Journal of Service Science and Management, 2011, 4, 227-233
doi:10.4236/jssm.2011.42027 Published Online June 2011 (http://www.SciRP.org/journal/jssm)
227
A Slacks-Based Measure of Efficiency of Electric
Arc Furnace Activity with Undesirable Outputs
Hao Zhang, Xiang Su, Shilun Ge
Economics & Management School, Jiangsu University of Science and Technology, Zhenjiang, China.
Received April 26th, 2010; revised May 23rd, 2011; accepted May 27th, 2011.
ABSTRACT
Efficiency reflects a scenario of higher quality product with fewer resource inputs and less pollution discharge. In terms
of a manufacturing firm, it means a win-win strategy in both economic and environmental categories. The paper ex-
tends the Electric Arc Furnace (EAF) activity management by taking pollution discharge as undesirable outputs. After
reviewing relative undesirable outputs DEA literatures and comparing their advantages and shortcomings, the current
paper introduces SBM models to treat undesirable outputs and then measures the efficiency of each EAF activity. Based
on resources input and activity quantity output and undesirable outputs, the DEA model can evaluate the efficiency of
EAF activity. By the input-output combination, the sensitivity analysis is done. At last, the paper demonstrates the ap-
plication of efficiency measurement in EAF activity of an iron & steel enterprise. The result implies objectivity, accu-
racy and practicability of the activity analysis and valuation method based on SBM-Undesirable model.
Keywords: Electric Arc Furnace, Slacks-based Measure, Activity Management, Undesirable Outputs
1. Introduction
Nearly 50% of the steel production these days is by Elec-
tric Arc Furnace (EAF) route, which uses high-current
electric arcs to melt steel scrap and convert it into liquid
steel of a specified chemical composition and tempera-
ture. The major charge material of electric-arc steelmak-
ing is scrap steel, and its availability at low cost and
proper quality is essential. Moreover, the electric power
used in EAF operation, however, is high, at 360 to 600
kilowatt-hours per ton of steel, and the installed power
system is substantial.
Activity-based costing (ABC) determines cost drivers
as activity measures to allocate overhead costs more ac-
curately than traditional cost systems. In contrast, activ-
ity-based management is not only concerned with allo-
cating overhead costs more precisely but tries also to
identify and improve inefficient activities. In this sense
activity-based management is the more comprehensive
concept. In identifying inefficient activities it is often
necessary to have reference activities for comparison.
These activities may correspond to other decision making
units (DMUs) such as other companies or divisions. Also,
one might want to compare different observations over
time corresponding to a single organizational unit. We
investigate the use of data envelopment analysis (DEA)
to benchmark activities. DEA provides each DMU with
an efficiency score that has to be viewed as its relative
efficiency in the set of all DMUs involved in the bench-
marking. We identify the pros and cons of DEA as being
applied to benchmark activities.
In the production theory approach, pollutants (also
called undesirable outputs) and desirable outputs are as-
sumed to be generated in the same production process.
So the paper extends the activity management by taking
pollution discharge as an undesi rabl e output.
The objective of this work is to provide a quantitative
model for EAF activity-based management (ABM) with
undesirable outputs. The model developed is based on
data envelopment analysis, an established operational
research technique for productivity and efficiency deter-
minations. Activity centers are considered as decision-
making units whose efficiencies are determined by solu-
tion of the ABM/DEA model. ABM follows from activ-
ity-based costing whereby ABC information is employed
in improvement and cost reduction plan. A real case
study of an iron and steel factory is used to illustrate the
application of the model.
2. Literature
Data envelopment analysis (DEA), developed by Char-

