two-stage data envelopment analysis
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Two-stage Data Envelopment Analysis
Foundation and recent developments
Dimitris K. Despotis, University of Piraeus, Greece
ICOCBA 2012, Kolkata, India
2
A Data Envelopment Analysis (DEA) primer
Opening the black-box
Two-stage processes: The two fundamental approaches
A novel additive efficiency-decomposition approach
Conclusions
3
Data Envelopment Analysis (DEA)(based on the seminal work of Farrell, 1957)
William W. Cooper 1914-2012
Abraham Charnes1917-1992
Edwardo Rhodes
Charnes, Cooper and Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research 2 (1978), pp. 429-444.
Banker, Charnes and Cooper, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science, 30 (1984), pp. 1078-1092.
Rajiv Banker
4
What is DEA
• DEA is a linear programming technique for evaluating the relative
efficiency of a set of peer entities, called Decision Making Units
(DMUs), which use multiple inputs to produce multiple outputs.
• DEA identifies an efficient mix of DMUs that achieve specified
levels of the outputs with the minimal deployment of resources
(inputs). The resources deployed by the efficient mix are then
compared with the actual resources deployed by a DMU to
produce its observed outputs. This comparison highlights whether
the DMU under evaluation is efficient or not.
5
Decision making units
•homogeneous•Independent•“black box”
▫internal structure unknown▫transformation mechanism (production
function) unknown
DMU
Inputs
Outputs
6
Efficiency
•The efficiency of a DMU is defined as the ratio of a weighted sum of the outputs yielded by the DMU over a weighted sum of its inputs
1
1
s
r rjr
j m
i iji
yE
x
s outputs: y1, y2, …,ys
m inputs: x1, x2, …, xm
Virtual output
Virtual input
7
Returns to scale
• Constant returns-to-scale (CRS–CCR model)• Variable returns-to scale (VRS – BCC model)
Input
Output
A
CB
CRS VRS
O
Production possibility set
8
Orientation • Input oriented model
▫ The objective is to minimize inputs while producing at least the given output levels
• Output oriented model▫ The objective is to maximize outputs while using no more than
the observed amount of any input
Input
Output
A
CB
O
Input oriented projection
Output oriented projection
DP Q R
Efficiency of unit D:
CRS: PQ/PD VRS: PR/PD
VRS ≥ CRS
9
The fractional form (CRS-input oriented)
0
0
0
1
1
1
1
max
. .
1, 1,...,
0, 0
s
r rjr
j m
i iji
s
r rjrm
i iji
i r
yE
x
s t
yj n
x
n DMUs
s outputs
m inputs
j0 the evaluated unit
10
Input oriented model - CRS
0
0
1
1
min
. .
, 1,...,
, 1,...,
0, 1,...,
n
j ij ijj
n
j rj rjj
j
s t
x x i m
y y r s
j n
0 0
0
1
1
1 1
max
. .
1
0, 1,...,
0, 0
s
j r rjr
m
i iji
s m
r rj i ijr i
i r
E u y
s t
v x
u y v x j n
v u
The multiplier form The envelopment form
At optimality: 0<θ≤1
Dual
11
Input oriented model - VRS
0
0
1
1
1
min
. .
, 1,...,
, 1,...,
1
0, 1,...,
n
j ij ijj
n
j rj rjj
n
jj
j
s t
x x i m
y y r s
j n
0 0
0
01
1
01 1
max
. .
1
0, 1,...,
0, 0
s
j r rjr
m
i iji
s m
r rj i ijr i
i r
E u y u
s t
v x
u y u v x j n
v u
The multiplier form The envelopment form
Dual
12
Projections on the frontier
0 0
* *
: * 0 : * 0
ˆ ˆj j
j j j j j jj j
x x y y
13
A Data Envelopment Analysis (DEA) primer
Opening the black-box
Two-stage processes: The two fundamental approaches
A novel additive efficiency-decomposition approach
Conclusions
14
Opening the black boxDMU
X Y
- DM subunits (DMSU)- (Sub)processes- Components
In some contexts, the knowledge of the internal structure of the DMUs can give further insights for the DMU performance evaluation
L. Castelli, R. Pesenti, W. Ukovich , A classification of DEA models when the internal structure of the Decision Making Units is considered, Ann Oper Res (2010)
15
A Data Envelopment Analysis (DEA) primer
Opening the black-box
Two-stage processes: The two fundamental approaches
A novel additive efficiency-decomposition approach
Conclusions
16
The fundamental two-stage production process
xj Stage 1 Stage 2zjyj
DMU j
The external inputs entering the first stage of the process are transformed to a number of intermediate measures that are then used as inputs to the second stage to produce the final outputs.
