The document describes data envelopment analysis (DEA), a quantitative technique for measuring the efficiency of decision making units (DMUs) that may have multiple inputs and outputs. DEA formulates a linear programming problem to determine efficiency scores for each DMU relative to other DMUs based on their input and output data. DMUs with a score of 1 are considered efficient, while those below 1 are inefficient. The example provided shows how DEA can be used to evaluate the efficiency of different production units.

1. DEPARTMENT OF COMMERCE
DEA
NORTH BENGAL UNICVERSITY
26-27 FEBRUARY, 2010
DATA ENVELOPMENT ANALYSIS
A QUANTITATIVE TECHNIQUE
TO
MEASURE EFFICIENCY
Click Mouse for Next

2. Single Input and Single Output
Units Inputs Outputs Output/ Input
A 2 1 0.5
B 3 3 1
C 3 2 0.666667
D 4 3 0.75
E 5 4 0.8
F 5 2 0.4
G 6 3 0.5
H 8 5 0.625
66
6
5 55 Efficient
4 Frontie r
Output
44
Regres sion Line
Output
3
33
2
1 22
0 11
Input
Input
0 2 4 6 8 10
00
0
0 11 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9

3. Two Input and Single Output
Units A B C D E F G H I
Input-1(x1) 4 7 8 4 2 5 6 5.5 6
Input-2(x2) 3 3 1 2 4 2 4 2.5 2.5
Output (y) 1 1 1 1 1 1 1 1 1

4. 4.5
4.5
4 G
4
E G
3.5 E
3.5
A
x2/y
A
x2 /y 3
3 B
B
H
22.5
.5 H I I
2
2 F
F
D D
Efficient Frontier Line
1. 5
1.5
1
1
C
0.5 C
0.5
0
00 1 2 3 4 5 6 7 8 9
0 2 4
x1/y
x1/y
6 8 10

5. 4.5
4 G
E
3.5
A
x2/y 3 B
2.5 P H I
2
F
1 .5 D
1
C
0.5
0
0 1 2 3 4 5 6 7 8 9
x 1/ y
Efficiency of A = OP
OA
= 0.8571

6. Production Possibility Set

7. 4.5
4 G
E
3.5
A
x2/y 3 A1 B
2.5 P H I
2
F
1 .5 D
1
C
0.5
0
0 1 2 3 4 5 6 7 8 9
x 1/ y
Improvement of efficiency

8. ONE INPUT AND TWO OUTPUTS
Units A B C D E F G
Input-1(x1) 1 1 1 1 1 1 1
Output y1 1 2 3 4 4 5 6
Output y2 5 7 4 3 6 5 2

9. 8 y2/x
7
6
5
4
3
2
1 y1/x
0
0 2 4 6 8

10. 8 y2/x
7 Efficient Frontier
6
5
4
3
2
1 y1/x
0
0 2 4 6 8

11. OD
B Efficiency − of − D = OP
= o.750
Q
OA
Efficiency − of − A = = 0.714
E OQ
F
A
P
C
D
G
Production Possibility Set

12. MULTIPLE INPUTS AND MULTIPLE OUTPUTS
Inputs Outputs
Units Vector Vectors
1 x1 y1
2 x2 y2
− − −
− − −
n xm ys
•Units in DEA are known as DMUs- Decision Making Units
•Generally a DMU is regarded as the entity responsible
for converting inputs into outputs.
•Data are assumed to be positive
•Measurement is unit invariant

13. ( DMUs j = 1 2 ...... n )
 x11 x12 ... x1n 
 
 x21 x22 ... x21 
X =
. . ... . 

x 
 m1 xm1 ... xmn 

 y11 y12 ... y1n 
 
 y21 y22 ... y21 
Y =
. . ... . 

y 
 s1 ys2 ... ysn 


14. For each DMU we have to compute a ratio
Output Virtual Output
=
Input Virtual Input
s
∑r yrk
u
θ= 1
m
; k = ,2,...., n
1
∑i xik
v
1
ur r = 1,2,..., s Output weights
vi i = 1,2,..., m Input weights