A Slacks-Based Measure of Efficiency of Electric Arc Furnace Activity with Undesirable Outputs 228
nes, Cooper, & Rhodes (1978), is a well-established non-
parametric approach used to evaluate the relative effi-
ciency of a set of comparable entities called decision
making units (DMUs) with multiple inputs and outputs
[1]. Numerous DEA theoretical and application studies
have been reported [2,3]. Efficiency is relatively meas-
ured meaning that efficiency of DMUs is subject to
analysis according to each other is measured. Following
the analysis, the DMU are grouped into two sets as effi-
cient and inefficient. The envelop line formed by the
efficient DMUs is named as efficient frontier, and it cov-
ers the inefficient DMUs as what an envelopment does.
Thus, this property gives the name of analysis. The effi-
ciency performance of the activity management has been
a critical research stream that draws considerable atten-
tion [4-7]. Most studies about the application of DEA-
based models to activity management performance
measurement assume that the reduction of undesirable
outputs (or inputs) and the increase of desirable outputs
are proportional. This implies that the slacks in inputs
and outputs are not accounted for when activity man-
agement performance is evaluated. Although the result-
ing efficiency measures have some good theoretical
properties, the assumption often leads to a lot of compa-
rable entities having the same efficiency scores of 1 and
hence difficult in making useful comparisons.
Therefore, it is meaningful to incorporate the input
excesses and output shortfalls into DEA-based models in
measuring activity management performance. Moreover,
most studies utilized the DEA approach as a tool for
evaluating accomplishments in the past. Although the
evaluation results highlight the status of the operational
performance and are helpful for planning future activities
for improving the performance, the ex post facto evalua-
tion might be a little late for an unsuccessful unit to find
its weaknesses and make the appropriate amendments.
There is one difficulty in doing an objective evaluation
of the performance of DMUs. The difficulty is how to
treat undesirable outputs jointly produced with desirable
outputs. Traditional literature only values the desirable
and simply ignores the undesirable. However, ignorance
of the undesirable is equal to saying that they have no
value in the final evaluation and may present misleading
results. It is therefore necessary to credit DMUs for their
provision of desirables and penalize them for their provi-
sion of undesirables. In the presence of undesirable out-
puts, however, technologies with more good (desirable)
outputs and less bad (undesirable) outputs relative to less
input resources should be recognized as efficient. In the
DEA literature, several authors have proposed methods
for this purpose [8-10].
In this research, we employ Slack Based Measurement
(SBM) for the involved DEA models [11]. In contrast to
the radial models, CCR and BCC which are based on the
proportional reduction (enlargement) of input (output)
vectors and which do not take account of slacks, the
SBM deals directly with input excess and outp ut shortfall.
SBM is non-radial and deals with input/output slacks
directly. The SBM returns an efficiency measure be-
tween 0 and 1 and gives unity if and only if the DMU
concerned is on th e frontiers of the production possibility
set with no input/ou tput slacks. In that resp ect, SBM dif-
fers from traditional radial measures of efficiency that do
not take account of the existence of slacks [12].
The main purpose of this paper is to propose one
slacks-based efficiency measures for modeling EAF ac-
tivity management performance.
In this research, we evaluate the performance of the
EAF activity management based on the input-output data
via slacks-based DEA model. The results can be used for
planning management activities in advance to enhance
the activities operational efficiency and increase a high
discriminating power for measuring activity management
performance. The rest of this paper is organized as fol-
lows: Section 2, we introduce the method of slacks-b ased
efficiency measures in DEA by taking pollution dis-
charge as an undesirable output, then in Section 3 we
discuss the input and output factors used to measure the
EAF activity efficiency. Sections 4 and 5 we utilize the
case of EAF, and finally, the results are discussed and
some conclusions drawn from the discussion.
3. Proposed SBM DEA Model
The SBM DEA model projects each unit onto the effi-
cient frontier and has many attractive features, among
them, units-invariance. The original SBM DEA model
computes the ratio of the average inputs reduction to the
average output increase. Minimizing that ratio implies
the simultaneous pursuit of improvements in both inputs
and outputs. It is, therefore, a non-oriented model. It is
also non-radial, i.e., it does not force the input and out-
puts to be improved uniformly or equal-proportionally,
letting the maximum possible improvement in each di-
mension be computed by the model. In addition, the
SBM efficiency score leaves no input or output slack
unaccounted, i.e. all possible improvements are ex-
hausted and properly taken into account in the objective
function.
In total, all the above explorations have effectively
broadened our understanding of efficiency evaluation of
DMUs. Based on the above, the model of this paper is
constructed.
Suppose that there are n DMUs(decision making units)
each having three factors: inputs, good outputs and
bad(undesirable)outputs, as represented by three vec-
tors ,
m
x
R 1
g
s
y
R and 2bs
y
R, respectively. We

