Data Envelopment Analysisksz.pwr.edu.pl/.../DEA-Seminar-27_11_2019-Despotis.pdf · efficiency of a...

DataEnvelopmentAnalysis

Presenter: Dimitris Despotis, University of Piraeus, Greece

Wroclaw UST, October 30, 2019

Overview

• Preliminaries• Understanding DEA• Basic models with explanatory illustrations


The founders of DEA and the seminal paper

• Abraham Charnes (1917-1992)• Finalist for the Nobel Prize in economics (1975)• The John von Neumann Theory Prize (1982)• Distinguished Public Service medal from the U.S. Navy for his contributions as an

operations analyst during World War II.

• William Cooper (1914-2012)• US Comptroller General Award (1986)

• Charnes, Cooper and Rhodes, (1978). Measuring the efficiency of decision making units, European Journal of Operational Research 2, pp. 429-444• Based on the seminal ideas of: M.J. Farrell (1957), The measurement of productive

efficiency, Journal of the Royal Statistical Society


What is DEA …

• DEA is a non-parametric 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. • The DMUs are assumed• homogeneous, i.e. they belong to the same system and they consume

different amounts of the same inputs to produce different amounts of the same outputs.• Independent• Black boxes

• The mathematical instrument used is linear programming.


DMUInputs Outputs

… What is DEA

• The basic logic behind the DEA is that it compares DMUs’ real status to ideal status by projecting their actual input-output bundle onto the production frontier.

• For each DMU, DEA identifies an efficient mix of the observed DMUs that can achieve the DMU’s levels of outputs with the minimal use of resources (inputs). The resources used by the efficient mix are then compared with the actual resources used by the evaluated DMU to produce its observed outputs. This comparison highlights whether the DMU under evaluation is efficient or not.

• To obtain the production frontier, all DMUs’ real inputs and outputs are used to construct a production possible set with certain axiomatic hypotheses, while the production frontier is an outside envelopment of the production possible set.


Understanding DEAThe elementary single input-single output case


Input (x)

A

C

B

O

D

E

F G

x

y

Output (y)

Absolute efficiencies: the individual slopes

Relative efficiencies: the individual slopes/max slope at B

Relative efficiency of B=1

Relative efficiency of the others < 1

Understanding DEA


Input

Output

A

C

B

O

D

E

F G

Τhe one-eyed man reigns over the blind

The production possibility set (PPS)


Input (x)

Output (y)

A

C

B

O

D

E

F G

The observed activities belong to PPS .

If an activity (x,y) belongs to PPS then the activity (λx, λy) belongs to PPS for any positive scalar λ (CRS assumption)

For any activity (x,y) in PPS any semipositive (x^,y^ ) with x^≥x and y^≤y is included in PPS.

Any semipositive linear combination of activities in PPS belongs to PPS.

CRS E

ff. fr

ontie

r

Returns to scale: CRS/VRS


Input

Output

A

C

B

O

Production possibility set

D

E

F GCR

S fro

ntie

r

VRS

fron

tier

Max slope

Orientation


Input

Output

A

C

B

O

D

E

F G

D΄P Input-oriented eff. of D=

' 1PDPD

£

D¨

Q

Output-oriented eff. of D='' 1QD

QD³

IO eff.=1/OO eff.

CRS / VRS efficiency scores


Input

Output

A

C

B

O

D

E

F G

VRS score ≥ CRS score

D΄ D΄΄P

.

.

PDCRS eff of D

PDPD

VRS eff of DPD

¢=

¢¢=

DEA (non-parametric) Vs regression (parametric)


Input (x)

Output (y)

A

C

B

O

D

E

F G

y=ax+b

Best

pra

ctic

e

Average

2 inputs/1 output


. OFEff of FOF

¢=

A and B are the referent units for F

Β and C are the referent units for G

Input 1/Output

Inpu

t 2/O

utpu

tA

CB

O

D

E

F

GF΄

Radial

reducti

on

G΄

Input excess

A B C D E F GInp1 10 17 30 22 45 20 27Inp2 50 20 15 65 15 35 26Out 1 1 1 1 1 1 1

1 input/2 outputs


Output 1/Input

Out

put 2

/Inpu

t

A

C

B

O

D

E

F

G

Radia

l impr

ovem

ent

. ODEff of DOD

¢=

A and B are the referent units for D

Β and C are the referent units for E

A is the referent unit for F

D΄

E΄

F΄

Output shortfall

Constructing an efficient mix to compare F


A B C D E F GInp1 10 17 30 22 45 20 27Inp2 50 20 15 65 15 35 26Out 1 1 1 1 1 1 1 * OF

OFq

¢=

θ* is the largest possible radial reduction of the inputs of DMU F that preserves the level of its output without stepping outside the PPS (F’ is the efficient mix of DMUs against which F is compared)

C

D

E

F

G

B

A

Input 1/Output

Inpu

t 2/

Out

put

F’

λD=0

λF=0

λG=0

λE=0λC=0

λA=0.23076923

λB=0.76923077

=0.76923077

C

D

E

F

G

B

A

Input 1/Output

Inpu

t 2/O

utpu

t F’

