anintroduction to measuring efficiency and productivity in agriculture by dea peter fandel slovak...

An Introduction to MeasuringEfficiency and Productivity

in Agriculture by DEA

Peter FandelSlovak University of Agriculture

Nitra, Slovakia

What are we going to cover?

Part 1 • Performance of a firm, productivity and efficiency

measurement• Introduction to DEA and DEA formulation• Input- and output orientation• Input- and output slacks• Returns to scale• Features of DEAPart 2• Software available

Variety of forms in customary analyses:• Cost per unit• Profit per unit• Satisfaction per unit• usually in ratio form:

• This is a commonly used measure of efficiency, but also of productivity

Performance of a firm

Input

Output

Production efficiency

Inputs Outputs

labour; production;

capital; sales;materials profit

Productivity• Partial productivity measures (output per worker

employed, output per worker hour, • Total factor productivity measures (all outputs, all inputs)

Transformation

Productivity and EfficiencySingle input and single output case

Farm A B C D E F G H

Employees 2 3 3 4 5 5 6 8

Sale 1 3 2 3 4 2 3 5

Sale/Empl.(productivity)

0.5 1 0.667 0.75 0.8 0.4 0.5 0.625

Efficiency –prod. relative to B

0.5 1 0.667 0.75 0.8 0.4 0.5 0.625

Sales per employee of others

Sales per employee of B0 ≤ ≤ 1

Relative efficiency: productivity / max. productivity

Source: Cooper, W.W. – Seiford, L.M., - Tone, K., 2002


0

1

2

3

4

5

6

0 2 4 6 8 10

Employee

Sal

es

A

B

C

D

E

F

G

HEfficiency frontier



0

0,5

1

1,5

2

2,5

3

3,5

0 0,5 1 1,5 2 2,5 3 3,5

A1

A

B

A2

Improvement of input

Improvement of output

Productivity and EfficiencyTwo inputs and one output case

A B C D E FEmployee x1 4 7 8 4 2 10Land x2 3 3 1 2 4 1Sale y 1 1 1 1 1 1

4; 3 7; 3

8; 1

4; 2

2; 4

10; 1

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0 2 4 6 8 10 12

Employee / Sales

Lan

d /

Sal

es

A

E

F

D

C

B

Efficiency frontier

Production possibility set

P(3,4;2,6)Efficiency of A =

0P / 0A = 0.8571

{ {


28.46.24.3 22

d(0,D) =

d(0,A) =

534 22

Productivity and EfficiencyOne input and two outputs case

534 22

3

204)

3

16( 22

A B C D E F GEmployee x 1 1 1 1 1 1 1Contracts y1 1 2 3 4 4 5 6Sales y2 5 7 4 3 6 5 2

Efficiency of D =

4; 6

5; 5

6; 2

4; 3

3; 4

1; 5

2; 7

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7

Contracts / Employee

Sa

les

/ E

mp

loy

ee A

C

D

G

F

E

BQ

P(16/3;4)

(1,4;7)

d(0,D)

d(0,P)= 0.75

d(0,D) =

d(0,P) =

534 22

Production possibility set


d(0,P)

d(0,D)= 1.33

Efficiency measurementwhen more inputs and more outputs

Efficiency =Output(1) + Output(2) + … + Output(s)

Input(1) + Input(2) + … + Input(m)

BUT• Firm outputs cannot be added together directly• Firm inputs cannot be added together directly


Efficiency = Output(1)*Weight(1) + Output(2)*Weight(2) + … + Output(s) )*Weight(s)

Input(1)*Weight(1) + Input(2)*Weight(2) + … + Input(m)*Weight(m)

BUT • It is necessary to estimate weights• When weights are known, it is easy to calculate efficiency measures


A B C D E FOutput1 U1= 1 100 150 160 180 94 230Output2 U2= 3 90 50 55 72 66 90Input1 V1= 5 20 19 25 27 22 55Input2 V2= 1 151 131 160 168 158 255

