Dobra Prezentacja o Dea

12007-05-30 – Ana Camanho

Performance Measurement using Data Envelopment Analysis (DEA) and

Malmquist indices: Issues, Challenges and Applications.

Ana Camanho University of Porto, Portugal

([email protected])

Accounting and Management Science SeminarsNorwegian School of Economics and Business Administration

30-05-2007

22007-05-30 – Ana Camanho ([email protected])

The School of Engineering


University of Porto - School of Engineering

15 Schools

60 graduate programs

120 master programs

100 doctoral programs

2200 lecturers and researchers

1600 administrative staff

27000 students, of which 3500 postgraduate

7 Departments

10 graduate programs

25 master programs

13 doctoral programs

450 lecturers and researchers

250 administrative staff

6000 students, of which 1000 postgraduate


Contents

Introduction do Data Envelopment Analysis (DEA)• Concept efficiency measurement • Input or output orientation• Returns to scale• By-products of a DEA analysis• DEA formulation (linear programming model)• Additional topics in DEA• DEA in practice: case study of retailing stores

- Efficiency analysis adjusting for environmental factors- Target setting using network DEA

Productivity change over time• Introduction to Malmquist index• DEA in practice: case study of retailing stores

- Analysis of productivity change using Malmquist index- Comparison of performance of stores with different configurations: The use of a

new Malmquist-type index.

Conclusions


Introduction to DEA

Model of efficiency analysis

Objective of a Data Envelopment Analysis (DEA) assessment:• Comparison of performance of homogeneous decision making units

(DMUs) that use multiple inputs for the production of multiple outputs.

• The efficiency measure compares the ratio output/input of the DMU assessed with the value of this ratio observed in the other DMUsanalysed.

Decision Making UnitsDecision Making UnitsDecision Making UnitsInputs Outputs


Introduction to DEA

Graphical illustration of the DEA concept• Single input and single output, assuming Constant Returns to Scale (CRS)

Output

Input

Inefficiency

A E

x

*E

Efficient frontier

Efficiency = *xE xE


Introduction to DEAOutput or input oriented analysis

• Input oriented measures keep output fixed- Input oriented efficiency indicates by how much can input quantities be proportionally reduced

holding output constant.• Output oriented measures keep input fixed

- Output oriented efficiency indicates by how much can output quantities be proportionally increased holding input constant.

Output

AE

x

*EEfficient frontier

Output efficiency = *xE

xE

y

Input efficiency =

yEyE **

**E

Input

Scope for output augmentation: EE*

Scope for input reduction: EE**


Introduction to DEA

Returns to scale assumptions• Constant Returns to Scale (CRS) or Variable Returns to Scale (VRS)

Output

Input

A

Efficient frontier (CRS)

Efficient frontier (VRS)

Increasing returns to scale

Decreasing returns to scale

Input efficiency (CRS) =

yEyE*CRS

Input efficiency (VRS) =

yEyE*VRS

EE*CRS E*VRS

Scale efficiency =VRS*

*CRS

yEyE

CRS efficiency = VRS efficiency × Scale efficiency

y


Introduction to DEAOutput or input oriented analysis

• Choice depends on analyst’s view over which variables (inputs or outputs) it is believed managers can exercise control.

• Input and output orientation will estimate the same frontier.

• Input and output oriented measures of efficiency are equivalent under CRS.

• Under VRS, input and output oriented analysis will give different measures of efficiency for DMUs with efficiency < 1.

Constant returns to scale or variable returns to scale• Analyst must understand the constraints of the sector analysed.

• Choice depends on the purpose of analysis and whether short-run or long-run efficiency is examined.

• A VRS assessment implies that firms are only compared to others firms of roughly similar size.

• VRS produces efficiency scores greater than or equal to CRS efficiency scores.


Introduction to DEAWhat else can we learn from efficiency analysis?

Cost efficiency target

Input 2 /output

DMU E

A

C

B

D

Isocost line

Efficient frontier

Technical efficiency target

Benchmarks

O

ET

Input 1 /output

Production possibility set

EC

Technical efficiency =

OEOET

Cost efficiency =

OEOEC

Allocativeefficiency = T

C

OEOE

Tech. Eff × Alloc. Eff= Cost Eff


Formulation of the DEA model

DEA is based on linear programming [Charnes et al., 1978]

Efficiency measure with one input and one output:

with multiple inputs and outputs

• But, firm outputs cannot be added together directly, and the same for the inputs…

• If we know the output weights and input weights, the job is done. There is no need for sophisticated analysis. A major contribution of DEA is to offer insights about the value of weights.

