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Ann Oper Res (2017) 255:323–346DOI 10.1007/s10479-015-1881-x
Units invariant DEA when weight restrictionsare present: ecological performance of US electricityindustry
Wade D. Cook1 · Juan Du2 · Joe Zhu3,4
Published online: 14 May 2015© Springer Science+Business Media New York 2015
Abstract Electricity generation currently is the main industrial source of air emissions inthe United States. Both researchers and practitioners are interested in conducting studies toevaluate the ecological performance of this industry, in order to propose solutions to curbemissions of air pollutants and to improve the efficiency of converting fossil resources intoelectric energy. In this paper, data envelopment analysis (DEA) is used to assess ecologicalefficiency where air emissions are used as undesirable outputs. Although conventional DEAdoes not require a priori information on the input and output weights, weight restrictions canbe incorporated to reflect a user’s preference over the performance metrics, or to refine theDEA results. Addingweight restrictions voids the fact that DEA scores are independent of theunits of measurement. To incorporate weight constraints in ecological efficiency assessment,this paper develops aDEAmodel that is units-invariant whenweight restrictions are imposed.Moreover, the proposed model is equivalent to the standard units-invariant DEAmodel whenweight restrictions are not present.
Keywords Data envelopment analysis (DEA) · Ecological efficiency · Air emissions ·Weight restrictions · Units-invariant
B Juan [email protected]
Wade D. [email protected]
1 Schulich School of Business, York University, Toronto, ON M3J 1P3, Canada
2 School of Economics and Management, Tongji University, 1239 Siping Road,Shanghai 200092, People’s Republic of China
3 International Center for Auditing and Evaluation, Nanjing Audit University,Nanjing 211815, People’s Republic of China
4 Robert A. Foisie School of Business, Worcester Polytechnic Institute,Worcester, MA 01609, USA
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324 Ann Oper Res (2017) 255:323–346
1 Introduction
According to theUSEnvironmental ProtectionAgency (EPA), electricity generation becomesthe dominant industrial source of air pollution emissions in the United States. Specifically,fossil fuel-powered plants are responsible for up to 67% of the nation’s sulfur dioxide emis-sions, 40% of manmade carbon dioxide emissions, and 23% of nitrogen oxide emissions.1
These air emissions generated from the electric industry affect the environment in a negativeway and increase the risk of climate change, thus causing concerns from both sides of aca-demics and practice. Moreover, as far as natural resources are concerned, the electric powergeneration industry is one of the major consumers of fossil fuel such as coal, oil and gas.
This study focuses on the assessment of the ecological performance of the electric utilities.According to Korhonen and Luptacik (2004), ecological efficiency involves with the rela-tionship between the desirable and undesirable outputs of the production process. Therefore,it can offer helpful insights into the operations and technological strategies of these utilities.Methodologically, we employ data envelopment analysis (DEA) to evaluate the ecologicalperformance of the 575 large-scale electricity-generating plants in the US. First introducedby Charnes et al. (1978), DEA provides performance evaluations for peer decision makingunits (DMUs) with multiple metrics classified as inputs and outputs. Various DEA-basedmethods are proposed to address environmental issues. Among them, Sarkis (2006) inves-tigates into how environmental performance relates to certain practice adoption. Goto et al.(2014) assess the operational and environmental efficiencies on Japanese regional industries.Chen (2014) does an analytical reexamination of two popular models which are widely-usedfor measuring eco-efficiency, and empirically compares them with a weighted additive DEAmodel through a supply-chain carbon emissions data set from 50 major US manufacturingcompanies. Lozano et al. (2009) use DEA to reallocate emission permits from a centralizedperspective. By using DEA radial measures, Sueyoshi and Goto (2012) provide a desirableprocedure for environmental assessment and planning, while on the other hand, Tone andTsutsui (2010) develop a dynamic DEA model based on non-radial measures (slacks) toevaluate the US electric utility operation.
The data for our application are publically available from the US EPA’s eGRID (theEmission & Generation Resource Integrated Database) system. In our assessment, variousair emissions, such as by-products from the electricity-generating process, are viewed asundesirable outputs. To allow for user preferences, weight restrictions, which reflect therelative importance of various performance metrics, are incorporated in the evaluation DEAmodel. However, adding weight restrictions voids the fact that DEA scores are independentof the units of measurement. For example, Sarrico and Dyson (2004) point out that directmultiplier restrictions are problematic since they are dependent on the units of measurementof the inputs and outputs. Cherchye et al. (2007) claim that while the DEA CCR efficiency isunaffectedwhen some performancemetric(s) are scaled by a factor, theweights are not. Otherstudies such as Thanassoulis et al. (2004) and Ruiz and Sirvent (2012) have also discussedthis issue concerning absolute weight constraints.
In the current paper, we formally develop a DEAmodel that is units invariant when weightrestrictions are imposed. The basic idea looks similar to data normalization, and we showthat the proposed model not only frees absolute weight restrictions from the dependence onthe units of measurement, but is equivalent to the standard CCR model (1) when weightrestrictions are not present.
1 http://www.epa.gov/cleanenergy/energy-and-you/affect/air-emissions.html.
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Ann Oper Res (2017) 255:323–346 325
Although normalization has been suggested to treat data in some studies, for example,Roll and Golany (1993), Cherchye et al. (2004), Ramón et al. (2010), and Ruiz and Sirvent(2012), they are not very clear about the real purpose in doing so. Actually, in their work,normalization is used but does not distinctly target the units-invariant issue in the presenceof multiplier constraints. There are no clear statements in any of these studies that if datanormalization is done, the resulting model is equivalent to the original one and is necessarilyunits invariant with absolute weight constraints. It is much more likely for them to normalizedata in order to bring all factors to a common scale, so that it is easier to set multiplierrestrictions.
The remainder of the paper is organized as follows. Section 2 develops a units-invariantDEA model in the presence of weight restrictions, which is equivalent to the standard DEAmodel when weight restrictions are not present. Section 3 applies the model to evaluate theecological performance of 575 large-scale electric utilities in the US. Section 4 provides withconcluding remarks.
