International Journal of Engineering Trends and Technology (IJETT) – Volume17Number5–Nov2014
Performance Measurement of OC Mines Using VRS Method
Dr.G.Thirupati Reddy
Professor, Dept of Mechanical Engineering,
Sree Visvesvaraya Institute of Technology & Science,
Mahabubnagar, Telengana state, INDIA
Abstract
Data envelopment analysis (DEA) is a linear
programming based technique for measuring the relative
performance of organizational units where the presence
of multiple inputs and outputs makes comparisons
difficult. Productivity improvement and cost control have
become key objectives of SCCL coal mines in recent
years. Selected various coal mines in SCCL and
calculated relative efficiency of mines by using Data
Envelopment Analysis (DEA) which helps to rank them
based on their efficiency score.
The efficiency score has been calculated based on
approaches Variable Return to Scale (VRS) and
comparison is made between ranking of the coal mines.
The comparison is made
For every inefficient coal mine, DEA identifies a set of
corresponding efficient coal mines that can be utilized as
benchmarks for improvement of performance and
productivity.
Methodology
As a result many research works have carried
out on productivity improvement in coal mines.
Operation Research techniques like Linear Programming
(LP), Non Linear Programming (NLP), Data
Envelopment Analysis (DEA), Fuzzy systems, Stochastic
Data Envelopment Analysis (SDEA) and Bench marking
etc are very popular tools in productive improvement
which can aggregate the input and output components in
such situations for obtaining an overall performance
measure to improve productivity.
Productivity in Coal Mines (OMS)
Historically and traditionally productivity
in coal mines all over the world is being measured in
terms of output per man shift (OMS). Thus, productivity
(OMS) in underground mines is = Tonnage of coal
produced / Manpower x Shifts
For opencast mines also productivity is
measured in terms of OMS, but additional efforts put to
excavate the overburden is also taken into account, The
OMS of opencast mines is determined as
The envelopment surface will differ depending on the
scale assumptions that underpin the model. Two scale
assumptions are generally employed: constant returns to
scale (CRS), and variable returns to scale (VRS). The
latter encompasses both increasing and decreasing
returns to scale. CRS reflects the fact that output will
change by the same proportion as inputs are changed
(e.g. a doubling of all inputs will double output); VRS
reflects the fact that production technology may exhibit
increasing, constant and decreasing returns to scale.
VRS (Variable to Returns Scale) or BCC Model
If the constraint is
is adjoined, they
are known as BCC (Banker, Cooper, 1984) models. This
added constraint introduces an additional variable,
into the (dual) multiplier problems. As this extra variable
makes it possible to effect returns-to-scale evaluations
(increasing, constant and decreasing). So the BCC model
is also referred to as the VRS (Variable Returns to scale)
model and distinguished form the CCR model which is
referred to as the CRS (Constant Returns to Scale)
model.
The CRS model is designed with the
assumption of constant returns to scale. This means that
there is no assumption that any positive or negative
economies of scale exist. It is assumed is that a small unit
should be able to operate as efficiently as a large one –
that is, constant returns to scale. In order to address this,
Banker, Charnes, and Cooper developed the BCC model.
It is also referred as VRS model.
In all variations of the DEA models, [17] the
DMU(s) with the best inherent efficiency in converting
inputs X1, X2,…,Xn
into outputs Y1, Y2,…,Ym is
identified, and then all other DMUs are ranked relative to
that most efficient DMU. For DMU 0, the basic CRS
Input Oriented model (so-called CCR after Charnes,
Cooper, and Rhodes) is calculated as follows:
P+1.4Q/M (1+1.4R)
Where P = Production of coal in tones
Q = Overburden removed in cubic meter
R = Stripping ratio in cubic meter of over burden per
tonne of coal
VRS Method
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The VRS model is closely related to the standard CRS
model as is evident in the dual of the BCC model:
Data collection
For our analysis, we have chosen four input variables
namely,
1.Wage Cost ( In Lakhs rupees per year),
2.Store Cost (In Lakhs rupees per year),
3.OBR Cost (In Lakhs rupees per year),
4.Other cost (In Lakhs rupees per year)
and one output variable namely
5.Production (in Lakh Tonnes per year),
The difference compared to the CRS model is the
introduction of the convexity condition
. This
additional constraint gives the frontiers piecewise linear
and concave characteristics.
The general Input oriented maximization using VRS
DEA model is used to obtain efficiency score. TORA
package and Data Envelopment Analysis Programme
(DEAP) has been used to solve the model.