InputOutput

=Efficiency

...InputInputInput....OutputOutputOutput

332211

332211

+×+×+×+×+×+×

=weightweightweight

weightweightweightEfficiency

...InputInputInput....OutputOutputOutput

321

321

++++++

=Efficiency


Formulation of the DEA modelFor each DMU, we have a model that maximizes the efficiency score, subject to all other DMUs having efficiencies less than or equal to one.

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Input oriented DEA model with Constant Returns to Scale

Linear programming model [Charnes et al., 1978]


Formulation of the DEA modelDEA input oriented model (with CRS).

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Duality of Linear Programming⇔


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Formulation of the DEA modelDEA input oriented model (with VRS) [Banker et al., 1984]

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Duality of Linear Programming⇔


DEA: evolution of the state-of-the-art (Seiford, 1996, Journal of productivity analysis)

Antecedents:Farrell (1957)

Pareto (1927)

Koopmans (1951)

Shephard (1970)

Malmquist (1953)DEA model

[Charnes et al,1978]

VRS model[Banker et al, 1984]

Non-radial models[Charnes et al., 1985]

Models with weight restrictions[Dyson and Thanassoulis, 1988]

Models with non-discretionary variables[Banker and Morey, 1986]

Analysis of panel data(Malmquist index) [Fare et al., 1994]

Comparison of groups (Program efficiency) [Charnes et al., 1981]

Statistical foundation for DEA [Banker, 1996]and Confidence intervals on DEA efficiencies using Bootstrapping [Simar and Wilson, 1998].Stochastic Frontiers

Aigner et al. (1977)

Target setting using Network DEA models (DEA models that account for interrelations between DMUs and sub-DMUs).[Thanassoulis and Dyson, 1992; Fare et al., 1997]

Other enhancements


DEA in practice: analysis of a retailing organizationThe structure of the retailing organisation (largest Portuguese retailing organisation)

Case study: 70 stores (14 hypermarkets and 56 supermarkets)

Objectives of the performance assessment • To analyse the impact of exogenous conditions (competition and population) on store performance.• To define targets for sales maximization, allowing for the reallocation of area among sections• To analyse the evolution of performance over time.• To compare the performance of Heavy Bazaar sections with different configurations.

T70HB70LB70P70G70Store 70

T1HB1LB1P1G1Store 1T2HB2LB2P2G2Store 2………………

Textilessection

Heavy Bazaarsection

Light Bazaarsection

Perishablessection

Grocerysection

Commercialmanagement

Operationalmanagement

T70HB70LB70P70G70Store 70

T1HB1LB1P1G1Store 1T2HB2LB2P2G2Store 2………………

Textilessection

Heavy Bazaarsection

Light Bazaarsection

Perishablessection

Grocerysection

Commercialmanagement

Operationalmanagement


DEA in practice:models with non-discretionary variables

Why consider non-discretionary variables in efficiency assessments?

• To allow fair comparisons: DMUs facing unfavourable exogenous conditions should not be penalised for producing less output or consuming more inputs than the other DMUs.

Examples of non-discretionary variables: • Competition and population density (e.g., affect bank branches,

supermarkets, restaurants):

• Cultural and economic level of families (e.g., school results)

• Fixed production quotas and farming areas (agriculture and fisheries)



The model by Banker and Morey, 1986.• Treats ND variables as a different set. For example, for an input

oriented analysis with ND inputs, the equiproportional input reductions are only looked for controllable inputs:

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Multi-stage models• Ray (1988, 1991): objective is to explain the differences in efficiency

scores based on the effect of non-discretionary variables.- 1st stage: run a DEA model without ND variables- 2nd stage: regress DEA scores on ND variables and correct the efficiency

score based on the impact of ND variables estimated with regression.

• Grosskopf et al. (1997): objective is to adjust the controllable input and output variables according to the effect of ND factors

- 1st stage: regression analysis is used to adjust the controllable input and output variables according to the effect of ND factors

- 2nd stage: run DEA with the adjusted variables

Various further methodological proposals – no generally accepted method.



Ruggiero (1996):• The PPS is defined only by controllable variables

• The comparison among DMUs is fair because it is ensured that the peers will always have ND factors that are equal or less favourablethan those of the DMU assessed.

