DATA ENVELOPMENT ANALYSIS
About Data Envelopment
Analysis (DEA):
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DEA
has become very important tool for quantitative analysis to check and evaluate
the performance of one, two or multiple firms and to understand how the efficiencies
of the firm can be improved with reference to the benchmark firms. This
approach can be applied to extensive variety of activities with respect to the
current requirement. Gradually DEA utilization is increasing in the current
market as it aids in meeting the todays demand.
Data
Envelopment Analysis (DEA) an extensive and useful technique was originally
developed by Rhodes and Charnes_Cooper in 1978 to majorly do the evaluation of public
sector organizations and NGOs. DEA has been utilized to improve the efficiency
to optimize the resources majorly in services which may not be easy to
calculate even based on the experience. Majority of service providers can take
advantage from this power technique and improve efficiency and productivity. More
people start using this it will help the research assistant to identify the
“benefits” and “banes” of this tool and may highlight the limitations, if any.
Research
on DEA and its fetched result will help to identify the areas where this tool
may not be effective as desired. DEA approach is not user-friendly or handy for
managers to understand and implement the technique. This is one of the bottlenecks
which is preventing DEA from entering the business. Aim is to focus on how this tool helps in
evaluating efficiency, to identify the areas to advance productivity, understand
limitations of DEA, and how to make use of this tool. This will assist applicators
to evaluate the importance of using it in services domain.
DEA
can be of help in the situations where a comparative performance of dissimilar
components is to be linked and assessed, like,
Ø To
check for inconsistency or inefficiency in the operations.
Ø DEA
can deal with intricate relation between multiple inputs and multiple outputs.
Ø
DEA techniques are associated
to linear programming concepts.
Efficiency
Measurement
To
calculate the efficiency of the various service units and demonstrate the same
is very critical and at the same time biggest concern on which technique to be
adopted. E.g. How to optimize the staff in big departmental stores, how to
aptly distribute the number of doctors/nurses on daily basis in hospital, how
to fix the no. of branches of banks in particular region etc.
Efficiency = Output / Input
This
formula though looks simple becomes complicated based on the number of outputs
and inputs given in the specified problem. If output is higher than the input
it suggests that efficiency is very high. Once the system/ unit reaches its
optimum efficiency level i.e. output /input ratio cannot be increased further,
it becomes evident that certain new method or technology needs to be adopted to
establish new benchmarks.
Technical and Scale Efficiency
For
example, in effectiveness of Portable charger, we might measure it as charging
rate i.e hours per full charge. We can define
the efficiency of “Charger” with the ideal chargers rate of charge. Let us take
that Full charge of I-phone 5S taken by Charger is 2 hours, however as
committed is 2.5 hours. We can say that charger is operating at 80% efficiency (2/2.5
Hours). To give the optimize results; charger shall perform at 125% (2.5/2
Hours) from its current level. This would reduce the time taken. Further, after
this efficiency is achieved for higher benchmarks, technology of charger needs
to be improved. It is to be ensured that charger of similar kinds are compared
to establish the realistic results.
Price Efficiency
If
A choose to use the cannon copier which generates 1000 B&W printout per
cartridge and cost of cartridge is INR 5000. Thus resulting cost is 5 rupees
per printout. B chooses to opt for HP printer which generates 1000 B& W
printouts per cartridge and the cost of cartridge is INR 4500 which derives the
price to 4.5 rupees per printout. A is less efficient than B but it is not due
to efficiency of copier but because of Price efficiency of HP cartridge.
Relative efficiency
measurement
In
this DMU has to be valued as 100% efficient after referring all the presented
confirmations, only in case if enactments of all presented DMUs cannot show /
increase the efficiency either by changing the inputs or outputs.
The
measurement of relative efficiency is used where there are numerous possible
insufficient inputs and outputs. A common measure for relative efficiency is,
Efficiency = Weighted sum
of Outputs
Weighted sum of inputs
Which
introducing the usual notation can be written as
Efficiency
of Unit = U1Y1j + U2Y2j +….