15. For k-th DMU DMU k
u1 y1k + u2 y2 k + .... + us ysk
θ=
v1 x1k + v2 x2 k + ... + vm xmk

16. Fractional Programming Problem
Maximize
u1 y1k +u2 y2 k +.... +u s ysk
θ=
v1 x1k + v2 x2 k +... + vm xmk
Subject to
u1 y11 + 2 y21 + + s y s1
u .... u
≤1
v1 x11 + 2 x21 + + m xm1
v ... v
u1 y12 + 2 y22 + + s y s 2
u .... u
≤1
v1 x12 + 2 x22 + + m xm 2
v ... v
... ... ... ... ... ...
u1 y1n + 2 y2 n + + s y sn
u .... u
≤1
v1 x1n + 2 x2 n + + m xmn
v ... v
u1 , u2 , ..., us ≥ 0 v1 , v2 , ..., vm ≥ 0

17. Linear Programming Problem Of kth DMU
Maximize
θ = µ1 y1k + µ 2 y2 k + .... + µ s ysk
Subject to
υ1 x1k + υ 2 x2 k + ... + υ m xmk = 1
µ1 y11 + µ 2 y21 + .... + µ s ys1 ≤ υ1 x11 + υ2 x21 + ... + υm xm1
µ1 y12 + µ2 y22 + .... + µ s ys 2 ≤ υ1 x12 + υ2 x22 + ... + υm xm 2
... ... ... ... ... ...
µ1 y1n + µ 2 y2 n + .... + µ s ysn ≤ υ1 x1n + υ2 x2 n + ... + υm xmn
µ,
1 µ,
2 ..., µs ≥ 0
υ,
1 υ,
2 ..., υm ≥ 0

18. • This is what is known as CCR model of DEA
• Proposed by Charnes, Cooper and Rhodes in 1978
CCR Efficiency :
DMUk is CCR-efficient if θ = 1
*
1.
and there exists at least one optimal (v*,u*)
with v* >0 and u* >0
2. Otherwise DMU is CCR-inefficient

19. Numerical Example
DMU A B C D E F
Input x1 4 7 8 4 2 10
x2 3 3 1 2 4 1
Output y 1 1 1 1 1 1
LPP of DMU A
C
B
Max θθθ=u u
Max = =u
Max
subject to
subject to
84v1++v3v=2==11
7v1 32 2 1
v
u u≤≤441v+++33222
u≤ 4vv11 3vvv u ≤ 7v1 + 3v2
u ≤ 7v1 + 3v2
u ≤ 7v + 3v u ≤ 8v1 + v 2
u ≤ 8v1 + v 2
1 2
u ≤ 10v1++2v22
u ≤ 4v1 + 2v2
4v1 v u ≤≤22v1++44v2
u v1 v2 u ≤≤4v1v+ + v2
u 10 2 v
1 2

20. LPP SOLUTION OF ALL DMUS
DMU CCR(θ*) Reference v1 v2 u
Set
A 0.8571 D,E 0.1429 0.1429 0.8571
B 0.6316 C,D 0.0526 0.2105 0.6316
C 1 C 0.0833 0.3333 1
D 1 D 0.1667 0.1667 1
E 1 E 0.2143 0.1429 1
F 1 C 0 1 1

21. • Optimal weights for an efficient DMU need not be unique
• Optimal weights for inefficient DMUs are unique except
when the line of the DMU is parallel to one of the
boundaries.
• If an activity (x,y)εP (Production Possibility set), then the
activity (tx.ty) belongs to P for any positive scalar t.

