepts defined above, the semi-disposability under VRS can be formulated as follows: Case 1: If Z Z ' , then undesirable outputs Z have strong disposability. Thus, in this case, the semi-disposability can be specified as follows. n
n
n
n
j 1
j 1
j 1
j 1
n
jYj Y , j Z j Z , j Z j Z , j X j X , j 1, j 0 j 1,
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PVm1 ( X ) Y , Z
j 1
, n ,
(16)
Then, the efficiency measure for a specific DMU k can be obtained as follows:
n
s.t.
x
xik
i 1,
,m
rj
km1 yrk
r 1,
,s
fj
z fk
f 1,
h
fj
fk z fk
f 1,
h
j ij
j 1 n
y j
j 1 n
z n
j
M
j 1
z j
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j 1 n
j 1
j
(17)
1
j 0
PT
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max km1
j 1,
, n.
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where km1 represents the efficiency of DMU k under the VRS assumption of semi-disposability in Case 1.
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Case 2: If Z Z ' , then undesirable outputs Z have weak disposability. Thus,
in this case, the semi-disposability can be specified as follows. n PVm 2 ( X ) Y , Z jY j Y , j 1
n n Z Z , X X , j , 0 1, j 0 j 1, , n , j j j j j 1 j 1 j 1 n
(18) Then, the efficiency measure for a specific DMU k can be obtained as follows:
19
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x
j ij
j 1 n
y j
j 1
rj
n
z j
j 1 n
j 1
j
fj
xik
i 1,
,m
km 2 yrk
r 1,
,s
fk z fk
f 1,
h
0 1 j 0 j 1,
, n.
(19)
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n
s.t.
where km 2 represents the efficiency of DMU k under the VRS assumption of
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semi-disposability in Case 2.
Obviously, the output set of semi-disposability under VRS is the union of Case 1 and Case 2, which can be specified as follows.
PVm ( X ) PVm1 ( X ) PVm 2 ( X )
(20)
can be obtained as follows:
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Therefore, the efficiency of DMU k under the VRS assumption of semi-disposability
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km max km1 , km 2
(21)
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where km represents the efficiency of DMU k under the VRS assumption of
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semi-disposability.
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5 Illustrative examples In this section, we provide two empirical examples to illustrate the validity and
rationality of our methods. The first empirical example is used to demonstrate the effectiveness of semi-disposability assumption. Due to insufficient data of the first
empirical example, the second empirical example is taken as a supplementary example to demonstrate the RPC method.
5.1 Environmental assessment of U.S. fossil fuel power plants 20
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This example illustrates the empirical differences among strong disposability, weak disposability, and semi-disposability during the environmental assessment of fossil fuel power plants. In this example, the nameplate capacity (x1) and fuel consumption (x2) are selected as the inputs, the net generation (y) is selected as the desirable output, and the emissions of SO2 (z1), NOx (z2), and CO2 (z3) are the
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undesirable outputs. The empirical dataset comprising 20 U.S. fossil fuel power plants in 2009 was taken from the study by Sueyoshi and Goto (2012a), and the descriptive statistics for this dataset are summarized in Table 1.
Table 1 Descriptive statistics for 20 power plants x2 (1000
(MW)
MMBtu)
2842 85 889.85 772.217
146042 61 38354.15 41244.99
y
Max Min Mean Std.dev
z1
z2
z3
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x1 Indicators
(GWh)
(Tons)
(Tons)
(Tons)
14620 2 3786.25 4285.682
37109.31 0.02 6828.79 9206.757
14346.41 0.91 3530.399 3706.735
15588.09 3.162 3959.786 4344.574
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In order to better compare the different disposability assumptions, we assume
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that the non-disposal degrees of z1, z2, and z3 for all the DMUs are the interval number [0.5, 0.8], i.e., 1Lj 2Lj 3Lj 0.5 and 1Uj 2Uj 3Uj 0.8 ( j 1,
, 20 ). Then,
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the efficiencies of all the DMUs can be obtained based on different disposability assumptions and the results are shown in Table 2.