• In practice, this model constructs several different frontiers, according to the values of the ND factors of the DMUs assessed.

C H

E F

B

G

AD

0

2

4

6

8

10

0 2 4 6 8 10 12

Input

Out

put

frontier 1

frontier 2

frontier 3

frontier 5

frontier 4

VRS DMU Efficiency Peers

A 100% λA =1 B 100% λB=1 C 100% λC=1 D 100% λD=1 E 100% λE=1 F 75% λE=1 G 89.58% λA =0.83, λC=0.17 H 75% λC=1

x1 (D) x2 (ND) y A 8 8 8 B 6 4.6 5 C 3 1.9 2 D 10 9 9 E 6 3.6 4.5 F 8 3.6 4.5 G 8 9 7 H 4 1.9 2


DEA in practice: models with non-discretionary variables

Model of retailing activity at the store level

• Objective: Maximize sales• The store assessment is based on the construction of a production possibility set defined

only by controllable factors. The effect on NC factors is taken into account by restricting the peers of each DMU to stores with similar or worst environmental conditions, as in Ruggiero, 1996. The ND factor considered is the ratio population/competition.

SalesOperational costs StoreStock

Staff costs

Area

Products spoiled

Population / competition



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The criteria for peer selection is having a ratio population/competition equal or lower than 1.2 times the value of this ratio in the store assessed. This correspond to α=20% in the new DEA model.



Results: example for one store, comparing observed inputs and outputs and DEA targets, accounting for ND factors.

Output:

Inputs:

Original valueObjective



Sales

AreaStock Op. costs staff costs prod. spoiled



Is the impact of NC factors on efficiency significant? • The differences in the inefficiency distributions corresponding to the model accounting for

ND factors and the model only with controllable factors were tested using the K-S test.

• The test revealed that the NC factors have a significant effect on store activity (p=0.004).

• When the NC factors are included in the model, the efficiency value increases in 50 stores. For these stores, the efficiency values increasing up to 15% (with the exception of two stores, with efficiency increase around 25% and 35%).

05

1015202530

0% 5% 10% 15% 20% 25% 30% 35%Difference between the results of the model without NC

factors and the new model with NC factors (%)

Freq

uenc

y


DEA in practice: Target setting using Network DEA

Model of retailing activity at the section level

• Objective: Maximize store sales taking into account the inter-relations among sections of the same store

• The sales target for each section should be defined allowing for the reallocation of floor space among sections within the store.

• Model used in based on Fare et al. (1997)

SalesNo. references

Store sections(grocery, perishables,

heavy bazaar, light bazaar, textiles)

Stock

Area

Products spoiled



Network DEA model with output orientation and Constant Returns to Scale.

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1jj textils][areajj textils][area

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1jj bazaar]heavy [areajj bazaar]heavy [area

n

1jj groceries] [areajj groceries] [area

n

1jj s]perishable [areajj s]perishable [area

sections allj section] [areaj area] [store

n

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Results: example of targets for one store, comparing standard DEA with the Network DEA model:

Original valueObjective DEAObjective Network DEA




Grocery Perishables Heavy B. Light B. Textiles Grocery Perishables Heavy B. Light B. Textiles

Products spoiled Stock No. references


Productivity change over time

The standard approach to the measurement of productivity change over time is the Malmquist index. (Caves et al, 1982; Fare et al, 1994)

• For use when we have panel data.

• Decomposes productivity change into efficiency change (firms moving closer to the frontier) and technological change (shifts in the frontier).

• No need for price data, no need for assumptions of cost minimisationor revenue maximisation.

• Input-based or output-based Malmquist index.

• Based on input or output distance functions.

• Calculates Total Factor Productivity (TFP) using DEA models.



Productivity change between 2 data points is calculated by ratios of distances of each point relative to a common technology.• The Malmquist index is a geometric mean of two indices, evaluated with

respect to period t and period t+1 technologies (Fare et al, 1994).

For output orientation: MO>1 → Productivity growthMO<1 → Productivity decline

• Decomposition of the index:

2/1

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Efficiency change Technological change



For output oriented analysis:• Distance function = DEA efficiency

Output 2 /input

a

O Output 1 /input

Frontier period t

Yt+1

Ytd

c

b

Frontier period t+1

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<1:efficiency decline

>1: technological improvement


DEA in practice: Retailing example: productivity change

Model of retailing activity at the store level

• Analysis of store productivity change between 2002 and 2004

• The Malmquist index was decomposed further to identify scale efficiency changes (Fare et al., 1994).