λD=0

λF=0

λG=0

λE=0λC=0

λA=0.23076923

λB=0.76923077

The input oriented CRS DEA model (LP) for DMU F


min $10'( + 17'+ + 30'- + 22'/ + 45'2 + 20'3 + 27'4 ≤ 20$50'( + 20'+ + 15'- + 65'/ + 15'2 + 35'3 + 26'4 ≤ 35$

'( + '+ + '- + '/ + '2 + '3 + '4 ≥ 1


!∗ = 0.76923077; ,- = 0.23076923 ,. = 0.76923077

,/ = 0 ,0 = 0 ,1 = 0 ,2 = 0 ,3 = 0

Optimal solution

F inefficient

'(, '+, '-, '/, '2, '3, '4 ≥ 0; $ ∈ ;

The input oriented CRS DEA model (LP) for DMU G


min $10'( + 17'+ + 30'- + 22'/ + 45'2 + 20'3 + 27'4 ≤ 27$50'( + 20'+ + 15'- + 65'/ + 15'2 + 35'3 + 26'4 ≤ 26$

'( + '+ + '- + '/ + '2 + '3 + '4 ≥ 1

A B C D E F GInp1 10 17 30 22 45 20 27Inp2 50 20 15 65 15 35 26Out 1 1 1 1 1 1 1 Optimal solution

C

D

E

F

G

B

A

Input 1/Output

Inpu

t 2/O

utpu

t

λD=0

λF=0

λG=0

λE=0λB>0

λC>0

λA=0

G΄

!∗ = 0.72938689; -. = 0

-/ = 0.79281184 -2 = 0.20718816

-3 = 0 -4 = 0 -5 = 0 -6 = 0

G inefficient

'(, '+, '-, '/, '2, '3, '4 ≥ 0; $ ∈ ;

The input oriented CRS DEA model (LP) for DMU B


min $10'( + 17'+ + 30'- + 22'/ + 45'2 + 20'3 + 27'4 ≤ 17$50'( + 20'+ + 15'- + 65'/ + 15'2 + 35'3 + 26'4 ≤ 20$

'( + '+ + '- + '/ + '2 + '3 + '4 ≥ 1

A B C D E F GInp1 10 17 30 22 45 20 27Inp2 50 20 15 65 15 35 26Out 1 1 1 1 1 1 1 Optimal solution

B efficient

!∗ = 1 %& = 0 () = * %+ = 0 %, = 0 %- = 0 %. = 0 %/ = 0

'(, '+, '-, '/, '2, '3, '4 ≥ 0; $ ∈ ;

What about E?



min $10'( + 17'+ + 30'- + 22'/ + 45'2 + 20'3 + 27'4 ≤ 45$50'( + 20'+ + 15'- + 65'/ + 15'2 + 35'3 + 26'4 ≤ 15$

'( + '+ + '- + '/ + '2 + '3 + '4 ≥ 1

Optimal solution

Is E efficient?

!∗ = 1 %& = 0 %( = 1 )* = + %, = 0 %- = 0 %. = 0 %/ = 0

'(, '+, '-, '/, '2, '3, '4 ≥ 0; $ ∈ ;

To get the full picture for a DMU …


min $10'( + 17'+ + 30'- + 22'/ + 45'2 + 20'3 + 27'4 ≤ 45$50'( + 20'+ + 15'- + 65'/ + 15'2 + 35'3 + 26'4 ≤ 15$

'( + '+ + '- + '/ + '2 + '3 + '4 ≥ 1

Phase I

Phase II max :;< + :=< + :;>10'( + 17'+ + 30'- + 22'/ + 45'2 + 20'3 + 27'4 + :;< = 45$∗50'( + 20'+ + 15'- + 65'/ + 15'2 + 35'3 + 26'4 + :=< = 15$∗

'( + '+ + '- + '/ + '2 + '3 + '4 − :;> = 1

!∗ = 1 %& = 0 %( = 1 )* = + %, = 0 %- = 0 %. = 0 %/ = 0 0+1 = +2 031 = 4 0+5 = 4

Optimal solution

'(, '+, '-, '/, '2, '3, '4 ≥ 0; $ ∈ E

'(, '+, '-, '/, '2, '3, '4 , :;<, :=<, :;> ≥ 0

When is a DMU efficient?

A DMU is efficient if and only if θ*=1 and all the slacks are zero


Multiple Inputs and outputs


A(λ1) B(λ2) C(λ3) D(λ4) E(λ5) F(λ6) G(λ7)Inp1 20 19 26 28 23 55 35

Inp2151 130 167 168 156 255 235

Out1 100 150 160 180 94 230 220

Out2 90 50 55 72 66 90 88

The input oriented CRS DEA model for unit A

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

min. .20 19 26 28 23 55 35 20151 130 167 168 156 255 235 151100 150 160 180 94 230 220 10090 50 55 72 66 90 88 90, , , , , , 0;

s t

R

q

l l l l l l l ql l l l l l l ql l l l l l ll l l l l l l

l l l l l l l q

+ + + + + + £

+ + + + + + £

+ + + + + + ³

+ + + + + + ³

³ Î

The VRS DEA model


A(λ1) B(λ2) C(λ3) D(λ4) E(λ5) F(λ6) G(λ7)Inp1 20 19 26 28 23 55 35

Inp2151 130 167 168 156 255 235

Out1 100 150 160 180 94 230 220

Out2 90 50 55 72 66 90 88

The input oriented VRS DEA model for unit A

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2

min. .20 19 26 28 23 55 35 20151 130 167 168 156 255 235 151100 150 160 180 94 230 220 10090 50 55 72 66 90 88 90