CRS Weak
Semi
( ks )
( kw )
( kmU )
( kmL )
( ks )
1.000 1.732 1.075 1.296 1.082 1.105 1.137 1.026 1.000
1.000 1.000 1.008 1.169 1.000 1.065 1.000 1.000 1.000
1.000 1.497 1.072 1.296 1.054 1.105 1.118 1.026 1.000
1.000 1.199 1.059 1.296 1.022 1.094 1.050 1.021 1.000
1.000 1.450 1.065 1.261 1.081 1.090 1.000 1.000 1.000
Strong
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Plant
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Table 2 Comparison of the results based on different assumptions
1 2 3 4 5 6 7 8 9
U
Semi
21
L
Strong
VRS Weak SemiU
SemiL
( kw )
( kmU )
( kmL )
1.000 1.000 1.003 1.169 1.000 1.063 1.000 1.000 1.000
1.000 1.316 1.057 1.261 1.054 1.090 1.000 1.000 1.000
1.000 1.126 1.027 1.261 1.022 1.088 1.000 1.000 1.000
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1.031 1.000 1.861 1.393 1.000 1.084 1.000 1.072 4.134 1.936 1.468
1.000 1.000 1.836 1.060 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.031 1.000 1.857 1.318 1.000 1.084 1.000 1.057 3.250 1.475 1.349
1.031 1.000 1.845 1.180 1.000 1.058 1.000 1.023 2.701 1.190 1.269
1.007 1.000 1.000 1.381 1.000 1.077 1.000 1.000 1.000 1.789 1.430
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.007 1.000 1.000 1.318 1.000 1.077 1.000 1.000 1.000 1.432 1.338
1.005 1.000 1.000 1.162 1.000 1.050 1.000 1.000 1.000 1.173 1.264
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10 11 12 13 14 15 16 17 18 19 20
According to Theorem 1, the strong and weak disposability assumptions represent the cases where all 0 and all 1 . In addition, kmU and kmL
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represent the efficiencies of DMU k in the cases where all 0.5 and all 0.8 under the CRS assumption of semi-disposability. ks , kw , kmU , and kmL represent the efficiencies under the VRS assumptions of different disposability. According to the results obtained under the CRS assumption in Table 2, we can
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see that for any DMU k , there is a decrease in efficiency with the decrease in the
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non-disposal degree, i.e., kw kL kU ks , which is consistent with Theorem 3. In particular, this influence is more obvious for inefficient plants, such as Plants 2 and 18.
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This is because according to Theorem 2, when the non-disposal degree is smaller, the potential for improving the efficiency is greater. This potential is reflected directly by
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the great differences between w and mL for these plants. In addition, these differences can be obtained within the scope of the current production technology, i.e.,
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after introducing advanced management and technology, the efficiency of these plants can be improved by reducing undesirable outputs with the given inputs and desirable outputs. Similarly, for some of the undesirable outputs that cannot be reduced freely, the impact on efficiency is reflected directly by the differences between s and mU , such as Plant 19. In this situation, reducing undesirable outputs with the given inputs and desirable outputs is impossible within the scope of the current production technology. 22
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Similar conclusions can be reached by comparing the results obtained with different
disposability
assumptions
under
VRS,
which
shows
that
the
semi-disposability assumption is still effective under VRS. According to the analysis given above, we can see that the assumption of weak disposability overestimates the efficiencies of DMUs, whereas the assumption of strong disposability underestimates these efficiencies. The assumption of semi-disposability gives a reasonable evaluation
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of the efficiencies of DMUs, and it also provides more information to facilitate improvement. Thus, the assumption of semi-disposability is a more scientific assumption in environmental assessments.
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5.2 Environmental assessment of Chinese regional industry
In order to further illustrate the effectiveness of the semi-disposability assumption with the RPC method, we describe the evaluation of the industrial
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efficiencies of 30 regions in mainland China in this section (due to a lack of data, Tibet is not considered in this study). It is worth noting that “industry” refers to the
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material production sector which is engaged in the extraction of natural resources and processing and reprocessing of minerals and agricultural products. A detailed of
“industry”
can
refer
to
China
Statistical
Yearbook
2014
PT
definition
(http://www.stats.gov.cn/tjsj/ndsj/2014/indexeh.htm).