A Slacks-Based Measure of Efficiency of Electric Arc Furnace Activity with Undesirable Outputs 229
define the matrices
X
、
g
Y and b Yas follows：


1,, mn n
X
xxR

,1
1,,
g
ggsn
n
YyyR




,
2
1,,
bbb sn
n
YyyR




, We assume0,X 0
g
Y
and 0
b
Y. The production possibility set (P) is defined
by：
 
,, |,,,0
gbggbb
PxyyxXyYyY  
Definition 1（Efficient DMU）:
A DMU0

,,
g
b
oo o
x
yy is efficient in the presence of
undesirable outputs if there is no vector

,,
gb
x
yy P

such that ,
g
g
oo
x
xy y and bb
o
y
y with at least
one strict inequality.
In accordance with this definition, we modify the SBM
in Tone (2001) as follows.
[SBM] 0
12
11
*
11
12 00
1
min 1
1
i
i
ms
mx
i
gb
ss
rr
gb
rr
rr
s
s
ss yy











s.t.
0
0
g
gg
x
Xs
yY s





0
0,0,0, 0
bb b
gb
yY s
sss





The objective function strictly decreases with respect
to
 
,g
ir
s
is r
 and


b
r
s
r

and the objective
value satisfies *
01  .Let an optimal solution of the
above program be

** **
,,,
gb
s
ss 
.
Then, we have:
Theorem 1 The DMU0 is efficient in the presence of
undesirable outputs if and only if **
1,. .,0,ie s 


*0 g
s and *0
b
s.
If the DMU0 is inefficient, i.e.,*1
 , it can be im-
proved and become efficient by deleting the excesses in
inputs and bad outputs, and augmenting the shortfalls in
good outputs via th e following SBM-projection:
***
,,
g
ggb bb
ooooo o
x
xsyy syys

 .
Using the transformation by Charnes and Cooper
(1962), we arrive at an equivalent linear program in
,, ,
g
tSS

 and b S as displayed below.


LP *
1
1
min m
i
iio
S
tmx 




Subject to
12
11
12
1
1gb
ss
rr
gb
rr
ro ro
o
gg g
o
SS
tss yy
xt XS
yt YS



 





0
0, 0,
0,0, 0.
bb b
g
b
yt YS
SS
St



 
Let an optimal solution of [LP] be


**** *
,, ,,
gb
tSSS

. Then we have an optimal solution
of [SBM] as defined by
*** **
,t   ,*** ,
s
St


***
,
gg
s
St***bb
s
St.
The existence of


**** *
,, ,,
gb
tSSS

 with *0t
is guaranteed by [LP].
4. Analytic Procedure and Selection of Input
and Output Items
4.1. Analytic Procedure
The analytic procedure for SBM-DEA mainly includes
three parts: first, the assessment objects should be de-
fined and selected. Second, relevant and appropriate in-
put, desirable and undesirable output items are deter-
mined, in order to facilitate the relative efficiency
evaluation for evaluation object. Third, based on the ap-
plication of DEA model, the experimental results will be
evaluated. Therefore this paper will take the above ana-
lytic procedure as the basis for DEA analysis.
4.2. The Selection of Input and Output Items
Without loss of generality, this paper will take electric
arc furnace (EAF) smelting activity efficiency analysis of
iron and steel enterprise as example. Solid substances
used in the electric arc furnace (scrap, ferrous alloys, ir on
slurry) are melted predominantly by electrical energy
which is inputted via the electrodes, as well as by fossil
energies in the presence of oxygen. Electric arc furnaces
have currently capacity of up 200 tones; the duration of
heat is in the region of 1 to 4 hours. Owing to their high
energy input, modern electric furnaces produces consid-
erable quantities of smoke and waste gas.
EAF activity efficiency is influenced by raw materials,
management, technology, operations, equipment status
and other factors.
So according to the applicability, simplicity and com-
parability of selection index, the input and output indexes
are determined.
The input data should more objectively reflect the ac-
tual situation of EAF operations. The most basic inputs
of EAF activity are iron and steel material consumption,
power consumption and energy consumption; and out-
puts include desirable outputs and undesirable outputs.
The desirable outputs include passing rate of molten steel,
and the undesirable outputs include wastewater emis-