Table 1: Normalized Data for Open-Cast mines
Normalized data of OC mines
Mines(DMU)
Wage Cost
Store Cost
OBR Cost
Other Cost
Production
OCM1
1.4159
1.3481
1.6260
1.5881
1.4980
OCM2
0.4178
0.2750
1.1271
0.6606
1.0283
OCM3
0.8347
0.3747
0.2395
0.2439
0.4547
OCM4
0.2877
0.0429
0.0886
1.4318
0.9398
OCM5
2.2116
2.7843
1.0544
1.9245
1.6182
OCM6
0.1794
0.3421
0.5946
0.3132
0.6900
OCM7
0.0900
0.0640
0.1193
0.0033
0.1348
OCM8
0.8788
0.6435
2.3050
0.6806
1.2584
OCM9
0.4472
0.3099
1.5266
0.3449
0.7523
OCM10
0.3140
0.1812
0.5095
0.1531
0.4167
OCM11
0.2761
0.0975
0.4884
0.2727
0.4347
OCM12
0.8668
0.4730
1.9179
0.5059
1.3427
OCM13
2.5188
3.8545
1.5713
2.2644
2.1494
OCM14
1.7423
1.7183
0.7791
0.7015
0.8720
OCM15
2.5188
2.4909
1.0527
3.9112
1.4102
This paper attempts in Benchmarking of 15 Open Cast
DEA with Variable Returns to Scale (VRS) Model
Mines for improving productivity (Output) using the
Unlike CRS model, variation in inputs may not lead to
Variable Return to Scale (VRS) Model (Input-oriented)
the same level of variation in the output in some
situations. In order to address this issue, an extension of
CRS model, popularly known as VRS model is used and
compared with CRS model (Banker et al., 1984).
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The methodology was explained
and the analysis
carried out by using TORA and DEA software’s .
Analysis of OC Mines using Input-oriented VRS model
The analysis carried out using same 15 OC
Mines used for CRS model analysis. The efficiency
scores, shadow values and peer groups shown in the table
2 after solving Input-oriented VRS Model.
Table 2: Efficiency Scores, Shadow values and peer groups for OC mines after solving Input – oriented VRS model
Benchmark or
Peer Group
DMU
Efficiency
Shadow Values
OCM1
96.50%
0.1407, 0.5965, 0.2628
4,12,13
OCM2
100%
1.0000
2
OCM3
100%
1.0000
3
OCM4
100%
1.0000
4
OCM5
94.20%
0.3302, 0.0903, 0.5795
4,6,13
OCM6
100%
1.0000
6
OCM7
100%
1.0000
7
OCM8
84.90%
0.2143, 0.042, 0.7437
2,4,12
OCM9
90.20%
0.1237, 0.1983, 0.3494, 0.3286
2,6,7,12
OCM10
93.30%
0.0378, 0.1459, 0.6779, 0.1383
2,6,7,12
OCM11
100%
1.0000
11
OCM12
100%
1.0000
12
OCM13
100%
1.0000
13
OCM14
86.50%
0.0636, 0.0825, 0.1138
4,6,13
OCM15
63.20%
0.6111, 0.3889
4,13
From the table 2 eight mines OCM2, OCM3, OCM4,
than CRS model. The efficiency comparisons of OC
OCM6, OCM7, OCM11, OCM12 and OCM13 out of 15
Mines are shown in fig:1.
have emerged benchmarking units for other coal mines.
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International Journal of Engineering Trends and Technology (IJETT) – Volume17Number5–Nov2014
120.00%
Efficiency
100.00%
80.00%
60.00%
40.00%
Efficiency
20.00%
OCM1
OCM2
OCM3
OCM4
OCM5
OCM6
OCM7
OCM8
OCM9
OCM10
OCM11
OCM12
OCM13
OCM14
OCM15
0.00%
OC Mines
Fig.1: OC Mines Vs Efficiency Score for Input – oriented VRS model
Table 3: Slacks assigned to input parameters of OC Mines after solving Input – oriented VRS model
DMU
Wage Cost
Store Cost
OBR Cost
Other Cost
Production
OCM1
0.147
0
0
0.434
0
OCM2
0
0
0
0
0
OCM3
0
0
0
0
0
OCM4
0
0
0
0
0
OCM5
0.513
0.345
0
0
0
OCM6
0
0
0
0
0
OCM7
0
0
0
0
0
OCM8
0
0.134
0.286
0
0
OCM9
0
0
0.448
0
0
OCM10
0.07
0
0
0
0
OCM11
0
0
0
0
0
OCM12
0
0
0
0
0
OCM13
0
0
0
0
0
OCM14
1.054
0.763
0
0
0
OCM15
0.436
0.049
0
0.716
0
The table 3 shows the assignment of Slack
variables to the Inputs of the in-efficient coal mines. The
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slack variable values of all inputs are zero in case of
efficient DMUs i.e. shown efficiency is 100%.