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The number of industrial employed persons (IEP), total investment in fixed assets of industry (FAI), and total energy consumption by industry (ECI) are selected
AC
as the inputs, which are used to reflect the investment of industry in labor, capital, and energy (Li & Shi, 2014; Chen & Jia, 2016, Chang et al., 2013). The industrial added value (IAV), which can reflect the development level of an industrial economy, is selected as the desirable output. The total volume of sulfur dioxide emissions by industry (SDEI) and total volume of chemical oxygen demand discharged by industry (CODDI) are selected as the undesirable outputs. SDEI and CODDI are the main pollutants monitored by government in air and water, respectively, and they can reflect the degree of environmental pollution caused by industry to a great extent. 23
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There are some limitations that should be noted: first, the industrial composition in different regions of China may be different; second, the industrial economies in different regions of China are related to each other by trade connections; third, FAI has a dependency on business cycles. These limitations affect the validity of empirical results, but do not affect this empirical example as a supplementary example to demonstrate the RPC method.
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The data were obtained from the China Statistical Yearbook 2014, China Energy Statistical Yearbook 2014, and China Statistical Yearbook on Environment 2014. The descriptive statistics for this dataset are summarized in Table 3.
Table 3 Descriptive statistics for Chinese regional industry in 2013 FAI (100
ECI
persons)
million yuan)
Max Min Mean Std.dev
1055.37 13.62 209.87 211.33
18371.77 355.24 6024.83 4485.10
IAV (100
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IEP (10000 Indicators
SDEI
CODDI
(10000 tce)
million yuan)
(Tons)
(Tons)
35357.59 1720.33 14249.71 8389.38
26894.54 472.38 9266.50 6960.40
1445348 31652 611688 363233
234281 6055 106468 59623
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In addition, the total volume of industrial waste gas emissions (IWGE) and total
ED
volume of industrial waste water discharge (IWWD) are selected to determine the non-disposal degree for SDEI and CODDI. The data came from the China Statistical
in Table 4.
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Yearbook on Environment 2014. Descriptive statistics for the data set are summarized
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Table 4
Descriptive statistics for determining the non-disposal degree SDEI
CODDI
IWGE (100
IWWD
(Tons)
(Tons)
million cu.m)
(10000 tons)
1445348 31652 611688 363233
234281 6055 106468 59623
79121 3692 22308 16153
220559 6744 69933 56128
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Indicators Max Min Mean Std.dev
Then, we let the optimal reference point c f * min c fj for the concentration of j1, , n
undesirable outputs. According to the data shown in Table 4, the non-disposal degree for SDEI and CODDI can be obtained by the RPC method, and the results are shown in Table 5. 24
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SDEI
Region Anhui Beijing Chongqing Fujian Gansu Guangdong Guangxi Guizhou Hainan Hebei
Non-disposal degree obtained by the RPC method
CODDI
0.42 0.48 0.13 0.32 0.18 0.26 0.33 0.21 1.00 0.45
0.46 0.88 0.36 0.72 0.13 0.41 0.29 0.20 0.30 0.35
Region Heilongjiang Henan Hubei Hunan Inner Mongolia Jiangsu Jiangxi Jilin Liaoning Ningxia
SDEI
CODDI
0.20 0.23 0.26 0.20 0.17 0.37 0.19 0.20 0.21 0.16
0.28 0.43 0.37 0.37 0.23 0.59 0.41 0.37 0.52 0.09
Region Qinghai Shaanxi Shandong Shanghai Shanxi Sichuan Tianjin Xinjiang Yunnan Zhejiang
SDEI 0.29 0.15 0.22 0.52 0.24 0.18 0.26 0.17 0.17 0.28
CODDI
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Table 5
0.11 0.21 0.77 1.00 0.34 0.34 0.40 0.10 0.14 0.53
Table 5 shows that Hainan and Shanghai are the optimal regions in terms of
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SDEI and CODDI, respectively, and they represent the current technical level for addressing SO2 and chemical oxygen demand in China, i.e., SDEI 1 for Hainan and
CODDI 1 for Shanghai. Most of the regions have great potential for reducing their
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undesirable outputs within the current technical level. For example, SDEI =0.32 for Fujian, which means that the SDEI can be reduced by 68% after improving the
ED
production process rather than reducing desirable outputs. Then, by applying model (6), the efficiencies of all the regions can be obtained
PT
based on the CRS assumption of semi-disposability and the results are shown in Table
Region
CE
6.