A Slacks-Based Measure of Efficiency of Electric Arc Furnace Activity with Undesirable Outputs 230
sions and dust emissions.
Index calculation methods are as follows:
Ferrous charges consumption:

M
iMw
Msi
M
es


In formula: Msi - ferrous charges consumption, kg/t;
Mi - Pig iron consumption, kg; Mw - Scrap steel con-
sumption, kg; Mes - qualified steel yield, t.
Process energy consumption:

Es Ep Eo
Eu Mes


In formula: Eu - process unit standard coal consump-
tion, kgce/t; Es - fuel consumption, kgce; Ep - power
consumption, kgce; Eo - Surplus energy recovery, kgce;
Mes - qualified steel yield, t.
General power consumption:

Ecp
Ec
M
es

Ecp - power consumption, kwh; Mes - qualified steel
yield, t.
Liquid steel qualified rate:
M
Md
Se
M


In formula: Se - liquid steel qualified rate, %; M - raw
material weight, t; Md -metal loss, t.
Contaminations:
Csl
Cl
M
s

In formula: Cl - unit discharge amount of the main
pollutant, kg/t; Csl - discharge amount of the main pol-
lutant, kg; Ms - liquid steel annual yield, t.
In summary, the input and output items of electric arc
furnace activity are shown in Table 1.
5. Case Study
5.1. Data Collection
In this paper, we analyze 15 furnace (capacity of 20 - 40 t)
Table 1. The input and output items of electric arc furnace
activity.
Items Unit
Steel scrap consumption kg/t
Power consumption kWh/t Inputs Process energy consumption kgce/t
Output Passing rate of molten steel %
Wastewater emissions m3/t
Undesirable
Outputs Dust emissions kg/t
in the September, 2009 in an iron & steel enterprise. The
original input and output data was obtained from the pro-
duction system. The main data information is shown in
Table 2.
5.2. Results and Discussion
According to the above model, we use the linear pro-
gramming software (lingo 9.0) to calculate platform, the
result is shown in Table 3.
According to the SBM-DEA model (Table 3): In in-
spected objects, only 4 DMU (DMU5, DMU6, DMU8,
DMU13 account for 26%) are integrated effective, the
synthetic validity (θ) is 0.729, it means the operation
performance of the EAF activity system has already
achieved certain level. Six DMUs (DMU1, DMU5,
DMU6, DMU8, DMU10, DMU13 account for 40%) is
technically effective and the synthetic validity (φ) is
0.776. That is to say, the technical validity is relatively
high. From tab 3, 2 DMUs (DMU1, DMU10 account for
13%) are only technical validity, not integrated effective.
It means these EAFs have already played its best tech-
nical level. But due to lack of organization o f production
and the impact of the order, these DMUS have failed to
increase production scale.
Other 9 DMUs have put too much resource and pro-
duced much pollutants, resulting in relatively non-
effective. Therefore raw materials, energy, and various
cost control must be considered in the future. By using
reasonable charge structure, increasing alloying elements
recovery, and reducing refining time and pollutants
emissions, better efficiency can be gotten to enhance the
EAF activity operational performance.
We applied the SBM-DEA model, with variable re-
turns to scale, to evaluate the technical efficiency of each
EAF activity. Also, the scale efficiency can be derived by
the ratio of overall efficiency to technical efficiency. Ta-
ble 3 summarizes the results. The four overall efficient
EAFs have the technical efficiency and the scale effi-
ciency. In particular, (DMU1, DMU10) has the technical
efficiency scores equal to 1 while their scale efficiency
scores are less than 1. It should adjust their scales of op-
eration to improve their scale efficiencies as well as
overall efficiencies. A DMU may be scale inefficient if it
exceeds the most productive scale size (thus experiencing
decreasing returns to scale), or if it is smaller than the
most productive scale size (thus having not taken the full
advantage of increasing returns to scale). Indeed, most of
the inefficient EAFs present increasing returns to scale
that can increase the scales to effectively improve their
efficiencies. In particular, seven of the scale inefficient
EAFs (i.e., DMU 2, 3, 4 7, 9, 11, 12, 14 and 15) had their
scale efficiency scores higher than the technical effi-
ciency scores, respectively. This implies that the overall