2 Units invariant weight restriction model
Consider a conventional DEA setting where there are a set of n decision making units(DMUs). Each unit j denoted by DMUj ( j = 1, . . . , n), is assumed to have m inputs ands outputs, denoted as xi j (i = 1, . . . ,m) and yr j (r = 1, . . . , s), respectively. The standardinput-oriented CCR ratio model can be expressed as (Charnes et al. 1978)
max
s∑
r=1ur yro
m∑
i=1vi xio
s.t.
s∑
r=1ur yr j
m∑
i=1vi xi j
≤ 1, j = 1, . . . , n
vim∑
i=1vi xio
≥ ε, i = 1, . . . ,m
urm∑
i=1vi xio
≥ ε, r = 1, . . . , s (1)
where restrictions vim∑
i=1vi xio
≥ ε, urm∑
i=1vi xio
≥ ε are imposed instead of vi , ur ≥ ε or vi , ur ≥ 0.
As pointed out in Cooper et al. (2011), such weight restrictions provide a fully rigorousdevelopment in the CCR model. However, our development to follow does not depend onthis particular choice of constraints on ε. In fact, such a choice ensures that the multipliersafter the Charnes–Cooper transformation (Charnes and Cooper 1962) are no less than thenon-Archimedean positive value ε.
DEA scores obtained from model (1) are invariant to the units of measurement of inputsand outputs. Or, we can say that the model (1) is units-invariant.
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326 Ann Oper Res (2017) 255:323–346
Weight restrictions can be added to model (1) to reflect the relative importance of variousinputs or outputs. In general, there are four types of weight restrictions (Thanassoulis et al.2004). These are (1) absolute weight restrictions, δi ≤ vi ≤ τi and ρr ≤ ur ≤ ηr , (2)assurance regions of type I (relative weight restrictions), κivi + κi+1vi+1 ≤ vi+2, wr ur +wr+1ur+1 ≤ ur+2, and αi ≤ vi
vi+1≤ βi , θr ≤ ur
ur+1≤ ζr , (3) assurance regions of type
II (input-output weight restrictions), γivi ≥ ur , and (4) restrictions on virtual inputs andoutputs, χi ≤ vi xi j∑m
i=1 vi xi j≤ πi and φr ≤ ur yr j∑s
r=1 ur yr j≤ ψr , where the Greek letters are user-
specified constants to reflect value judgments of the user (decision maker). For example, ifinput 1 is regarded as 3 times more important than input 2, then one imposes the restrictionv1 ≥ 3v2. However, the units-invariant property is no longer true under such restrictions.Suppose inputs 1 and 2 are bothmeasured inUSdollars. Now, if themeasurement unit of input2 is changed to 100 US dollars, then v1 ≥ 3v2 should be adjusted to v1 ≥ 0.03v2. In otherwords, weight restrictions on inputs and outputs have to consider the units of measurement.
Sarrico and Dyson (2004), aware of the problem with directly-restricted multipliers, sug-gest using weight restrictions on virtual inputs and outputs, since these are independent ofthe units of measurement. However, restrictions on the virtual input and output weights havereceived relatively little attention in the DEA literature.
While it is easy to adjust weight restrictions when inputs and outputs are measured inmonetary units, it is difficult to make such adjustments when inputs and outputs are notmeasured in monetary units. For example, suppose in a banking evaluation input 1 representsthe number of employees and input 2 the percentage of bad business loans. Assume thedecision maker believes that input 1 is at least four times as important as input 2, meaningthat the required restriction on the multipliers should be v1 ≥ 4v2. However, input 1 may bemeasured in units of 100, or 1,000 employees, meaning that it may now be difficult to expressdifferences with respect to the relative importance between the percentage of bad loans andthe number of employees when the latter is measured on these larger scales.
To address the problem of having to consider the units of measurement when weightrestrictions are present, we modify the CCR ratio model (1) into the following:
max
s∑
r=1
urmaxj
{yr j } yrom∑
i=1
vimaxj
{xi j } xio
s.t.
s∑
r=1
urmaxj
{yr j } yr jm∑
i=1
vimaxj
{xi j } xi j≤ 1, j = 1, . . . , n
vi/maxj
{xi j
}
m∑
i=1vi xio/max
j
{xi j
}≥ ε, i = 1, . . . ,m
ur/maxj
{yr j
}
m∑
i=1vi xio/max
j
{xi j
}≥ ε, r = 1, . . . , s (2)
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Ann Oper Res (2017) 255:323–346 327
If we let vi = vimaxj
{xi j } for i = 1, . . . ,m and ur = urmaxj
{yr j } for r = 1, . . . , s, then model
(2) is equivalent to model (1). Therefore model (2) produces exactly the same efficiencyscores as the original CCR model (1).
Applying the Charnes–Cooper transformation by letting υi = tvi , i = 1, . . . ,m and
μr = tur , r = 1, . . . , s where t =(
m∑
i=1
vi xiomaxj
{xi j })−1
, we get the multiplier model as
maxs∑
r=1
μr
maxj
{yr j
} yro
s.t.s∑
r=1
μr
maxj
{yr j
} yr j −m∑
i=1
υi
maxj
{xi j
} xi j ≤ 0, j = 1, . . . , n
m∑
i=1
υi
maxj
{xi j
} xio = 1
υi
maxj
{xi j
} ≥ ε, i = 1, . . . ,m
μr
maxj
{yi j
} ≥ ε, r = 1, . . . , s (3)
Without loss of generality, we now suppose assurance region (AR) type I of (input) weightrestrictions, say, v1 ≥ α1v2, are imposed in model (2), where α1 is a user-specified constant,and have the following model (4).
max
s∑
r=1
urmaxj
{yr j } yrom∑
i=1
vimaxj
{xi j } xio
s.t.
s∑
r=1
urmaxj
{yr j } yr jm∑
i=1
vimaxj
{xi j } xi j≤ 1, j = 1, . . . , n
v1 ≥ α1v2
vi/maxj
{xi j
}
m∑
i=1vi xio/max
j
{xi j
}≥ ε, i = 1, . . . ,m
ur/maxj
{yr j
}
m∑
i=1vi xio/max
j
{xi j
}≥ ε, r = 1, . . . , s (4)
Here these weight restrictions are specified by the user or decision maker based upon the“normalized” data set, namely the transformed data through respectively dividing by themaximum value of each metric. In doing so, this “normalized” data set consists of absolute
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328 Ann Oper Res (2017) 255:323–346
values without the influence of units or magnitudes. However, if the weight restrictions aredetermined according to the current (original) data set measured by various units, in order toaccommodate model (2), the weight restrictions need to be adjusted to preserve the originalpreferences. Consider the AR restriction v1 ≥ α1v2 as an example. If it is derived from theoriginal data set with units of measurement, in order to convey the same meaning to be usedin model (2), we should change it into v1
maxj
{x1 j } ≥ α1v2maxj
{x2 j } and incorporate this new AR
restriction to formmodel (4). In such a case, themultiplier restrictions inmodel (4) vary alongwith changes made to the maximum values of various metrics. To fix the weight restrictionsin our method, in this study they are supposed to be decided based on the “normalized” data.