DMUs are homogeneous.
17
Profitability and marketability of the top 55 U.S. Commercial Banks (Seiford and Zhu, 1999)
Profits Profitability Marketability
Revenues
Employees
Assets
Equity
Market value
Total returns to investors
Earnings per share
Stage 1 Stage 2
18
The multiplicative approach 1/5Kao and Hwang (2008)
Stage 1X Z Stage 2 Y
Stage -1 efficiency
Stage-2 efficiency
Overall DMU efficiency = stage 1 . stage 2
A series relationship is assumed between the stages.
The value of the intermediate measures Z is assumed the same, no matter they are considered as outputs of the first stage or inputs to the second stage.
19
The multiplicative approach 2/5Kao and Hwang (2008)
0 0
0 0
0 0
0
0 0 0
0
11 2 1
1 1
1 2 1
1
,
q s
p pj r rjp r
j jm q
i ij p pji p
s
r rjo rj j j m
i iji
z ye e
x z
ye e e
x
20
The multiplicative approach 3/5Kao and Hwang (2008)
0 0 0
0
0 00
1 1 1
1 11
1 1
1 1
max
. .
0, 1,...,
0, 1,...,
0, 0, 0
q s s
p pj r rj r rjpo r r
j m q m
i ij i ijp pji ip
qs
r rj p pjr p
q m
p pj i ijp i
i p r
z y ye
x xz
s t
y z j n
z x j n
21
The multiplicative approach 4/5Kao and Hwang (2008)
0 0
0
1
1
1 1
1 1
max
. .
1
0, 1,...,
0, 1,...,
0, 0, 0
soj r rj
r
m
i iji
qs
r rj p pjr p
q m
p pj i ijp i
i p r
e u y
s t
v x
u y w z j n
w z v x j n
v w u
0
0
0
0
0
0
*
11
*
1
*
2 1
*
1
*
1 2 1
*
1
o
o
o o o
q
p pjp
j m
i iji
s
r rjr
j q
p pjp
s
r rjo rj j j m
i iji
w z
e
v x
u y
e
w z
u y
e e e
v x
22
The multiplicative approach 5/5Kao and Hwang (2008)
•The multiplicative model is not extendable to VRS situations
•Chen, Cook and Zhu (2010) provide a modeling framework to derive the efficient frontier
23
The additive approach 1/4Chen, Cook, Li and Zhu (2009)
Stage 1X Z Stage 2 Y
Stage -1 efficiency
Stage-2 efficiency
Overall DMU efficiency = t1 . stage 1 + t2 . stage 2 (t1+t2=1)
A series relationship is assumed between the stages.
The value of the intermediate measures Z is assumed the same, no matter they are considered as outputs of the first stage or inputs to the second stage.
24
The additive approach 2/4Chen, Cook, Li and Zhu (2009)
0 0
0 0
1 11 2
1 1
1 1
1 1
max
. .
0, 1,...,
0, 1,...,
0, 0, 0
q s
p pj r rjp rm q
i ij p pji p
qs
r rj p pjr p
q m
p pj i ijp i
i p r
z yt t
x z
s t
y z j n
z x j n
0
0 0
0
0 0
11
1 1
12
1 1
m
i iji
qm
i ij p pji p
q
p pjp
qm
i ij p pji p
xt
x z
z
t
x z
25
The additive approach 3/4Chen, Cook, Li and Zhu (2009)
0 0
0 0
1 1
1 1
1 1
1 1
max
. .
1
0, 1,...,
0, 1,...,
0, 0, 0
qs
r rj p pjr p
qm
i ij p pji p
qs
r rj p pjr p
q m
p pj i ijp i
i p r
u y w z
s t
v x w z
u y w z j n
w z v x j n
v w u
0 0 0
0 * 1 * 21 2j j jt t
26
The additive approach 4/4Chen, Cook, Li and Zhu (2009)
•The additive decomposition approach is extendable to VRS situations
•Does not comply with the rule that VRS efficiency scores >= CRS scores
•Does not provide sufficient information to derive the efficient frontier
27
A Data Envelopment Analysis (DEA) primer
Opening the black-box
Two-stage processes: The three fundamental approaches
A novel additive efficiency-decomposition approach
Conclusions
28
An alternative additive model
Stage 1X Z Stage 2 Y
Stage -1 efficiency
Stage-2 efficiency
Overall DMU efficiency = ½ stage 1 + ½ stage 2
A series relationship is assumed between the stages.