A Slacks-based Measure of Efficiency of Electric Arc Furnace Activity with Undesirable Outputs 231
Table 2. Statistical analysis of input-output data.
Items MaxMinMean SD
Steel scrap consumption 11209201032.40 71.93
Power consumption 321232275.40 31.17
Inputs Process energy consumption 12766101.67 18.61
Outputs Passing rate of molten steel 1009799 0.845
Wastewater emissions 1.30.40.91 0.277
Undesirable Outputs Dust emissions 41.92.8 0.65
Table 3. The efficiency and returns to scale for DMUs.
Number Overall
efficiency Technical efficiency Scale efficiencyReturns to scaleReference set Reference times Rank
1 0.748 1.000 0.748 irs 1
2 0.638 0.645 0.990 irs 5
3 0.559 0.571 0.980 drs 13
4 0.580 0.587 0.989 drs 13
5 1.000 1.000 1.000 - 5 4 2
6 1.000 1.000 1.000 - 6
7 0.645 0.660 0.978 irs 13
8 1.000 1.000 1.000 - 8
9 0.656 0.663 0.989 irs 13
10 0.623 1.000 0.623 irs 10
11 0.679 0.687 0.988 irs 13
12 0.608 0.615 0.989 irs 5
13 1.000 1.000 1.000 - 13 7 1
14 0.642 0.642 1.000 - 5
15 0.569 0.569 1.000 - 13
Mean 0.729 0.776 0.952
inefficiency is primarily due to the technical inefficiency.
Only DMU 3 and 4 present the decreasing returns to
scale that can decrease their scales to possibly improve
their efficiencies. On the other hand, one overall ineffi-
cient EAF (i.e., DMUs 1 and 10) is mainly due to the
scale inefficiency because their scale inefficiency scores
are lower than technical efficiency scores. The technical
inefficient EAF should improve their productivity and
make better use of their resources.
One way for increasing their efficiency is to adjust
their scales by transferring resources from EAFs operat-
ing at decreasing returns to scale to those operating at
increasing returns to scale.
Taking the number 2 furnace as the example, the
meaning of reference set is explained. Number 2 EAF
overall efficiency is 0.638 as the SBM-DEA non-
effective unit. So it must take number 5 furnace as a
benchmark, adjusting the input and output to achieve the
DEA effective with reference to their production units.
Which is referenced more times (such as the DMU 13) is
a more powerful efficiency unit.
Non-DEA efficient units do not achieve the technical
efficiency is largely due to excessive investment in the
number of resources. By slack variable analysis, input
and output adjustment amount of the non-DEA unit is
shown in Table 4.
For the slack variable analysis, there are 6 EAFs at the
efficiency frontier with input and output slack variables
of 0. Among the inefficient EAFs, DMU 14 had the
greatest excess in the input variable ‘Steel scrap con-
sumption’. DMU 4 had the greatest excess in the input
variables ‘Power consumption’ and ‘Dust emissions’.
DMU 3 had the greatest excess in undesirable output
‘Wastewater emissions’.
6. Sensitivity Analysis
6.1. Input/Output Indicators Combinations
Sensitivity analysis is a very important aspect of DEA to
evaluate the robustness of the results. Since DEA is a
data based analysis, any error in the data set can change
the results. Sensitivity analysis has been carried out in a
number of ways in the literatures. In this study, we as-
sume that the data set is correct and precise, as it is taken
from various sources. So, we have carried out the sensi-
tivity analysis based on removal of variables one by one
from the data set, and finding out the efficiency scores to
check the robustness of the DEA results.
Table 5 summarizes the results of sensitivity analysis
carried out to check the robustness of the results (effi-
ciency scores).
6.2. Efficiency Value Analysis of Different
Indicators Combinations
The efficiencies of all EAFs were calculated under each
of the 3 specifications. The full list of specifications can
be seen in the heading of the columns in Table 5. Table

