two-stage data envelopment analysis

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





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





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


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


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


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 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 -1 efficiency

Stage-2 efficiency

Overall DMU efficiency = t1 . stage 1 + t2 . stage 2 (t1+t2=1)



24


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


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 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



Two-stage processes: The three fundamental approaches


Conclusions

28

An alternative additive model


Stage -1 efficiency

Stage-2 efficiency

Overall DMU efficiency = ½ stage 1 + ½ stage 2



29



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



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


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


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


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


•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!

two-stage data envelopment analysis

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