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International Journal of Engineering Trends and Technology (IJETT) – Volume17Number5–Nov2014
Table 4 : Ranking and Peer count of OC Mines after solving Input – oriented VRS model
Benchmark or
Peer Group
DMU
Efficiency
Ranking
Peer Count
OCM1
96.50%
4,12,13
6
0
OCM2
100%
2
3
4
OCM3
100%
3
5
1
OCM4
100%
4
1
6
OCM5
94.20%
4,6,13
7
0
OCM6
100%
6
2
5
OCM7
100%
7
4
3
OCM8
84.90%
2,4,12
11
0
OCM9
90.20%
2,6,7,12
9
0
OCM10
93.30%
2,6,7,12
8
0
OCM11
100%
11
5
1
OCM12
100%
12
2
5
OCM13
100%
13
2
5
OCM14
86.50%
4,6,13
10
0
OCM15
63.20%
4,13
12
0
For example, OCM5 having efficiency score of 94.20%
whereas OCM3 and OCM11 for one time respectively.
can refer OCM4, OCM6 and OCM13 can assign a
OCM4 ranked as 1 due to maximum number of peer
weightage of 0.3302 to OCM4, 0.0903 to OCM6 and
count (6) among the efficient mines and OCM6, OCM12
0.5795 to OCM13 to become a benchmark unit. One
and OCM13 are ranked as 2. OCM4, OCM6, OCM12
DMU (e.g. OCM4) have become the peer unit six times
and OCM13 are called as dominant peers which are the
while OCM6, OCM12 and OCM13 become the referring
Benchmarking for the other mines for improving
institute for five times, respectively. OCM2 and OCM7
production.
becomes the referring institute for four and three times
OCM5 has got efficiency score less than 1(0.9420).
Virtual Efficient (Target) Inputs of OC Mines
OCM4, OCM6 and OCM13 are in the peer group of
From the results, it is clear that 8 mines (OCM2, OCM3,
OCM5 and their corresponding shadow values are
OCM4, OCM6, OCM7, OCM11, OCM12 and OCM13)
0.3302, 0.0903 and 0.5795. Its virtual producer is a linear
have an efficiency score of 100%. The usage of
combination of inputs of
combinations of efficient DMUs is called virtual
OCM6 and OCM13 (peer group which have a relative
producers corresponding to the inefficient ones. The
efficiency 1 with respect to OCM5).
“shadow values” and “peer groups” are helpful in
The virtual efficient wage cost for OCM5 is:
constructing the virtual producers. For example mine
0.2877*0.3302+ 0.1794* 0.0903+ 2.5188*0.5795.