V1X1j + V2X2j +……
U1
= Weight of Output i
Y1j=
Amount of Output 1 from unit j
V1
= Wright given to input 1
X1j=
Amount of input 1 to unit j
(Efficiency usually lie in
the range 0,1).
E.g.
Input
Output
Student
Hours of Study for
DS
marks out of 100
Marks per Hour of
Study
Relative efficiency
A
6
80
13.333
78.4%
B
7
74
10.571
62.2%
C
5
85
17.000
100.0%
D
4
65
16.250
95.6%
DEA
process
Output/
Efficiency of the system can be evaluated by two approaches.
v Partial
efficiency measures
v Total
factor efficiency measures
Partial
efficiency approach does not account all the output and input factors, whereas,
total factor effectiveness approach is designed to consider all the data
outputs and inputs.
To
contemplate the available data, a technique is must with ability to address and
account the mentioned critical areas.
v To
arrive at single ratio with multiple outputs and multiple inputs.
v How
to prioritize or to understand the criticality of one attribute with respect to
others.
v Addressing
the challenge of varied variables and constraints?
SINGLE
INPUT AND SINGLE OUTPUT
If
we refer to the single output to single input case let us understand one simple
example. Cricket Tournament has 6 Teams which are T1 to T6
The
number of Bowlers and wicket taken are observed for further evaluation of
performance. First line is the number of bowlers who bowled for the team and
second line is the number of wickets taken by the total number of bowlers in
the respective team. The last line of the below mentioned table reflects “Wickets
taken per bowler” measure of efficiency or effectiveness of a Bowler.
Cricket Team
T1
T2
T3
T4
T5
T6
Bowlers
2
3
3
5
6
5
Wickets
1
3
1
3
3
2
Wickets/Bowler
0.5
1
0.33
0.6
0.5
0.4
_____ Frontier line ______
Regression line
By
this we are able to see that T2( 1) is the most efficient team in terms of
bowling and T3(0.33) is the team least
efficient . Highest slope is achieved by the Frontier line and it is
also known as “efficient frontier” It is necessary/must that this line should
touch at least one point and all other points should either be below or above
this line and thus it is known as envelopment .
Regression
line passes from the (0,0) and is usually determined by statistical approach
and it goes from the centre of all the plotted points such that residual value
is always zero.. Frontier line highlights the performance of the team T2. DEA helps
to point out the benchmark for others to move towards improvements.
We
can assess the effectiveness of other teams relative to T2 and can organize
them in order by referring the output:
0<= Wickets/Bowler of others<= 1 Wickets/Bowler of B 1=T2>T4>T5=T1>T6>T3=0.33
Thus
Team T3 has the worst efficiency i.e. 0.33 *100% =33% of T2 efficiency
TWO
INPUTS AND ONE OUTPUT CASE
Let
us look at two inputs and single output case and its handling, table shows the
performance of 9 teams each having two inputs and one output. Input x1 is the
number of bowlers, Input X2 the wickets and Output Y1 number of wins However, number
of wins has been normalized to 1 .
Cricket Team
T1
T2
T3
T4
T5
T6
T7
T8
Bowlers(Input
X1)
4
7
8
4
2
5
6
5
Wickets(Input
X2)
3
3
1
2
4
2
4
2.5
Wins (
Output) Y1
1
1
1
1
1
1
1
1
Bowlers/Win
4
7
8
4
2
5
6
5
wickets/win
3
3
1
2
4
2
4
2.5
T1
T3
T4
T5
____ Efficient Frontier
With
efficiency in mind it is but obvious that the team which utilizes less inputs
to get the same output is more efficient compared to others. Thus the Frontier
line drawn which shows all points(teams)
lying above needs to be more efficient No team mentioned on the frontier line can get better its input values with no decline
of the other. We call this region the production possibility set.