A Slacks-based Measure of Efficiency of Electric Arc Furnace Activity with Undesirable Outputs 232
Table 4. Potential improvement for inefficient DMU.
inputs outputs(+) Undesirable outputs(–)
Number Steel scrap
consumption
Power
consumption
Process energy
consumption
Passing rate of
molten steel
Wastewater
emissions Dust emissions
2 145 35 29 0 0.4 0.9
3 138 59 38 2 0.6 2.1
4 30 83 60 1 0.5 1.0
7 20 62 44 2 0.4 0.4
9 118 40 33 1 0.5 0.4
11 119 29 39 1 0.4 0.2
12 128 74 38 1 0.5 0.6
14 160 7 52 0 0.3 1.2
15 150 60 61 0 0.3 1.9
Table 5. Combination of different input/output indicators.
inputs outputs(+) Undesirable outputs(-)
Group Steel scrap
consumption Power consumption Process energy consumptionPassing rate of molten steel Wastewater emissions Dust emissions
1    
2      
3     
Table 6. Efficiency value of different input/output indicator combinations.
Overall efficiency
Number Group 1 Group 2 Group 3
1 0.848 0.748 0.730
2 0.780 0.638 0.653
3 0.754 0.559 0.592
4 0.737 0.580 0.584
5 1.000 1.000 1.000
6 1.000 1.000 1.000
7 0.775 0.645 0.623
8 1.000 1.000 1.000
9 0.796 0.656 0.635
10 0.831 0.623 0.661
11 0.795 0.679 0.645
12 0.752 0.608 0.598
13 1.000 1.000 1.000
14 0.796 0.642 0.677
15 0.728 0.569 0.599
Mean 0.839 0.729 0.733
0
0.2
0.4
0.6
0.8
1
1.2
1 2 3 4 5 6 7 8 9101112131415
Overall efficiency
Group 1
Overall efficiency
Group 2
Overall efficiency
Group 3
Figure 1. Changes in operating efficiency trend of various indicators combinations.

A Slacks-based Measure of Efficiency of Electric Arc Furnace Activity with Undesirable Outputs 233
6 shows, fo r each of the 15 EAFs and every specification,
the efficiency score achieved.
It can be observed that, given a particular EAF, effi-
ciency depends on the specification estimated. Visual
inspection of Table 6 reveals some interesting features.
Take DMU 14 as an example, the removal of ‘dust
emissions’ would drop the efficiency value from 0.677 to
0.642. This shows that ‘dust emissions’ were the more
sensitive variables than ‘wastewater emissions’ for the
operational performance of this EAF. DMU 11 is 0.679
efficient under the Original combination 2, but only
0.645 efficient under combination 3. It is clear that this
EAF activity has a strong at dust emissions.
7. Conclusions
Evaluation of the operational performance of EAFs is
important for ensuring efficiency in the iron & steel in-
dustry. By adopting SBM-DEA models and considering
the undesirable output, iron & steel industry can accu-
rately assess aspects of their own performance that re-
quire improvement, and can gain an enhanced under-
standing of the current EAF operational status and future
improvement. Here SBM-DEA analysis has provided
several useful insights.
First, efficiency value analysis has established that
while some major EAFs are of optimal scale, others are
not.
Secondly, slack variable analysis is seen as potentially
providing an understanding of how EAFs can improve
their operational performance by enabling managers can
focus on a limited number of variables for short-term and
long-term improvement.
Thirdly, sensitivity analysis shows that the ‘dust emis-
sions’ and the ‘Wastewater emissions’ are two undesir-
able output variables that have higher sensitivity with
respect to EAF efficiency. Moreover DEA can be based
on non-financial evaluations, it is appropriate to compare
DMUs with different cost conditions.
8. Acknowledgements
This work was financially supported by National Natural
Science Foundation of China (No. 70971056), and the
Department of Science and Technology, Jiangsu Provin-
cial People’s Government (BK2009728) of the People’s
Republic of China.
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