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efficient mines of OCM4,
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International Journal of Engineering Trends and Technology (IJETT) – Volume17Number5–Nov2014
Table 5: Improvements in Inputs and Output of OC Mines after solving Input – oriented VRS model
DMU
OCM1
OCM2
OCM3
OCM4
OCM5
OCM6
OCM7
OCM8
OCM9
OCM10
OCM11
OCM12
OCM13
OCM14
OCM15
Wage Cost
Actual to
Target
1.4159 to
1.2194
0.4178 to
0.4178
0.8347 to
0.8347
0.2877 to
0.2877
2.2116 to
1.5708
0.1794 to
0.1794
Store Cost
Actual to
Target
1.3481 to
1.3011
OBR Cost
Other Cost
Production
Actual to Target
Actual to Target
Actual to Target
1.626 to 1.5694
1.5881 to 1.0984
1.498 to 1.498
0.275 to 0.275
0.3747 to
0.3747
0.0429 to
0.0429
2.7843 to
2.2787
0.3421 to
0.3421
1.1271 to 1.1271
0.6606 to 0.6606
1.0283 to 1.0283
0.2395 to 0.2395
0.2439 to 0.2439
0.4547 to 0.4547
0.0886 to 0.0886
1.4318 to 1.4318
0.9398 to 0.9398
1.0544 to 0.9935
1.9245 to 1.8133
1.6182 to 1.6182
0.5946 to 0.5946
0.3132 to 0.3132
0.69 to 0.69
0.09 to 0.09
0.8788 to
0.7463
0.4472 to
0.4035
0.314 to
0.2229
0.2761 to
0.2761
0.8668 to
0.8668
2.5188 to
2.5188
1.7423 to
0.4526
2.5188 to
1.1554
0.064 to 0.064
0.6435 to
0.4125
0.3099 to
0.2796
0.1812 to
0.1691
0.0975 to
0.0975
0.1193 to 0.1193
0.0033 to 0.0033
0.1348 to 0.1348
2.305 to 1.6716
0.6806 to 0.5779
1.2584 to 1.2584
1.5266 to 0.9292
0.3449 to 0.3112
0.7523 to 0.7523
0.5095 to 0.4756
0.1531 to 0.1429
0.4167 to 0.4167
0.4884 to 0.4884
0.2727 to 0.2727
0.4347 to 0.4347
0.473 to 0.473
3.8545 to
3.8545
1.7183 to
0.7228
2.4909 to
1.5252
1.9179 to 1.9179
0.5059 to 0.5059
1.3427 to 1.3427
1.5713 to 1.5713
2.2644 to 2.2644
2.1494 to 2.1494
0.7791 to 0.6736
0.7015 to 0.6065
0.872 to 0.872
1.0527 to 0.6652
3.9112 to 1.7556
1.4102 to 1.4102
The comparisons are made between Actual Inputs and Target Inputs are shown in figs 2,3 & 4.
3
Wage Cost
2.5
2
1.5
1
Actual Wage Cost
0.5
Target Wage Cost
OCM1
OCM2
OCM3
OCM4
OCM5
OCM6
OCM7
OCM8
OCM9
OCM10
OCM11
OCM12
OCM13
OCM14
OCM15
0
OC Mines
Fig 2: Actual Wage Cost Vs Target Wage Cost for Input – oriented VRS model
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4
3
2
1
Actual Store Cost
0
Target Store Cost
OCM1
OCM2
OCM3
OCM4
OCM5
OCM6
OCM7
OCM8
OCM9
OCM10
OCM11
OCM12
OCM13
OCM14
OCM15
Store Cost
5
OC Mines
Fig 3: Actual Store Cost Vs Target Store Cost for Input – oriented VRS model
2.5
1.5
1
0.5
Actual OBR Cost
0
Target OBR Cost
OCM1
OCM2
OCM3
OCM4
OCM5
OCM6
OCM7
OCM8
OCM9
OCM10
OCM11
OCM12
OCM13
OCM14
OCM15
OBR Cost
2
OC Mines
Other Cost
Fig 5.5 Actual OBR Cost Vs Target OBR Cost for Input – oriented VRS model
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Actual Other Cost
Target Other Cost
OC Mines
Fig 4: Actual Other Cost Vs Target Other Cost for Input – oriented VRS model
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International Journal of Engineering Trends and Technology (IJETT) – Volume17Number5–Nov2014
Conclusions
However, there is a scope for improvement of Open cast
The efficiency score of these DMU’s approaches unity
mines because mean efficiency score for all DMUs
while that of DEA-inefficient DMU’s is less than unity.
shows 0.9392 (93.92%) and it is greater than Input-
The corresponding shadow values assigned to the each
oriented CRS model mean efficiency score of 0.8178
inefficient mines to improve their performance with
(81.78%).
reference to the concerned allotted efficient peer group.
OCM9 and OCM10 allotted 4 peer groups and OCM1,
OCM5, OCM8, OCM14 allotted 3 peer groups and
References
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mines increased in VRS model.
There is no adjustment of slack required for those
efficient mines but adjustment of slack required in case
of in-efficient mines. For example the efficiency of
OCM1 is 96.50% and the slacks assigned are 0.147 to
wage cost and 0.434 to other cost respectively for
improving the efficiency
DEA efficiency ranking finds that 8 DMUs out of 15
DMUs have emerged as benchmarking units for the other
7 DMUs. The benchmarking units are listed as OCM2,
OCM3, OCM4, OCM6, OCM7, OCM11, OCM12 and
OCM13 as shown in Table 4.8. The efficiency score for
these DMUs approaches unity while that of DEAinefficient DMUs is less than unity. The inefficient units
can refer the peer groups given with the corresponding
weightage shown in table1 for improvement in
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