The
effectiveness of teams not coinciding on frontier line can be calculated by the
frontier point. For illustration, T1 is unproductive. To calculate its ineffectiveness
let OT1, the line from Origin (0, 0) to T1, cross the frontier line at B, Then,
the effectiveness of T1 will be formulated by OB/ OT1
This
way the inadequacy of T1 is to be evaluated by a arrangement of T4 and T5
because the point B is on the line linking these two points. T4 and T5 are
called the position set for T1. This set for an unproductive team may differ.
For
example, T1 can be successfully enhanced by progress to B because these are the
coordinates of P, the point on the efficient frontier that we beforehand recognized
with the line segment OT1. However, any point on this segment may be used to
improve the efficiency of Team T1. M is achieved by reducing X1 (Bowlers) and N
is achieved by dipping Input X2 Wickets
N
B
M
ONE
INPUT AND TWO OUTPUTS CASE
Tbl
below mentions the number of Bold per bowler and LBW per bowler for 6 teams. To
get a frontier for this scenario, outputs are divided by the inputs (Bowlers).
Bowlers are normalized to one for calculation effectiveness. The proficient
frontier will have the line connecting T2, T5 and T6.
Team
T1
T2
T3
T4
T5
T6
Bowler
X
1
1
1
1
1
1
Bold
Y1
1
2
3
4
4
6
LBW
Y2
5
7
4
3
6
2
Bold/Bowler
1
2
3
4
4
6
LBW/Bowler
5
7
4
3
6
2
The
production set is the region enclosed by the axis and the frontier line. Team
T1, T3 and T4 are unproductive and their effectiveness can be calculated by
referring to the frontier lines.
Efficiency
of T1 can be improved by changing the outputs and keeping the input normalized.
FIXED
AND VARIABLE WEIGHTS
The
cases used to this point have been very narrow in the number of inputs and
outputs used. Idea is to build up methods that ensure that it is feasible to treat
such applications with no load on the executer to solve this with undue scrutiny
or working out and with no large numbers of suppositions.
think,
for example, the condition is which
records actions planned to serve up as a basis for evaluating the relation
efficiency of 12 teams in terms of two inputs, number of bowlers and number of batsmen, and two outputs acknowledged
as number of wins in Test and One day
Team
T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
T11
T12
Bowler
X1
5
4
6
6
4
5
4
7
4
5
4
6
Batsmen
X2
8
6
7
8
8
7
7
6
7
6
7
5
Test Win
Y1
4
5
7
5
4
2
4
6
5
3
6
4
One day Win
Y2
6
7
3
4
6
3
2
1
3
5
4
2
CCR
is calculated by forming the equation for all the teams as per the criteria and
to calculate the efficiency using the solver.
Eg.
U1 is for output 1 and U2 is output 2. V1 is for input 1 and V2 is for input 2
Efficiency
is output/ Input within the range 0,1. For team T1
0<= 4U1 +6U2/ 5V1 +8 V2<=1 0<= 4U1 +6U2 <=5V1 +8 V2 4U1 +6U2 - 5V1 -8 V2 <=0 ----- (1) Similarly, all 11 equations are created. Input is normalized and fixed to any value . Solving this with solver gives the CCR T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 CCR 0.69 1.00 1.00 0.68 0.86 0.37 0.67 1.00 1.00 0.71 1.00 0.81 DEA, by difference or exclusively, uses variable weights. In meticulous way, the weights are derived using solver directly from the Excel. Moreover, the weights are selected in modes that assign a best set of weights to each team. The term "best" is referred here to denote that the ensuing input-to output ratio for each team is maximized relative to all other teams. The row labelled CCR in above mentioned table shows results calculated from DEA using "CCR model". Furthermore, this "best ratio" result is obtained under the following conditions: (1) All data derived are nonnegative (2) The consequential ratio must lie between the range0,1 (3) These weights for the target entity (=team) are useful to all. 69% for Team T1 means that it is 31% inefficient. That is, in comparison to the team lying on the efficient frontier, it is achievable to recognize a technical inefficiency of 31%—and other possible inefficiencies.