It is important to note that this weight-restricted model (4) is not equivalent to the standardCCR model (1) with the same AR restriction v1 ≥ α1v2 incorporated. In fact, the restrictionv1 ≥ α1v2 of model (4) should be modified into v1 max
j
{x1 j
} ≥ α1v2 maxj
{x2 j
}to fit for
model (1), where vi , i = 1, . . . ,m and ur , r = 1, . . . , s denotes the multipliers of model (1).In other words, weight restriction v1 max
j
{x1 j
} ≥ α1v2 maxj
{x2 j
}, rather than v1 ≥ α1v2,
should be imposed in CCR model (1) to make it identical to our model (4).If we change the unit of measurement of input 2 by increasing or decreasing k-fold (k
is a positive constant), then in the original CCR model (1), we need to impose v1 ≥ α1v2k .
However, note that model (4) becomes
max
s∑
r=1
urmaxj
{yr j } yrom∑
i=1i �=2
vimaxj
{xi j } xio + v2maxj
{x2 j /k} (x2o/k)
s.t.
s∑
r=1
urmaxj
{yr j } yr jm∑
i=1i �=2
vimaxj
{xi j } xi j + v2maxj
{x2 j /k}(x2 j/k
)≤ 1, j = 1, . . . , n
v1 ≥ α1v2
vi/maxj
{xi j
}
m∑
i=1i �=2
vi xiomaxj
{xi j } + v2(x2o/k)maxj
{x2 j /k}≥ ε, i = 1, . . . ,m, i �= 2
v2/maxj
{x2 j/k
}
m∑
i=1i �=2
vi xiomaxj
{xi j } + v2(x2o/k)maxj
{x2 j /k}≥ ε
ur/maxj
{yr j
}
m∑
i=1i �=2
vi xiomaxj
{xi j } + v2(x2o/k)maxj
{x2 j /k}≥ ε, r = 1, . . . , s (5)
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Ann Oper Res (2017) 255:323–346 329
Note that the item v2maxj
{x2 j /k}(x2 j/k
)in model (5) can be simplified to v2
maxj
{x2 j } x2 j , whichrenders model (5) equivalent to the weight-restricted model (4). Furthermore, there is noneed to adjust the weight restrictions.
One should note that dividing the data under various performancemeasures by their respec-tive maximum values will not affect their essential quantitative characteristics within eachmetric. Hence these multiplier restrictions, which are determined based on the “normalized”data, reflect the user’s preference on various performance metrics, and do not require knowl-edge of the specific units. Onemay argue that there is no such thing as “absolute importance”.For example, Keeney et al. (2006) claim that weights should be related to the units and rangesof the measures. They illustrate with an example in evaluating academic programs that it isreasonable to state that one published book in an area is four times the contribution of onepublished article in that area, but it does not make sense to say that books are four times moreimportant than articles. However, as for our model (4), this “absolute importance” does existbecause the weight restrictions are decided by user from the “normalized” data set withoutthe need to consider units. The decision maker only needs to determine his/her preferenceor the relative importance of one dimensionless unit of one normalized metric compared tothat of another.
Therefore, no matter what units input or output measures take, weight-restricted model(4) actually deals with the constant weight restrictions and the same set of data, whichare obtained through a transformation similar in essence with “normalization”. By usingmodel (4), the decision maker only needs to apply his/her intuitive preferences on var-ious metrics, rather than adjusting them every time there is any change on units. Thisimplies that our newly-proposed efficiency model (2) is units-invariant even under weightrestrictions.
While the above discussion is based upon the input-oriented CCRModel, the same devel-opment and model (2) can be applied to other DEA models. Such development can also bebased upon the multiplier models.
For example, if we impose υ1 ≥ α1υ2 in model (3), and suppose that the unit of input 2is changed by k-fold, then we have
maxs∑
r=1
μr
maxj
{yr j
} yro
s.t.s∑
r=1
μr
maxj
{yr j
} yr j −m∑
i = 1i �=2
υi
maxj
{xi j
} xi j − υ2
maxj
{x2 j/k
}(x2 j/k
) ≤ 0, j =1, . . . , n
m∑
i=1i �=2
υi
maxj
{xi j
} xio + υ2
maxj
{x2 j/k
} (x2o/k) = 1
υ1 ≥ α1υ2υi
maxj
{xi j
} ≥ ε, i = 1, . . . ,m, i �= 2
υ2
maxj
{x2 j/k
} ≥ ε
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330 Ann Oper Res (2017) 255:323–346
μr
maxj
{yr j
} ≥ ε, r = 1, . . . , s (6)
The item υ2maxj
{x2 j /k}(x2 j/k
)inmodel (6) is equal to υ2
maxj
{x2 j } x2 j ,makingmodel (6) equivalent
tomodel (3)with the sameweight restrictionwithout the need to adjust theweight restrictions.In summary, we provide the following application procedure/steps for using our units-
invariant DEA model:
Step 1. Normalize data (via dividing the data of each metric by its maximum value, oraverage, range, etc.);
Step 2. Specify weight restrictions based on the normalized data (without considering theunits of measurement);
Step 3. Apply units-invariant model (4).
However, in practice it may be that determining weight restrictions from normalizeddata, without considering units, is impractical. It is highly possible that in a real case, theuser comes up with the weight restrictions while considering the units of measurement, forexample, the book versus journal article example mentioned previously. If that is the case, theuser’s version of weight restrictions needs to be adjusted to preserve the original preferences.Take the AR restriction v1 ≥ α1v2 for example. If it is determined from the original data setwith units under consideration, in order to convey the same meaning, we should transformit into v1
maxj
{x1 j } ≥ α1v2maxj
{x2 j } and incorporate this new AR restriction in model (4). Note
that maxj
{xi j
}could be substituted by mean
j
{xi j
}or range
j
{xi j
}if the average or range
is used as the divisor for normalization. Therefore, the application procedure for using ourunits-invariant model is modified to fit such a case as:
Step 1. Specify weight restrictions based on the current/original data set measured by units;Step 2. Transform the weight restrictions determined in Step 1 to fit for model (2) or (4)
(by dividing each multiplier by its corresponding divisor used in model (2) or (4),which could be the maximum value, average, or range, etc. of each performancemetric);
Step 3. Incorporate the transformed weight restrictions obtained in Step 2 into model (2) toform units-invariant model (4), and apply it to the data set.