1,

s tq


l l l l l l ll l

+ + + + + + £

+ + + + + + £

+ + + + + + ³

+ + + + + + ³

+ + + + + + =

3 4 5 6 7, , , , , 0; Rl l l l l q³ Î

The output oriented DEA model: CRS


1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

max. .20 19 26 28 23 55 35 20151 130 167 168 156 255 235 151100 150 160 180 94 230 220 10090 50 55 72 66 90 88 90, , , , , , 0;

s t

R

q

l l l l l l ll l l l l l ll l l l l l l ql l l l l l l q

l l l l l l l q

+ + + + + + £

+ + + + + + £

+ + + + + + ³

+ + + + + + ³

³ Î

A(λ1) B(λ2) C(λ3) D(λ4) E(λ5) F(λ6) G(λ7)Inp1

20 19 26 28 23 55 35

Inp2151 130 167 168 156 255 235

Out1100 150 160 180 94 230 220

Out290 50 55 72 66 90 88

The output oriented DEA model: VRS


1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

max. .20 19 26 28 23 55 35 20151 130 167 168 156 255 235 151100 150 160 180 94 230 220 10090 50 55 72 66 90 88 90, , , , , , 0;

s t

R

q

l l l l l l ll l l l l l ll l l l l l l ql l l l l l l q

l l l l l l l q

+ + + + + + £

+ + + + + + £

+ + + + + + ³

+ + + + + + ³

³ Î

A(λ1) B(λ2) C(λ3) D(λ4) E(λ5) F(λ6) G(λ7)Inp1

20 19 26 28 23 55 35

Inp2151 130 167 168 156 255 235

Out1100 150 160 180 94 230 220

Out290 50 55 72 66 90 88

!" + !$ + !% + !& + !' + !( + !) = 1

The dual LP (The multiplier CRS/IO model)


1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

min. .20 19 26 28 23 55 35 20151 130 167 168 156 255 235 151100 150 160 180 94 230 220 10090 50 55 72 66 90 88 90, , , , , , 0;

s t

R

q


l l l l l l l q

+ + + + + + £

+ + + + + + £

+ + + + + + ³

+ + + + + + ³

³ Î

1 2

1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2

max 100 90. .20 151 1100 90 20 151 0 ( )150 50 19 130 0 ( )160 55 26 167 0 ( )180 72 28 168 0 ( )94 66 23 156 0 ( )230 90 55 255 0 ( )220 88 35

AE u us tv vu u v v Au u v v Bu u v v Cu u v v Du u v v Eu u v v Fu u

= +

+ =+ - - £+ - - £+ - - £+ - - £+ - - £+ - - £+ - 1 2

1 2 1 2

235 0 ( ), , , 0

v v Gu u v v

- £³

CRS /IO

The dual LP (The multiplier CRS/IO model)


1 2

1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2

max 100 90. .20 151 1100 90 20 151 0 ( )150 50 19 130 0 ( )160 55 26 167 0 ( )180 72 28 168 0 ( )94 66 23 156 0 ( )230 90 55 255 0 ( )220 88 35

AE u us tv vu u v v Au u v v Bu u v v Cu u v v Du u v v Eu u v v Fu u

= +

+ =+ - - £+ - - £+ - - £+ - - £+ - - £+ - - £+ - 1 2

1 2 1 2

235 0 ( ), , , 0

v v Gu u v v

- £³

1 2

1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

100 90max

20 151. .(100 90 ) (20 151 ) 1 ( )(150 50 ) (19 130 ) 1 ( )(160 55 ) (26 167 ) 1 ( )(180 72 ) (28 168 ) 1 ( )(94 66 ) (23 156 ) 1 ( )(230 90 ) (55 255 ) 1

Au uEv v

s tu u v v Au u v v Bu u v v Cu u v v Du u v v Eu u v v

+=

+

+ + £+ + £+ + £+ + £+ + £+ + £

1 2 1 2

1 2 1 2

( )(220 88 ) (35 235 ) 1 ( ), , , 0

Fu u v v G

u u v v+ + £

³

Linear form Fractional form

Interpretation of the multiplier form of the DEA model

• DEA estimates optimal weights for the evaluated DMU that maximize its efficiency score

• Each DMU selects its own optimal weights so as to show itself in the best possible light relatively to the other DMUs


When is a DMU efficient?

A DMU is efficient if and only if E=1 and there is an optimal solution with non-zero weights


A DMU is efficient if and only if θ*=1 and all the slacks are zero

Or equivalently

Thank you for your attention


Data Envelopment Analysisksz.pwr.edu.pl/.../DEA-Seminar-27_11_2019-Despotis.pdf · efficiency of a...

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