2/1

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tO

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O

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tO

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xyxy

Technical efficiency change (VRS)

Technological changeScale efficiency change

SalesOperational costs StoreStock

Staff costs

Area

Products spoiled


DEA in practice: Retailing example: productivity change

Efficiency status of the stores:

Productivity change: (no. of stores with an index greater than 1 [improvement], equal to 1, or smaller than 1 [decline])

10 stores promoted innovation (5 supermarkets and 5 hypermarkets)

• These stores define the frontier in t+1 and are beyond the production possibility set of period t.

Year 2002 Year 2004No. efficient stores 17 16No. inefficient stores 53 54No. stores remained in the frontier between 2002 and 2004Average efficiency (for inefficient stores) 87% 87%

11

Malmquist index

Technological change

Efficiency change (VRS)

Scale efficiency change

improve (index >1) 12 7 13 414 hypermarkets decline 2 2 1 6

equal 0 5 0 4improve 24 21 16 14

56 supermarkets decline (index <1) 32 29 37 41equal 0 6 0 1


DEA in practice: Comparison of performance of sections with different configurations

Objective: • To explore the differences in performance between two groups of

heavy bazaar sections (selling electrical appliances, electronics,…)

• Although all sections are from the same organisation:- Sections in Group A are within hypermarkets and large supermarkets

located in large cities- Sections in Group B are within smaller supermarkets, located in small

cities/towns.- Sample of 18 stores in each group

• The research question is: which group of stores performs better?- Better performance implies having:

- Less efficiency spread within the group- A more productive frontier.


DEA in practice: Comparison of performance of stores with different configurations

Malmquist-type index for group comparisons: [Camanho and Dyson, 1996]

• The new index (IAB) compares the performance of groups of DMUs operating under different conditions (A and B).

- The index focuses on comparisons in a static setting (i.e., for a given moment in time).

- The index handles directly all the observations corresponding to individual DMUs.

• This index can be decomposed into two sub-indexes:- Comparison of efficiency spread between groups (IEAB).- Comparison of productivity differences between the group best-practice frontiers

(IFAB).

ABABAB IFIEI ×=



Overall Malmquist-type index for comparison of group performance(index IAB, with α DMUs in group A and β DMUs in group B)

• For an output oriented analysis, a value of IAB > 1 indicates that group A performs better than group B.

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Comparison of efficiency spread between groups(index IEAB, with α DMUs in group A and β DMUs in group B)

For an output oriented analysis, a value of IEAB > 1 indicates that the efficiency spread is smaller in DMUs of group A than in group B

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Comparison of productivity differences between the group frontiers(index IFAB, with α DMUs in group A and β DMUs in group B)

For an output oriented analysis, a value of IFAB > 1 indicates greater productivity of the frontier of group A than the frontier of group B

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=

=

=

ββ

ββ

αα

αα

j

Bj

Bj

A

j

Bj

Bj

B

j

Aj

Aj

A

j

Aj

Aj

B

yxd

yxd

yxd

yxd


Model of heavy bazaar activity:

• Heavy bazaar sections from group A have better performance (IAB>1).- The frontier of group A is more productive (IFAB>1), but the efficiency spread is larger (IEAB<1)

which indicates that there is still scope for efficiency improvements in group A.• Non-parametric tests were used to test the statistical significance of the differences

between groups, captured by the indices IEAB and IFAB. - The Kolmogorov-Smirnov test showed that the position of the frontiers and the efficiency spreads

are different between the stores of groups A and B.


SalesStockNo. of references

Area

Products wasted

IAB IEAB IFAB

Heavy Bazaar

Output

Input

Group AGroup B


Conclusions: Advantages and disadvantages of DEA

Advantages:• Easy to use

• Allows multiple inputs and multiple outputs

• Does not require specification of functional form for the frontier

• Does not require a priori specification of weights for inputs and outputs

• Inputs and outputs can be expressed in different measurement units

Disadvantages:• No account for measurement error / random noise (all shortfall in the

input-output ratio of a DMU is inefficiency).