3 Ecological efficiency of US electric utilities
3.1 Data
The data source utilized in this study is the EPA’s Emissions & Generation Resource Inte-grated Database (eGRID)2, which is a comprehensive source of data on the environmentalcharacteristics of almost all electric power generated in the US. For this application, we useeGRID2012 Version 1.0, which is released in April 2012 and includes the data of year 2009.Data reported in that version consists of nearly 5500 power plants, and include informationfrom up to 25 dimensions, such as generation in megawatt-hour (MWh), the total heat con-sumed in millions of British thermal unit (MMBtu), and emissions in tons for carbon dioxide
2 http://www.epa.gov/airmarkets/egrid/.
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Ann Oper Res (2017) 255:323–346 331
(CO2), nitrogen oxides (NOx), and sulfur dioxide (SO2). These emissions are treated asundesirable outputs. There are a number of ways that one can deal with undesirable outputs(Zhu 2014). In the current paper, we adopt the approach of Seiford and Zhu (2002) to treat theemissions. i.e., we convert undesirables to desirables via subtracting the undesirable valuesfrom a large number. We should note that such a conversion is only solution invariant underthe variable returns to scale (VRS) assumption. However, the current paper’s interest is not onobtaining an invariant solution, and we just use this linear translation to treat our undesirablemeasures.
In order to evaluate the ecological efficiency, we use the same set of environmentaloutputs as in Sarkis and Cordeiro (2012), which analyzes the data of year 1996 through1998 from eGRID 2000. Specifically, these environmental outputs, consisting of the annualemissions in tons for CO2, NOx, and SO2, are incorporated in our DEA models as undesir-able metrics, along with a regular desirable output—the annual net electricity generated inMWh. Two input measures are selected for this application, namely, the total heat used forgeneration measured in MMBtu, and the net generator (nameplate) capacity in megawatts(MW).
We follow the same filtering principles adopted by Sarkis and Cordeiro (2012) to select asample. First of all, all non-fossil fuel-generated plants are eliminated. Then smaller plantswith an annual generation under 106 MWh are also eliminated. These lead to a total of 575plants left in the final data set. Descriptive statistics for these utilities are demonstrated inTable 1. Note that deleting the smaller-size plants makes the final data set exhibit a higherdegree of homogeneity, which is a desirable feature in DEA applications.
For the assessment of environmental performance, we believe that the annual emissions ofany of the three air pollutants are no less important than the annual net generation. Therefore,restrictions ur ≥ u1, r = 2, 3, 4 should be imposed on the output multipliers. Currently,input 2 the net generator capacity and output 1 the annual net generation are measured inMW and MWh, respectively. If they are instead measured in kilowatts (kW) and kilowatt-hour (kWh), where one MW equals 1000 kW and one MWh equals 1000 kWh, all theoriginal values for input 2 and output 1 should be multiplied by 1000, respectively. If theconventional CCR model (1) is used, the AR restrictions ur ≥ u1, r = 2, 3, 4 should beadjusted to ur ≥ 1000u1, r = 2, 3, 4 to correctly reflect the relative importance of theconcerned measures. However, as far as model (4) is concerned, there is no need to changethe original AR restrictions ur ≥ u1, r = 2, 3, 4 and thus it can be directly applied to thenew data set (measured by the changed units) for efficiency calculation.
First of all, wemaintain the current units for each input/output measure, and use weighted-restricted model (4) to evaluate the ecological efficiency of each plant. Multiplier restrictionsur ≥ u1, r = 2, 3, 4 are incorporated in model (4), and the results are listed in Table 2.
From Table 2, we note that only 3, or 0.52%, of all electric utilities are ecologicallyefficient. The eco-efficiency scores of all 575 observations exhibit a fairly diversified distri-bution, ranging from the lowest 0.0276 to the highest 1 with an average of 0.2611. Figure 1illustrates the frequency of all plants falling into various efficiency intervals. The distrib-ution of eco-efficiency demonstrates an uneven trend. An overwhelming majority (550) ofall 575 plants locate in the efficiency intervals below 0.6, totally accounting for 95.65%,among which 42.26% (or 243 plants) have a score less than 0.2. Only 25 or 4.35% ofall plants are assessed with an eco-efficiency beyond the medium-level 0.6. Moreover,only 8 (1.39%) utilities have a high-level environmental performance above the score 0.8.These imply that most plants do rather poorly in terms of ecological performance. Theremarkable majority of the observed plants have high levels of CO2, NOx, and SO2 annualemissions, and due to our emphasis on the relative importance of these undesirable out-