370 300 325 396 292 500251 226 285 303 268 530

1,47 1,33 1,14 1,31 1,09 0,941,00 0,90 0,77 0,89 0,74 0,64

Productivity (TFP)Efficiency score

FirmsWeights

Weighted outputsWeighted inputs

When fixed weights are available:



When fixed weights are not available:

• Linear programming (LP) is used to calculate both efficiency measure and the weights for each firm by comparison with other firms

• The specific application of LP is called Data Envelopment Analysis - DEA

Introduction to DEA and DEA formulation

0

1

2

3

4

5

6

0 2 4 6 8 10

Employee

Sal

es

A

B

C

D

E

F

G

HEfficiency frontier

Regression line

y=0.662x


Introduction to DEA and DEA formulation

• Technical efficiency (TE)

Maximisation of outputs for given set of inputs• Allocative efficiency (AE)

use of inputs in optimal proportions given their respective prices and production technology

• Economic efficiency

Combination of TE and AE• DMU – Decision Making Unit

DEA formulation

m

iii

s

rrr

xv

yu

10

10

max : TEi = 0 ≤ ≤ 1

yr = quantity of output r;vr = weight attached to output r;yi = quantity of input i;vi = weight attached to input ifor s outputs and m inputs

DEA formulation Fractional programming problem

max θ =

≤ 1 (j = 1, …, n)

u1y10+u2y20+ … + usys0

v1x10+v2x20+ … + vmxm0

u1y1j+u2y2j+ … + usysj

v1x1j+v2x2j+ … + vmxmj

v1,v2, … , vm ≥ 0

u1,u2, … , um ≥ 0

subject to

DEA formulation (primal)Transformation to a linear programming problem

max θ =

(j = 1, …, n)

u1y10+u2y20+ … + usys0

subject to v1x10+v2x20+ … + vmxm0 = 1

u1y1j+u2y2j+ … + usysj ≤ v1x1j+v2x2j+ … + vmxmj

v1,v2, … , vm ≥ 0

u1,u2, … , um ≥ 0

DEA formulation (primal)An example of linear programming problem for firm C

max θ = 160u1+55u2

subject to 25v1+160v2 = 1

A: 100u1+90u2 ≤

19v1+131v2

C: 160u1+55u2 ≤

20v1+151v2

B: 150u1+50u2 ≤

27v1+168v2

E: 94u1+66u2 ≤

25v1+160v2

D: 180u1+72u2 ≤

22v1+158v2

A B C D E FOutput1 100 150 160 180 94 230Output2 90 50 55 72 66 90Input1 20 19 25 27 22 55Input2 151 131 160 168 158 255v2

u1u2v1

FirmsWeights

F: 230u1+90u2 ≤ 55v1+255v2

v1,v2 ≥ 0 u1,u2 ≥ 0

Primal DEA results1. A firm (DMU0) is efficient if θ* = 1 and there exists at

least one optimal solution (u*, v*), with u* > 0 and v* > 02. Otherwise a firm (DMU0) is inefficient3. If a firm (DMU0) is inefficient, at least one constraint of

the LP problem must be satisfied as an equation. Firms for which constraints are of this character are called reference set, peer group, or benchmark.

4. Optimal θ* is the technical efficiency measure. It says to what extend inputs of DMU0 should be equiproportionally reduced, or what level of possible outputs is DMU0

generating from its inputs.

Primal optimum for the firm CLinear programFirm C u1 u2 v1 v2

0,003583 0,005625 0 0,00625 ThetaMax THETA 160 55 0,882708

LHS Rel RHSNormalization 25 160 1 = 1Firm A 100 90 -20 -151 -0,07917 <= 0Firm B 150 50 -19 -131 4,54E-14 <= 0Firm C 160 55 -25 -160 -0,11729 <= 0Firm D 180 72 -27 -168 -9,8E-13 <= 0Firm E 94 66 -22 -158 -0,27942 <= 0Firm F 230 90 -55 -255 -0,26333 <= 0