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Anhui Beijing Chongqing Fujian Gansu Guangdong Guangxi Guizhou Hainan Hebei
Table 6
Results of the environmental assessments
m RPC
m0.8
Region
m RPC
m0.8
Region
m RPC
m0.8
1.002 1.000 1.240 1.049 1.679 1.000 1.080 1.377 1.463 1.014
1.002 1.000 1.200 1.049 1.513 1.000 1.031 1.099 1.340 1.014
Heilongjiang Henan Hubei Hunan Inner Mongolia Jiangsu Jiangxi Jilin Liaoning Ningxia
1.356 1.259 1.165 1.000 1.000 1.000 1.020 1.063 1.094 1.712
1.355 1.251 1.165 1.000 1.000 1.000 1.007 1.063 1.094 1.164
Qinghai Shaanxi Shandong Shanghai Shanxi Sichuan Tianjin Xinjiang Yunnan Zhejiang
1.424 1.100 1.147 1.000 1.625 1.141 1.000 1.617 1.476 1.009
1.233 1.040 1.147 1.000 1.482 1.124 1.000 1.345 1.242 1.004
25
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Note that
m represents the efficiency based on the CRS assumption of semi-disposability in RPC
the case where
SDEI and CODDI are determined using the RPC method. In addition, m0.8
represents the efficiency based on the CRS assumption of semi-disposability in the case where all
SDEI CODDI 0.8 .
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m Table 6 shows that there are seven efficient regions where RPC 1 according to
the industrial environmental assessments based on the CRS assumption of semi-disposability using the RPC method, where most of them are the developed areas in China, such as Beijing, Tianjin, and Jiangsu. Thus, they provide a benchmark for the surrounding inefficient regions. In addition, the inefficient regions are usually
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distributed in the undeveloped and developing areas, which have great potential for improvement, such as Shanxi, Yunnan, and Gansu. These regions can introduce advanced technology, equipment, and management skills from the efficient regions to increase the desirable outputs and reduce undesirable outputs using the given inputs.
M
Moreover, to illustrate the effectiveness of the RPC method, we let all 0.8 and the evaluation results are also shown in Table 6. By comparing the results, we can
ED
m see that the values of m0.8 are always smaller than the values of RPC , such as
PT
Guizhou and Xinjiang because the actual non-disposal degrees for SDEI and CODDI in most of the regions are usually smaller than 0.8. Obviously, the estimation error in
CE
the non-disposal degree may cause some deviations in the results of environmental assessments. The RPC method proposed in this study can provide an objective
AC
criterion for determining the non-disposal degree based on empirical data, and thus environmental assessments can be performed scientifically based on the assumption of semi-disposability.
6 Conclusion The assumptions of strong and weak disposability on undesirable outputs have long dominated previous studies of data envelopment analysis (DEA) in 26
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environmental assessments. However, some studies have noted that different undesirable outputs may have quite different technical features in a production system. In addition, these diverse features are difficult to describe based only on the strong or weak disposability assumption. In fact, weak disposability usually overestimates the efficiencies of decision-making units, whereas strong disposability underestimates these efficiencies. This is because some undesirable outputs may be reduced freely
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within the scope of the current production technology, whereas others cannot.
In this study, we reviewed the concepts of strong and weak disposability according to their conceptual and methodological implications. Moreover, based on their disadvantages,
we proposed
a new disposability assumption called
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semi-disposability to replace the strong and weak disposability assumptions in DEA for environmental assessments. Semi-disposability introduces the concepts of a non-disposal degree and interval non-disposal degree, which are determined using a new reference point comparison method to describe the diverse technical features of
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different undesirable outputs, so the evaluation results are closer to reality. Finally, we provided two empirical examples to illustrate the validity and rationality of the
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semi-disposability assumption.