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332 Ann Oper Res (2017) 255:323–346
Table1
Descriptiv
estatisticsforthefin
aldataseto
f575plants
Descriptiv
estatistic
sInpu
t1Inpu
t2Outpu
t1Outpu
t2Outpu
t3Outpu
t4Ann
ual
heat
inpu
t(M
MBtu)
Netgenerator
capacity
(MW)
Annualn
etgeneratio
n(M
Wh)
Ann
ualC
O2
emissions(tons)
Ann
ualN
Ox
emissions(tons)
AnnualS
O2
emissions(tons)
Mean
4,12
,07,51
7.7
1026
.044
,19,72
5.1
38,29,42
9.06
3316
.41
9559
.67
SD3,91
,76,31
7.0
695.5
39,37,46
4.3
42,18,44
0.46
4663
.27
1656
3.96
Maxim
um24
,26,39
,972
.743
92.5
2,29
,77,98
0.0
2,48
,94,85
2.43
42,510
.52
1,13
,138
.95
Minim
um90
4.0
136.9
10,04,74
2.0
73.70
0.14
0.09
Range
24,26,39
,068
.742
55.6
2,19
,73,23
8.0
2,48
,94,77
8.73
42,510
.38
1,13
,138
.86
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Ann Oper Res (2017) 255:323–346 333
Table2
Efficiency
results
viamodel(4)with
originalunits
andARrestrictions
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
10.36
5611
60.69
2823
10.45
0234
60.41
6946
10.42
02
20.52
7111
70.46
5123
20.07
8434
70.20
0446
20.23
89
30.04
3411
80.09
3623
30.24
2034
80.26
8746
30.18
40
40.24
3211
90.15
3423
40.50
1634
90.25
2746
40.29
57
50.13
8912
00.03
2923
50.40
7535
00.14
0746
50.18
00
60.22
9712
10.34
0123
60.09
1735
10.21
5246
60.23
76
70.21
3112
20.31
0523
70.33
0135
20.52
3446
70.42
88
80.05
2412
30.15
2923
80.59
4935
30.30
4546
80.53
12
90.09
8612
40.06
9323
90.09
2735
40.28
5346
90.40
52
100.17
7112
50.24
2224
00.43
8835
50.25
9347
00.27
52
110.10
8212
60.26
4024
10.09
6935
60.20
8847
10.07
25
120.04
0312
70.11
8424
20.21
0035
70.21
1047
20.20
52
130.30
9412
80.02
7824
30.21
2035
80.13
9847
30.10
53
140.18
5112
90.07
5224
40.08
8635
90.15
0947
40.21
66
150.43
2713
00.19
0024
50.32
8336
00.83
4947
50.32
18
160.09
6113
10.57
2124
60.50
1836
10.57
1547
60.28
90
170.23
3213
20.09
5524
70.32
2336
20.38
3247
70.25
53
180.27
1513
30.66
7424
80.29
4336
30.24
7947
80.40
70
190.06
7313
40.21
5324
90.07
0836
40.26
7647
90.16
92
200.37
8613
50.70
5425
00.15
6136
50.17
4748
00.11
29
210.77
3913
60.35
0225
10.59
8736
60.19
6848
10.19
80
220.13
6313
70.44
2625
20.86
4336
70.06
9948
20.35
38
230.06
5913
80.11
5425
30.83
6536
80.07
1648
30.43
26
240.22
0613
90.19
9325
40.39
3536
90.09
6548
40.25
30
250.11
3414
00.38
5925
50.07
3337
00.04
8148
50.13
39
123
334 Ann Oper Res (2017) 255:323–346
Table2
continued
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
260.15
1314
10.16
5525
60.36
0037
10.24
1948
60.13
58
270.49
7514
20.48
1125
70.88
8437
20.04
5948
70.24
36
280.10
6214
30.17
4225
80.20
9937
30.19
5448
80.55
74
290.40
7114
40.07
1625
90.02
7637
40.11
7648
90.30
07
300.21
5914
50.64
2726
00.41
6637
50.08
6849
00.15
47
310.11
2114
60.70
2726
10.22
4537
60.07
4549
10.28
27
320.05
4714
70.06
6726
20.15
1737
70.04
8349
20.06
41
330.18
1414
80.13
0126
30.07
4737
80.09
1349
30.24
64
340.18
6714
90.22
0426
40.15
7637
90.08
7349
40.24
96
350.33
9115
00.24
0826
50.22
5738
00.51
8449
50.04
05
360.07
4215
10.30
9226
60.28
2838
10.38
0049
60.16
86
370.38
1015
20.16
4626
70.11
3638
20.50
6649
70.05
30
380.20
7415
30.30
2326
80.04
7738
30.44
1149
80.29
86
390.38
1315
40.24
4026
90.52
7538
40.11
5749
90.25
93
400.47
9515
50.43
2327
00.58
3238
50.20
5650
00.30
49
410.26
4315
60.10
5627
10.12
8038
60.29
0350
10.20
96
420.16
6615
70.36
9627
20.11
3738
70.14
4550
20.18
60
430.12
8015
80.11
2827
30.38
1638
80.29
8450
30.21
45
441
159
0.29
7127
40.04
6238
90.05
9750
40.28
65
450.24
3116
00.09
3627
50.12
0639
00.06
0350
50.29
94
460.28
8716
10.31
5327
60.22
1839
10.21
2650
60.18
31
470.14
4016
20.10
2027
70.10
8239
20.15
7950
70.50
00
480.19
1116
30.06
2827
80.10
1339
30.22
7950
80.28
70
490.34
8916
40.16
2827
90.24
7139
40.14
7550
90.10
38
500.78
6916
50.09
9628
00.50
4239
50.10
5951
00.31
25
123
Ann Oper Res (2017) 255:323–346 335
Table2
continued
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
510.13
0716
60.20
0628
10.10
6939
60.20
9951
10.35
31
520.24
9316
70.22
9128
20.44
5839
70.54
5351
20.20
77
530.46
3216
80.06
0828
30.54
1739
80.52
2251
30.64
81
540.27
8216
90.20
5128
40.11
2039
90.25
4951
40.31
12
550.58
9717
00.11
2528
50.52
0140
00.23
1151
50.31
93
560.14
0317
10.08
7028
60.27
3440
10.28
9251
60.35
87
570.25
5517
20.51
0928
70.20
2940
20.31
3351
70.15
91
580.18
2217
30.36
4928
80.21
8740
30.23
4651
80.26
22
590.38
1217
40.55
8228
91
404
0.34
5851
90.52
52
600.23
8017
50.03
8129
00.44
3340
50.04