Weights

Peers for the firm C: firm B and firm D

Primal optimum for all firms

u1 u2 v1 v21. Firm A 0,003407 0,007326 0,005917 0,005839 1 1,2,42. Firm B 0,003883 0,00835 0,006744 0,006655 1 1,2,43. Firm C 0,003583 0,005625 0 0,00625 0,882708 2,44. Firm D 0,002987 0,006422 0,005187 0,005119 1 1,2,45. Firm E 0,003407 0,007326 0,005917 0,005839 0,763499 1,2,46. Firm F 0,002248 0,003529 0 0,003922 0,834771 2,4

WeightsTHETA Peers

DEA formulation (dual)

Disadvantages of primal DEA:• Usually more optimal solutions• Too many constraints (the number is

equivalent to the number of firms evaluated)

• Complicated way of efficient DMU identification

DEA formulation (dual)Primal DEA problem for the firm C (adapted)

max θ = 160u1+55u2

θ : 25v1+160v2 = 1

λ1: 100u1+90u2

-19v1-131v2 ≤ 0

λ3 : 160u1+55u2

-20v1-151v2 ≤ 0

λ2 : 150u1+50u2

-27v1-168v2 ≤ 0

λ5 : 94u1+66u2

-25v1-160v2 ≤ 0

λ4 : 180u1+72u2

-22v1-158v2 ≤ 0

λ6 : 230u1+90u2 -55v1-255v2 ≤ 0

v1,v2 ≥ 0 u1,u2 ≥ 0

DEA formulation (dual)Dual DEA problem for the firm C

min θsubject to

100 λ1+150 λ2+160 λ3+180 λ4+94 λ5+230 λ6 ≥ 160

90 λ1+ 50 λ2+ 55 λ3+ 72 λ4+66 λ5+ 90 λ6 ≥ 55

25θ - 25 λ1 - 19 λ2 - 25λ3 - 27 λ4 - 22 λ5 - 55λ6 ≥ 0

160θ - 151 λ1 - 31λ2 - 160 λ3 -168 λ4 -158 λ5 - 255 λ6 ≥ 0

λ1, λ2, λ3, λ4, λ5, λ6 ≥ 0

θ - free

DEA formulation (dual)general formulation

min θ

subject to

yr1 λ1+ yr2 λ2 + … + yrn λn ≥ yr0 , r = 1, 2,…, s

- θxi0 + xi1 λ1 + xi2 λ2 + … + ximλn ≤ 0 , i = 1,2, …, m

λ1, λ2, …, λn ≥ 0

θ - free

Dual DEA results1. A firm (DMU0) is efficient if θ* = 1, all λj =0, except λ0=12. A firm (DMU0) is inefficient if θ* < 1. 3. If a firm (DMU0) is inefficient, nonzero λj point at peers.

A convex combination of peer inputs and outputs with λj gives a virtual DMU at the frontier

4. Optimal θ* in this case gives so called Farrell input oriented efficiency measure

5. Constant returns to scale are assumed6. Output oriented measure φ = 1/ θ

Dual DEA resultsall firms

θ λ1 λ2 λ3 λ4 λ5 λ6Firm A 1,0000 1,0000 0,0000 0,0000 0,0000 0,0000 0,0000Firm B 1,0000 0,0000 1,0000 0,0000 0,0000 0,0000 0,0000Firm C 0,8827 0,0000 0,9000 0,0000 0,1389 0,0000 0,0000Firm D 1,0000 0,0000 0,0000 0,0000 1,0000 0,0000 0,0000Firm E 0,7635 0,5794 0,0572 0,0000 0,1526 0,0000 0,0000Firm F 0,8348 0,0000 0,2000 0,0000 1,1111 0,0000 0,0000

Firms A, B, D are efficientFirms C, E, F are inefficientTarget values of inputs and outputs

Input and output efficiency

Input oriented measures keep output fixed• input oriented technical efficiency (TEi) by how much

can input quantities be proportionally reduced holdingoutput constant

Output oriented measures keep input fixed• output oriented technical efficiency (TEo) by how

much can output quantities be proportionally expandedholding input constant

Input and output efficiency

0

0,5

1

1,5

2

2,5

3

3,5

0 0,5 1 1,5 2 2,5 3 3,5

A1

A

B

A2

Input and output orientated DEA

Input oriented DEAmin θ s.t.