22. CCR model and importance of Dual
PRIMAL DUAL
max uyk min θ
subject to subject to
vxk = 1 θxk − Xλ ≥ 0
− vX + uY ≤ 0 Yλ ≥ yk
v≥0 u≥0 λ ≥ 0 θ unrestricted

23. Input excesses and output shortfalls
min θ
subject to
θ k −Xλ≥0
x
Yλ≥ yk
λ≥0 θ unrestrict
min θ
subject to
θ k −Xλ−s =0
x −
Yλ−s + =yk
λ≥0 θ unrestrict

24. LPP of shortfalls and excesses
− +
max w = es +es
subject to
−
s =θ k − Xλ
x k
+
s =Yλ − yk
− +
λ ≥0 s ≥0 s ≥0

25. COMPUTATIONAL PROCEDURE OF
CCR MODEL
Phase-I: We solve the dual first to obtain
Ө*. This Ө* is called “Farrell Efficiency” and
optimal objective value of LP
Phase-II: Considering this Ө* as given, we
solve the LPP of output shortfalls and
Input excesses. (max slack solution)

26. Refine Definition of CCR-Efficiency
If an optimal solution (θ*,λ*,s-*,s+*) of two
LPs satisfies θ*=1 and s-*=0 and s+*=0 then
DMUk is CCR-Efficient.
DMU
DMU CCR(θ*)
CCR(θ*) Reference
Reference v1 v
Excess2 u
Shortfall
Set
Set S1 -
S2 - S+
A
A 0.8571
0.8571 D,E
D,E 0.1429
0 0.1429
0 0.8571
0
B
B 0.6316
0.6316 C,D
C,D 0.0526
0 0.2105
0 0.6316
0
C
C 1
1 C
C 0.0833
0 0.3333
0 1
0
D
D 1
1 D
D 0.1667
0 0.1667
0 1
0
E
E 1
1 E
E 0.2143
2 0.1429
0 1
0
F
F 1
1 C
C 0 1
.6667 1
0

27. EXTENSION OF TWO-PHASE
Two-Phase process aims to obtain the maximum sum of slacks
(input excesses+output shortfalls). For the projection of an
inefficient DMU on efficient frontier,Itxmay result a mix which is far
DMU x1 x1 1 y
from the observed mix A 1 2 1 1
B 1 1 2 1
C 2 10 5 1
The Phase-III process is recently proposed (Tone K., “An
Extension of Two Phase Process”,ORS 1
D 2 5 10 Con, 1999 ) and
implemented in few software 2 10 groups similar DMUs in a
E which 10 1
subset within peer set.

28. INPUT ORIENTED AND OUTPUT ORIENTED MODELS
We discuss so far input-oriented models whose objective
is to minimize inputs while producing atleast given level of
Outputs.
There is another type of model that attempts to maximize
outputs while using no more than the given (observed)
inputs. This is known as output-oriented model.

29. Comparing Input-oriented CCR Model
with Output-oriented CCR model
If µ, η are variable vector of output oriented CCR problem
then at the optimum the following are the equivalences with
Optimal solution of Input-oriented CCR problem λ* and θ*:
* 1
η= * =
λ
µ θ *
*
θ *
Slacks of the output-oriented model are related to the slacks
of input-oriented models
−
t −*
=s *
t +*
=s+*
θ *
θ*

30. It suggests that an input-oriented CCR model will be efficient
for any DMU iff it is also efficient when the output –oriented
CCR model is used to evaluate its performance.

31. BCC MODEL
CCR model has been developed on the assumption of constant
return to scale. A variation of DEA model is BCC (Banker,
Charnes and Cooper) which is based on
variable return to scale.
• Increasing return to scale
• Decreasing return to scale
• Constant return to scale
The BCC model has its production frontiers spanned by the
Convex Hull of the existing DMUs

32. Production Frontier of BCC Model
6
6 Production Frontier
of BCC Model
4
4
Output
2
2
0 Input
0
0 5 10
0 2 4 I n put
6 8 10

33. Acknowledgement:
Cooper, Seiford & Tone: Data Envelopment Analysis – A Comprehensive
Text with Models, Applications, References
Kluwer Academic Publishers, London (Fifth Ed, 2004)
Thanks