6952
00.10
78
610.32
5317
60.13
2129
10.15
8340
60.20
1652
10.38
08
620.51
0517
70.11
9129
20.45
8140
70.06
8452
20.03
05
630.27
8317
80.24
7429
30.32
6540
80.16
6852
30.07
05
640.46
4517
90.29
2429
40.26
2740
90.12
5352
40.36
94
650.35
3718
00.05
1529
50.23
1741
00.27
6252
50.19
47
660.16
2918
10.04
0129
60.05
7841
10.17
2852
60.23
80
670.10
3518
20.20
6929
70.05
2441
20.05
3252
70.27
18
680.09
2918
30.17
9329
80.77
4041
30.05
2652
80.69
92
690.17
3418
40.11
7629
90.22
9841
40.33
7052
90.32
54
700.35
2018
50.17
1130
00.06
4041
50.06
7053
00.08
91
710.28
8418
60.48
8130
10.33
0441
60.26
2953
10.13
38
720.49
9218
70.38
6430
20.17
8141
70.33
6053
20.07
97
730.24
2418
80.06
0330
30.13
5741
80.30
6553
30.63
36
740.21
9618
90.07
6430
40.20
8141
90.23
2853
40.38
50
750.47
5719
00.23
6430
50.25
3542
00.06
7553
50.16
07
123
336 Ann Oper Res (2017) 255:323–346
Table2
continued
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
760.21
6219
10.36
7030
60.06
6042
10.07
8253
60.07
01
770.56
9619
20.40
9330
70.18
0842
20.22
4553
70.31
71
780.63
7419
30.45
3630
80.18
5242
30.22
6653
80.15
90
790.30
2119
40.11
2030
90.74
8442
40.36
4553
90.28
46
800.43
7919
50.19
3131
00.05
1242
50.36
3154
00.16
88
810.26
5919
60.38
2531
10.13
8942
60.26
0854
10.31
55
820.24
2519
70.23
7531
20.09
5542
70.51
2754
20.23
10
830.36
0019
80.16
5931
30.27
4442
80.44
3654
30.38
77
840.21
9819
90.20
1731
40.17
5142
90.32
4154
40.59
70
850.22
1920
00.29
2931
50.15
3843
00.05
5054
50.41
09
860.42
8720
10.05
7631
60.70
6143
10.42
0954
60.55
06
870.06
1620
20.08
1731
70.08
1643
20.17
5554
70.55
49
880.38
5920
30.33
1631
80.08
3243
30.26
9354
80.07
46
890.17
2720
40.22
0231
90.19
4343
40.24
2754
90.12
04
900.28
8120
50.07
1432
00.57
2343
50.17
4455
00.16
70
910.46
5720
60.04
7232
10.19
5743
60.19
2755
10.38
43
920.13
0220
70.22
7732
20.26
2943
70.10
4655
20.33
23
930.03
7020
80.06
5032
30.45
6743
80.27
1655
30.65
70
940.32
2120
90.13
2932
40.20
0143
90.10
3955
40.11
00
950.33
0521
00.37
8032
50.14
3144
00.18
4855
51
960.08
0821
10.06
1032
60.18
1044
10.05
0855
60.26
88
970.09
0621
20.36
4532
70.26
0044
20.07
9155
70.32
59
980.07
6721
30.17
6632
80.16
4844
30.15
5755
80.11
11
990.44
7421
40.48
0232
90.31
6144
40.08
4355
90.11
26
100
0.13
6121
50.30
0333
00.51
7344
50.20
7556
00.11
96
123
Ann Oper Res (2017) 255:323–346 337
Table2
continued
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
101
0.12
2821
60.57
2933
10.52
7044
60.88
3456
10.06
40
102
0.06
9321
70.29
1133
20.05
4244
70.74
2456
20.04
33
103
0.04
5121
80.19
6633
30.23
5944
80.48
8356
30.30
14
104
0.22
5821
90.36
3033
40.39
4544
90.29
0856
40.33
82
105
0.11
1022
00.20
5733
50.07
0845
00.66
1856
50.08
32
106
0.29
7622
10.59
1133
60.29
4945
10.19
2256
60.10
42
107
0.14
4222
20.14
6933
70.11
6345
20.10
1256
70.07
95
108
0.33
6022
30.38
4733
80.36
8045
30.56
3356
80.20
07
109
0.17
5722
40.22
2133
90.24
1845
40.45
6156
90.09
61
110
0.41
3622
50.38
2634
00.24
8645
50.30
1057
00.14
57
111
0.50
9522
60.09
7234
10.22
8945
60.26
4857
10.05
07
112
0.07
3222
70.11
5034
20.21
2845
70.46
1557
20.23
45
113
0.08
7922
80.28
0434
30.29
7845
80.35
1057
30.11
38
114
0.09
4022
90.17
5834
40.18
6345
90.20
5357
40.16
43
115
0.10
7223
00.38
5334
50.56
4646
00.20
8957
50.35
53
123
338 Ann Oper Res (2017) 255:323–346
00.05
0.10.15
0.20.25
0.30.35
0.40.45
0-0.2 0.2-0.4 0.4-0.6 0.6-0.8 0.8-1 1
Frequency in Eco-efficiencyFrequency
Efficiency intervals
Fig. 1 Eco-efficiency distribution from model (4) with AR restrictions
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0-0.2 0.2-0.4 0.4-0.6 0.6-0.8 0.8-1 1
Frequency in Eco-efficiencyFrequency
Efficiency intervals
Fig. 2 Eco-efficiency distribution from model (4) without AR restrictions
puts via multiplier restrictions, they are not able to eliminate or lessen the negative impactfrom those environmental outputs on eco-efficiency by choosing their favorable multipliersfreely.
However, if theAR restrictionsur ≥ u1, r = 2, 3, 4 are dropped frommodel (4),we obtaina much better group of efficiency scores, ranging from the lowest 0.1543 to the highest 1with an average of 0.5515. There are 6 eco-efficient plants, 47 or 8.17% have a high-levelperformance (above the score 0.8), and 233 or 40.52% are operating at amedium level (abovethe score 0.6). The number of plants with an eco-efficiency below 0.6 is 342, accounting for59.48%. The eco-efficiency distribution exhibits a symmetrical trend, which is displayed inFig. 2. The efficiency intervals with the top two frequencies are [0.4, 0.6) and [0.6, 0.8),covering 37.39 and 32.35% of all plants, respectively. Then the frequencies are successivelydecreasing towards either the lower efficiency intervals or the higher efficiency intervals.