• Sensitive to outlier data


Software available: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/

DEAP version 2.1 – Centre for Efficiency and Productivity Analysis, University of New England, Australia, by Tim Coelli

• Available freely at http://www.uq.edu.au/economics/cepa/software.htm

Frontier Analyst, version 4 – Banxia Software in Glasgow• Commercially available at http://www.banxia.com/famain.html

Performance Improvement Management (PIM DEA SoftV1.), by Emmanuel Thanassoulis & Ali Emrouznejad (developers of Warwick DEA software)

• Commercially available at http://www.deasoftware.co.uk/• Thanassoulis, E. (2001) “Introduction to the theory and Application of Data Envelopment Analysis: A

foundation text with integrated software, Kluwer Academic Publishers.

DEAFrontier – Joe Zhu• Details at http://www.deafrontier.com/software.html• Zhu, J. (2003) “Quantitative models for performance evaluation and bechmarking: Data Envelopment

Analysis with Spreadsheets and DEA excel solver”, Kluwer Academic Publishers.

DEA-Solver-PRO version 5 – SAITECH, Inc.• Commercially available at http://www.saitech-inc.com/Products/Prod-DSP.asp• Cooper, W.W., Seiford, L.M., Tone, K. (2006), “Introduction to Data Envelopment Analysis and Its Uses:

With DEA-Solver Software and References”, Springer.

OnFront version 2 – Economic Measurement and Quality AB (EMQ AB)• Available at http://www.emq.com/software.html


References:Aigner, D., Lovell, C. A. K., and Schmidt, P. (1977). Formulation and estimation of stochastic frontier production function models. Journal of Econometrics, 6(1):21–37.

Banker, R. D., Charnes, A., and Cooper, W. W. (1984). Some models for estimating technical and scale inefficiencies in Data Envelopment Analysis. Management Science, 30(9):1078-1092.

Banker, R. D. and Morey, R. C. (1986). Efficiency analysis for exogenously fixed inputs and outputs. Operations Research, 34(4):513-521.

Banker, R. (1996). Hypothesis tests using Data Envelopment Analysis. Journal of Productivity Analysis, 7(2-3):139-159.

Caves, D. W., Christensen, L. R., and Diewert, W. E. (1982). The economic theory of index numbers and the measurement of input, output and productivity. Econometrica, 50:1393–1414.

Camanho, A. S. and Dyson, R. G. (2006). Data envelopment analysis and Malmquist indices for measuring group performance. Journal of Productivity Analysis, 26(1):35–49.

Charnes, A., Cooper, W., and Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6):429-444.

Charnes, A., Cooper, W.W., Rhodes, E. (1981). Evaluating program and managerial efficiency: an application of data envelopment analysis to program follow through. Management Science, 27(6):668–697.

Charnes, A., Cooper, W.W., Golany, B., Seiford, L., Stutz, J. (1985). Foundations of Data Envelopment Analysis for Pareto-Koopmans Efficient Empirical Production Functions. Journal of Econometrics, 30 (1/2), 91-107.

Dyson, R., Thanassoulis, E. (1988). Reducing weight flexibility in Data Envelopment Analysis. Journal of the Operational Research Society, 39 (6), 563-576.

Färe, R., Grabowski, R., Grosskopf, S., and Kraft, S. (1997). Efficiency of a fixed but allocatable input: A non-parametric approach. Economics Letters, 56:187-193.

Fare, R., Grosskopf, S., Lindgren, B., Roos, P. (1994) Productivity developments in swedish hospitals: a Malmquist output index approach. In: Charnes A, Cooper WW, Lewin AY, Seiford LM (eds). Data envelopment analysis: theory, methodology and applications. Kluwer Academic Publishers, Boston, pp 253–272.

Grosskopf, S., Hayes, K., Taylor, L., and Weber, W. (1997). Budget-constrained frontier measures of scale equality and efficiency in schooling. The review of Economics and Statistics, 79:116124.

Ray, S. C. (1988). Data Envelopment Analysis, nondiscretionary inputs and efficiency: an alternative interpretation. Socio-Economic Planning Sciences, 22(4):167–176.

Ray, S. C. (1991). Resource-use efficiency in public-schools - a study of Connecticut data. Management Science, 37(12):1620–1628.

Ruggiero, J. (1996). On the measurement of technical efficiency in the public sector. European Journal of Operational Research, 90:553–565.

Simar, L. and Wilson, P.W. (1998). Sensitivity analysis of efficiency scores: how to bootstrap in nonparametric models. Management Science, 44(1):49–61.

Thanassoulis, E. and Dyson, R. (1992). Estimating preferred target input-output levels using Data Envelopment Analysis. European Journal of operational research, 56 (1), 80-97.

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Dobra Prezentacja o Dea