The value of the intermediate measures Z is assumed the same, no matter they are considered as outputs of the first stage or inputs to the second stage.
29
An alternative additive model
Stage 1X Z Stage 2 Y
0
0
0
11
1
1 1
1min
. .
1
0, 1,...,
0, 0
m
i ijij
q
p pjp
q m
p pj i ijp i
i p
v xE
s t
w z
w z v x j n
v w
0 0
0
2
1
1
1 1
max
. .
1
0, 1,...,
0, 0
s
j r rjr
q
p pjp
qs
r rj p pjr p
p r
E u y
s t
w z
u y w z j n
w u
Output oriented Input oriented
30
An alternative additive model
Stage 1X Z Stage 2 Y
Output oriented Input oriented
0
0
1
1
1 1
1 1
min
. .
1
0, 1,...,
0, 1,...,
0, 0, 0
m
i iji
q
p pjp
q m
p pj i ijp i
qs
r rj p pjr p
i p r
v x
s t
w z
w z v x j n
u y w z j n
v w u
0
0
1
1
1 1
1 1
max
. .
1
0, 1,...,
0, 1,...,
0, 0, 0
s
r rjr
q
p pjp
q m
p pj i ijp i
qs
r rj p pjr p
i p r
u y
s t
w z
w z v x j n
u y w z j n
v w u
Common constraints, bi-objective LP
31
An alternative additive model
0 0
0
1 1
1
1 1
1 1
max
. .
1
0, 1,...,
0, 1,...,
0, 0, 0
s m
r rj i ijr i
q
p pjp
q m
p pj i ijp i
qs
r rj p pjr p
i p r
F u y v x
s t
w z
w z v x j n
u y w z j n
v w u
0
0
1
*
1
1j m
i iji
e
v x
0 0
2 *
1
s
j r rjr
e u y
0 0 0
1 2( ) / 2oj j je e e
Simple average …
Stage-1Stage-2
32
An alternative additive model
0 0 0
1 21 2
oj j je a e a e
1 1 1 11 2
1 1 1 1 1 1 1 1
,
qn m n
ij pjj i j p
q qn m n n m n
ij pj ij pjj i j p j i j p
x z
a a
x z x z
… or a weighted average
a1, a2 user defined weights,or weights reflecting the “size” of the stages with respect to the portion of total resources used in each stage (in raw quantities)
33
An alternative additive model
0
0
0
1
1 1
1
min
. .
, 1,...,
0, 1,...,
, 1,...,
0, 0, 1,..., ;
n
j ij ijj
n n
j pj j pj pjj j
n
j rj rjj
j j
s t
x x i m
z z z p q
y y r s
j n free
The dual model
0 0
0
1 1
1
1 1
1 1
max
. .
1
0, 1,...,
0, 1,...,
0, 0, 0
s m
r rj i ijr i
q
p pjp
q m
p pj i ijp i
qs
r rj p pjr p
i p r
F u y v x
s t
w z
w z v x j n
u y w z j n
v w u
The primal model
34
An alternative additive model
•The model is extendable to VRS situations•The new model suffers from the same
irregularities with other additive-decomposition models
35
Deriving the efficient frontiers
0
0
0
1
1
1 1
1
min
. .
, 1,...,
, 1,...,
0, 1,...,
0
0, 0, ,
n
j ij ijj
n
j rj rjj
n n
j pj j pj pj pj j
q
pp
j j p
s t
x x i m
y y r s
z z z a p q
a
a free
0 0
0
1 1
1
1 1
1 1
1
ˆmax
. .
1
0, 1,...,
0, 1,...,
0, 1,..., 1
0,
s m
r rj i ijr i
q
p pjp
q m
p pj i ijp i
qs
r rj p pjr p
p p
i p
F u y v x
s t
w z
w z v x j n
u y w z j n
w w p q
v w
0, 0ru
Dual Primal
36
Deriving the efficient frontiers
•The assumption that the weights of the intermediate measures are equal is sufficient to drive the efficiency assessments in two-stage DEA processes in compliance with the DEA standards
37
Extensions - Conclusions • Two-stage DEA: A
fundamental approach
• Extensions to multi-stage processes
• Other two-stage schemes
…..X YZ1 Zk
X YZ
E
H
38
Thank you for your attention!
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