Due to the fact that model (4) is independent of units, if input 2 (the net generator capacity)is measured in kW instead of MW and output 1 (the annual net generation) is measured inkWh instead of MWh, there is no need to modify the original AR restrictions according to
123
Ann Oper Res (2017) 255:323–346 339
Table3
Efficiency
results
viamodel(1)with
originalunits
andARrestrictions
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
10.36
7211
60.69
2823
10.45
0234
60.41
6946
10.42
02
20.52
7111
70.46
5123
20.08
5034
70.20
3546
20.23
93
30.05
1411
80.11
3923
30.24
5034
80.27
3946
30.18
56
40.24
4611
90.15
3423
40.50
1634
90.25
2746
40.29
57
50.14
3412
00.03
7223
50.41
0535
00.14
6746
50.18
96
60.23
2712
10.34
2223
60.09
7535
10.21
8946
60.24
33
70.21
3912
20.31
3123
70.33
0135
20.52
8346
70.43
09
80.06
3612
30.15
2923
80.59
4935
30.30
6346
80.53
12
90.09
8712
40.08
6023
90.09
6835
40.28
5346
90.41
37
100.18
3112
50.24
8124
00.43
8835
50.26
6047
00.28
07
110.11
3812
60.26
4024
10.09
9435
60.21
1347
10.07
66
120.04
5512
70.12
5024
20.21
7335
70.21
1647
20.21
73
130.31
0512
80.03
4924
30.21
2035
80.14
1447
30.11
15
140.19
0112
90.07
4824
40.09
0935
90.15
0947
40.21
92
150.43
2613
00.19
2924
50.33
1036
00.83
4847
50.32
46
160.09
7513
10.57
2124
60.50
5536
10.57
1547
60.29
40
170.23
7913
20.10
8924
70.32
5836
20.38
4447
70.26
15
180.27
4013
30.66
7424
80.30
1136
30.24
8247
80.41
59
190.07
6013
40.22
2724
90.07
8836
40.27
1147
90.17
43
200.37
8613
50.70
8225
00.15
6136
50.18
8748
00.12
10
210.77
5113
60.35
2125
10.59
8736
60.19
8948
10.20
21
220.13
6213
70.44
2625
20.86
4336
70.07
0848
20.35
47
230.07
6813
80.12
5125
30.83
6536
80.08
0348
30.43
26
240.22
0613
90.20
5425
40.40
4236
90.11
1548
40.26
06
250.11
3514
00.38
8825
50.08
0137
00.04
9748
50.13
39
123
340 Ann Oper Res (2017) 255:323–346
Table3
continued
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
260.15
7714
10.16
5525
60.36
3137
10.24
1948
60.14
04
270.49
7514
20.48
1125
70.88
8437
20.05
3148
70.24
85
280.10
9514
30.17
6625
80.20
9937
30.19
5448
80.55
74
290.40
7114
40.07
4325
90.04
0137
40.12
2548
90.30
07
300.21
5614
50.64
2726
00.41
6637
50.09
2449
00.16
10
310.11
8514
60.70
2726
10.23
2537
60.08
8049
10.28
26
320.05
4714
70.06
9326
20.15
9137
70.06
3749
20.06
87
330.18
5014
80.13
0126
30.08
3237
80.09
5049
30.25
15
340.18
9514
90.22
0626
40.17
0637
90.10
2549
40.25
26
350.33
9115
00.24
0826
50.22
5738
00.51
8449
50.05
12
360.07
5715
10.30
9226
60.28
2838
10.38
0049
60.16
90
370.38
1015
20.16
7526
70.11
8538
20.50
6649
70.06
24
380.20
7415
30.30
2326
80.05
1238
30.44
1149
80.30
21
390.38
1315
40.24
4026
90.52
7538
40.12
4749
90.25
93
400.47
9515
50.43
2327
00.58
3238
50.21
0450
00.30
49
410.27
0115
60.10
9627
10.12
8038
60.29
5850
10.21
12
420.17
2715
70.36
9627
20.11
3738
70.14
9850
20.18
60
430.12
7615
80.11
7227
30.38
1638
80.30
5650
30.21
52
441
159
0.29
7127
40.05
5538
90.06
7950
40.29
01
450.24
8416
00.09
7527
50.12
5039
00.07
1150
50.30
27
460.29
1316
10.31
5327
60.23
2439
10.21
3350
60.18
31
470.14
5116
20.10
5727
70.11
1239
20.16
1050
70.50
00
480.19
6616
30.06
9827
80.10
3739
30.22
7950
80.28
70
490.34
8916
40.16
5027
90.25
5039
40.14
7550
90.10
51
500.78
6916
50.10
2128
00.50
7039
50.11
5951
00.31
61
123
Ann Oper Res (2017) 255:323–346 341
Table3
continued
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
510.13
6916
60.20
3428
10.12
2339
60.22
1751
10.35
31
520.25
4616
70.22
9228
20.44
5839
70.54
5451
20.22
03
530.46
3116
80.06
9128
30.54
1739
80.52
3651
30.65
16
540.28
1816
90.20
5128
40.11
4539
90.25
9451
40.31
19
550.59
7917
00.11
2528
50.52
0140
00.23
6151
50.32
70
560.14
6117
10.10
3528
60.27
3440
10.29
4651
60.35
87
570.26
0717
20.51
0928
70.20
2940
20.31
7751
70.16
53
580.18
8517
30.36
4928
80.22
3740
30.23
7951
80.26
19
590.38
1217
40.57
0228
91
404
0.34
5851
90.52
52
600.24
3617
50.04
1129
00.44
3340
50.04
8852
00.11
62
610.32
6617
60.13
5429
10.16
4240
60.20
9352
10.38
08
620.51
9117
70.12
2629
20.45
8140
70.07
1252
20.03
30
630.27
9217
80.24
7429
30.32
9740
80.16
6852
30.07
68
640.46
4617
90.29
9929
40.26
2740
90.13
1552
40.36
94
650.36
2718
00.05
6329
50.23
5341
00.27
8252
50.20
04
660.16
3418
10.05
2529
60.06
4641
10.17
5552
60.24
12
670.10
3518
20.21
6829
70.05
4141
20.07
5452
70.27
18
680.09
2918
30.18
5629
80.77
4041
30.06
6352
80.70
51
690.17
7318
40.12
6229
90.22
9841
40.33
7052
90.32
88
700.35
2518
50.17
1130
00.06
3441
50.07
1553
00.08
91
710.28
8418
60.48
8130
10.33
6441
60.26
6953
10.13
38
720.51
3018
70.38
6430
20.17
8141
70.33
5553
20.08
15
730.24
6418
80.06
0330
30.13
8641
80.30
6953
30.63
36
740.21
9618
90.08
2430
40.21
9141
90.24
2153
40.38
65
750.47
6119
00.23
6430
50.26
3842
00.08
7353
50.16
29
123
342 Ann Oper Res (2017) 255:323–346
Table3
continued
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
760.22
0619
10.37
0930
60.06
6942
10.08
2553
60.07
67
770.56
9619
20.40
9330
70.18
0342
20.22
4753
70.31
71
780.63
7419
30.45
6030
80.18
5242
30.23
7453
80.15
90
790.30
6119
40.12
1930
90.74
8442
40.36
4553
90.28
46
800.43
7919
50.20
0731
00.05
1942
50.36
3154
00.16
84
810.26
6919
60.39
0631
10.14
6442
60.26
5454
10.31
55
820.24
8019
70.23
7531
20.10
4442
70.51
3454
20.23
10
830.36
0019
80.16
9331
30.29
0542
80.44
3654
30.38
77
840.22
8519
90.20
1731
40.19
8642
90.32
3854
40.59
70
850.22
1920
00.29
5731
50.17
6543
00.05
4154
50.41
09
860.42
8720
10.05
8131
60.70
6143
10.46
1254
60.55
06
870.06
1620
20.08
1431
70.09
4143
20.18
0154
70.55
66
880.38
5920
30.33
1631
80.09
1643
30.26
9254
80.07
46
890.17
5720
40.22
0231
90.20
2743
40.24
7854
90.12
66
900.28
8120
50.07
7332
00.57
7343
50.17
5255
00.17
07
910.46
5720
60.05
2432
10.20
0343
60.19
7155
10.38
57
920.13
5020
70.22
7732
20.26
7543
70.10
4455
20.34
02
930.05
3920
80.07
4132
30.45
6743
80.28
9455
30.65
92
940.32
2220
90.13
2932
40.20
4043
90.10
5655
40.11
00
950.33
0521
00.37
8032
50.14
7344
00.18
4855
51
960.08
6521
10.06
8832
60.18
7944
10.05
1455
60.26
88
970.09
6421
20.36
8232
70.26
0044
20.08
3355
70.32
67
980.08
4021
30.18
0432
80.16
6744
30.16
6455
80.11
28
990.44
7421
40.48
0232
90.31
6144
40.09
1955
90.11
52
100
0.14
6121