Yλ ≥ y0

- θx0 +Xλ ≤ 0 λ ≥ 0

0 ≤ θ ≤ 1

Output oriented DEAmax φ s.t.

- φy0 + Yλ ≥ 0

Xλ ≤ x0

λ ≥ 0

1 ≤ φ ≤ +∞

• Output- and input-oriented models will estimate exactly the same frontier

• The same set of DMUs will be identified as efficient

• Efficiency measures of inefficient DMUs may differ

Input slacks

4; 3 7; 3

6; 1

4; 2

2; 4

10; 1

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0 2 4 6 8 10 12

Employee / Sales

Lan

d /

Sal

es

A

E

F

D

C

B

Q

P(3,4;2,6)

Farrell efficiency vs Pareto-Koopmans efficiencyAdapted from: Cooper, W.W. – Seiford, L.M., - Tone, K., 2002

Output slacks

4; 6

5; 5

6; 2

4; 3

3; 4

1; 5

2; 7

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7

Contracts / Employee

Sa

les

/ E

mp

loy

ee A

C

D

G

F

E

BQ

P(16/3;4)

(1,4;7)

Farrell efficiency vs. Pareto-Koopmans efficiencySource: Cooper, W.W. – Seiford, L.M., - Tone, K., 2002

Treatment of slacks

Input orientated DEAmin θ– ε ∙1's+ - ε ∙1's

- s.t.

Yλ - s+ = y0

- θx0 +Xλ + s - = 0 λ ≥ 0

0 ≤ θ ≤ 1

Output orientated DEA

max φ

s.t.

- φy0 + Yλ - s+ = 0

Xλ + s - = x0

λ ≥ 0

1 ≤ φ < +∞

DMU is efficient if and only if θ = 1 and all slacks s+ = 0, s - = 0

y

PC

x0

QR

PAPV

CRS Frontier

NiRS Frontier

VRS Frontier

B

TECRS = APC/AP, TEVRS = APV/AP, ER = APC/APV

Returns to scale


Features of DEA• We use LP to solve DEA formulations• It assigns weights to each DMU to put them in• the best possible light• DEA constructs a piecewise linear frontier which

envelops the other inefficient DMUs (intersecting planes in 3D-space

• DEA measures inefficiency as the radial distance from the inefficient unit to the frontier

• The inefficiency score is unit invariant• DEA is a data driven approach

Features of DEA

Advantages:• Easy to use• Allows multiple inputs and multiple outputs• Does not require specification of functional form• Does not require a prior specification of weightsDisadvantages:• No account of error / random noise• Non-parametric method – no goodness of fit

measures, model specification measures

Software• DEAP version 2.1 by Tim Coelli

– Centre for Efficiency and Productivity Analysis (CEPA)– Coelli, T.J. (1996), “A Guide to DEAP Version 2.1: A Data Envelopment Analysis (Computer) Program”, CEPA Working Paper 96/8, Department of Econometrics, University of New England, Armidale NSW Australia.– Freely available at http://www.uq.edu.au/economics/cepa/software.htm

• EMS: Efficiency Measurement System version 1.3 – University of Dortmund, by Holger Scheel– Available freely at http://www.wiso.uni-dortmund.de/lsfg/or/scheel/ems/– Uses Excel or ASCII data files

• DEAFrontier – by Joe Zhu– Zhu, J. (2003) Quantitative Models for Performance Evaluation and Benchmarking Data Envelopment Analysis with Spreadsheets and DEA Excel Solver, Kluwer Academic Publishers: Boston.– Excel Solver– Details at http://www.deafrontier.com/software.html

References

1. Cooper, W.W. – Seiford, L.M., - Tone, K. 2002. Data Envelopment Analysis. A Comprehensive Text with Models, Applications, References and DEA-Solver Software

2. Jacobs Rovena, 2005. An Introduction to Measuring Efficiency and Productivity in Public Sector, Workshop material, Data Envelopment Analysis workshop, Centre for Health Economics, University of York, 10-11 January 2005

anintroduction to measuring efficiency and productivity in agriculture by dea peter fandel slovak...

Documents