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Acknowledgments
This research is supported by National Natural Science Foundation of China
CE
(#71371053 and #71471125), Humanities and Social Science Foundation of the Ministry of Education (#14YJC630056), and Social Science Planning Fund project of
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Fujian Province (#FJ2016C199).
References Adler N., Volta N., 2016. Accounting for externalities and disposability: A directional economic environmental distance function. European Journal of Operational Research 250(1), 314-327. Arabi B., Munisamy S., Emrouznejad A, 2015. A new slacks-based measure of 27
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Malmquist-Luenberger index in the presence of undesirable outputs. Omega 21, 29-37. Baležentis T., Li T., Streimikiene D., et al., 2016. Is the Lithuanian economy approaching the goals of sustainable energy and climate change mitigation? Evidence from DEA-based environmental performance index. Journal of Cleaner Production 116, 23-31.
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Bian Y.W., Liang N.N., Xu H., 2015. Efficiency evaluation of Chinese regional industrial systems with undesirable factors using a two-stage slacks-based measure approach. Journal of Cleaner Production 87, 348-356.
Boussemart J.P., Leleu H., Shen Z.Y., 2015. Environmental growth convergence
AN US
among Chinese regions. China Economic Review 34, 1-18.
Chang Y.T., Zhang N., Danao D., et al., 2013. Environmental efficiency analysis of transportation system in China: A non-radial DEA approach. Energy policy 58:
M
277-283.
Chen, L., Jia, G., 2016. Environmental efficiency analysis of China's regional industry:
ED
a data envelopment analysis (DEA) based approach. Journal of Cleaner Production,
PT
http://dx.doi.org/10.1016/j.jclepro.2016.01.045. Chen P.C., Yu M.M., et al., 2015. The enhanced Russell-based directional distance
CE
measure with undesirable outputs: Numerical example considering CO2 emissions. Omega 53, 30-40.
AC
Dyckhoff H., Allen K., 2011. Measuring ecological efficiency with data envelopment analysis (DEA). European Journal of Operational Research 132(2), 312-325.
Färe, R., Grosskopf, S., 2003. Nonparametric Productivity Analysis with Undesirable Outputs: comment. American Journal of Agricultural Economics 85(4), 1070-1074. Färe R., Grosskopf S., 2004. New directions: efficiency and productivity. Kluwer Academic Publishers, Boston. Färe R., Grosskopf S., Lovell C.A.K., 1989. Multilateral productivity comparisons 28
ACCEPTED MANUSCRIPT
when some outputs are undesirable: a nonparametric approach. The Review of Economics and Statistic 71, 90-98. Gorobets A., 2014. Eco-centric policy for sustainable development. Journal of Cleaner Production. Journal of Cleaner Production 61(1), 654-655. Hailu A., Veeman T.S., 2001. Non-parametric productivity analysis with undesirable
of Agricultural Economics 83, 605-616.
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outputs: an application to the Canadian pulp and paper industry. American Journal
Hang Y., Sun J.S., et al., 2015. Measuring energy inefficiency with undesirable outputs and technology heterogeneity in Chinese cities. Economic Modelling 49, 46-52.
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Korhonen P., Luptacik M., 2004. Eco-efficiency analysis of power plants: An extension of data envelopment analysis. European Journal of Operational Research 154, 437-446.
Kuosmanen T., 2005. Weak disposability in nonparametric production analysis with
M
undesirables. American Journal of Agricultural Economics 87(4), 1077-1082. Kuosmanen T., Matin R.K., 2011. Duality of weakly disposable technology. Omega
ED
39, 504-512.
Leleu H., 2013. Shadow pricing of undesirable outputs in nonparametric analysis.
PT
European Journal of Operational Research 231, 474-480. Li H., Shi J.F., 2014. Energy efficiency analysis on Chinese industrial sectors: an
CE
improved Super-SBM model with undesirable outputs. Journal of Cleaner Production 65, 97-107.