50.30
0333
00.58
9844
50.20
7556
00.13
19
123
Ann Oper Res (2017) 255:323–346 343
Table3
continued
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
DMU
Efficiency
101
0.12
7321
60.57
8733
10.52
7044
60.88
4656
10.06
55
102
0.07
5121
70.29
5933
20.05
7944
70.74
2456
20.04
33
103
0.05
1721
80.19
6633
30.24
0244
80.48
8356
30.30
47
104
0.22
7821
90.36
8933
40.39
4544
90.29
1556
40.34
62
105
0.11
1022
00.20
5733
50.07
0845
00.66
3856
50.08
18
106
0.29
9822
10.59
1133
60.29
8445
10.19
6956
60.10
29
107
0.14
5922
20.14
6933
70.12
2445
20.10
8056
70.07
94
108
0.33
6022
30.38
4733
80.36
8045
30.56
5456
80.20
80
109
0.17
5722
40.22
7633
90.24
6445
40.45
6156
90.09
62
110
0.41
3622
50.38
2634
00.25
2345
50.30
2257
00.15
77
111
0.50
9522
60.10
1534
10.23
4745
60.26
4857
10.05
47
112
0.08
0022
70.11
7834
20.21
2845
70.46
1557
20.23
45
113
0.09
1622
80.28
4834
30.29
9045
80.35
3057
30.11
38
114
0.09
6322
90.18
1334
40.18
8945
90.21
5257
40.18
45
115
0.10
7523
00.38
5334
50.56
4646
00.22
1757
50.36
16
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344 Ann Oper Res (2017) 255:323–346
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0-0.2 0.2-0.4 0.4-0.6 0.6-0.8 0.8-1 1
Frequency in Eco-efficiency
Efficiency intervals
Fig. 3 Eco-efficiency distribution from model (1) with AR restrictions
the changing units of measurement. Therefore, we apply our model (4) with the original ARrestrictions, directly to the transformed data set under the new units (kW and kWh), andobtain exactly the same efficiency results as in Table 2.
If adding the same AR restrictions directly to the conventional CCRmodel (1) to calculatethe same data set, we obtain another group of efficiency scores and report them in Table 3. Theeco-efficiency from Table 2 and 3 are similar but not exactly the same, because the proposedweight-restricted model (4) is not strictly equal to model (1) with the same AR restrictions.One should modify the multiplier restrictions accordingly in order to make them identicalto each other. We have explained this in detail above directly after proposing model (4) inSect. 2.
The eco-efficiency distribution of Table 3 is very similar to that of Table 2, which isdiversified and also demonstrates an uneven trend. The efficiency scores from Table 3 rangefrom the lowest 0.0330 to the highest 1 with an average of 0.2646.Only 3, or 0.52%, ofall electric utilities are ecologically efficient. A significant majority (550, or 95.65%) of all575 plants perform poorly with respect to environment with an eco-efficiency below 0.6,among which 41.22% (or 237) are evaluated worse than 0.2. We present the frequency ofeach efficiency interval in Fig. 3.
However, if we change the units of measurement in the same way, namely input 2and output 1 are respectively re-measured in kW and kWh, the original AR restrictionsur ≥ u1, r = 2, 3, 4 must be adjusted into ur ≥ 1000u1, r = 2, 3, 4 to preserve the samepreference of the concernedmeasures. If they are not changed accordingly, the eco-efficiencyscores obtained via CCR model (1) under the new units of measurement vary significantly,compared with the results in Table 3. The maximum and average absolute efficiency differ-ences are 0.8628 and 0.2865, respectively. This indicates that it is tremendously importantto adjust restrictions on multipliers according to the units of measurement; otherwise onewould get erroneous results.
4 Conclusions
This study uses DEA to evaluate the ecological performance of the US electricity industry,which is believed to be the main industrial source of air emissions in this country. Altogether
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Ann Oper Res (2017) 255:323–346 345
575 large-scale electric utilities are selected to compose the sample, and the environmentaloutputs, namely the annual emissions of CO2, NOx, and SO2, are treated as undesirablemeasures in our DEA models.
In order to reflect users’ preferences or the relative importance of the various performancemetrics, weight restrictions should be incorporated in the evaluation model. However, thisvoids the fact that DEA scores are independent of the units of measurement. To settle thisproblem, this paper develops a DEA model that is units-invariant when weight restrictionsare imposed.Moreover, the proposedmodel is equivalent to the standard units-invariant DEAmodelwhenweight restrictions are not present. Therefore, changing the units ofmeasurementwill not cause any problem for our weighted-restricted model. However, if adding weightrestrictions to the standard DEAmodels, one needs to adjust the restrictions according to theunits of measurement, or one would obtain erroneous efficiency results.
The eco-efficiency distribution of the US electricity industry is diversified and uneven. Interms of environmental performance, most (more than 95%) plants observed do rather poorlywith efficiency scores below 0.6, due to their extensive emissions of harmful air pollutants.
Finally, in terms of model development, note that model (2) uses maximum values ofinputs and outputs. Alternatively, the averages, ranges or minimum values (if zero data arenot present) can also be used. This can be viewed as input and output data “normalization”.However, with the units-invariant property in DEA, it can be argued that data normalizationis not required. Given the fact that our newly-proposed model (2) is equivalent to the originalCCR model without weight restrictions, and that weights restrictions are dependent on theinput and output units, we recommend that normalization be applied to the DEA data anduse model (2).
Acknowledgements The authors are grateful for the comments and suggestions made by two anonymousreviewers. Dr. Juan Du thanks the support by the National Natural Science Foundation of China (Grant No.71471133 & 71432007) and the Youth Project of Humanities and Social Science funded by Tongji University(Grant No. 20141872). This paper is partially supported by the Priority Academic Program Development ofJiangsu Higher Education Institutions (China).
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