AC
Liang L., Wu J., et al., 2008. The DEA game cross-efficiency model and its Nash equilibrium. Operations Research 56, 1278-1288.
Liu W.B., Zhou Z.B., et al., 2015. Two-stage DEA models with undesirable input-intermediate-outputs. Omega 56, 74-87. Maghbouli M., Amirteimoori A., Kordrostami S., 2014. Two-stage network structures with undesirable outputs: a DEA based approach. Measurement 48, 109-118. Mariano E.B., Sobreiro V.A., et al., 2015. Human development and data envelopment analysis: A structured literature review. Omege 54, 33-49. 29
ACCEPTED MANUSCRIPT
Martinez-Núñez M., Pérez-Aguiar W.S., 2014. Efficiency analysis of information technology and online social networks management: An integrated DEA-model assessment. Information & Management 51(6), 712-725. Sahoo B.K., Luptacik M., Mahiberg B., 2011. Alternative measures of environmental technology structure in DEA: an application. European Journal of Operational Research 215, 750-762.
European Journal of Operational Research 142, 16–20.
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Seiford L.M., Zhu J., 2002. Modeling undesirable factors in efficiency evaluation.
Song M.L., Wang S.H., Liu Q.L., 2013. Environmental efficiency evaluation considering the maximization of desirable outputs and its application.
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Mathematical and Computer Modelling 58(5-6), 1110-1116.
Sueyoshi T., Goto M., 2012a. DEA radial measurement for environmental assessment and planning: desirable procedures to evaluate fossil fuel power plants. Energy Policy 41: 422-432.
M
Sueyoshi T., Goto M., 2012b. Weak and strong disposability vs. natural and managerial disposability in DEA environmental assessment: Comparison between
34, 686-699.
ED
Japanese electric power industry and manufacturing industries. Energy Economics
PT
Sueyoshi T., Goto M., 2014. Environmental assessment for corporate sustainability by resource utilization and technology innovation: DEA radial measurement on
CE
Japanese industrial sectors. Energy Economics 46, 295-307. Sueyoshi T., Goto M., Snell M.A., 2013. DEA environmental assessment:
AC
Measurement of damages to scale with unified efficiency under managerial disposability or environmental efficiency. Applied Mathematical Modelling 37, 7300-7314.
Vlontzos G., Niavis S., Manos B., 2014. A DEA approach for estimating the agricultural energy and environmental efficiency of EU countries. Renewable and Sustainable Energy Reviews 40, 91-96. Wang Y.M., Greatbanks R., Yang J.B., 2005. Interval efficiency assessment using data envelopment analysis. Fuzzy Sets and Systems 153, 347-370. 30
ACCEPTED MANUSCRIPT
Wanke P., Barros C.P., Emrouznejad A., 2016. Assessing productive efficiency of banks using integrated Fuzzy-DEA and bootstrapping: A case of Mozambican banks. European Journal of Operational Research 249, 378–389. Woo C.W., Chung Y.H., et al., 2015. The static and dynamic environmental efficiency of renewable energy: a Malmquist index analysis of OECD countries. Renewable and Sustainable Energy Reviews 47, 367-376.
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Wu J., An Q.X., et al., 2013. Congestion measurement for regional industries in China: A data envelopment analysis approach with undesirable outputs. Energy policy 57, 7-13.
Wu J., Yin P.Z., Sun J.S., et al., 2016. Evaluating the environmental efficiency of a
AN US
two-stage system with undesired outputs by a DEA approach: An interest preference perspective. European Journal of Operational Research 254(3), 1047-1062.
Yang H.L., Pollitt M., 2010. The necessity of distinguishing weak and strong
M
disposability among undesirable outputs in DEA: environmental performance of Chinese coal-fired power plants. Energy Policy 28, 4440-4444.
ED
Zha Y., Zhao L., Bian Y., 2016. Measuring regional efficiency of energy and carbon dioxide emissions in China: A chance constrained DEA approach. Computers &
AC
CE
PT
Operations Research 66, 351-361.
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