AnIntroduction to Measuring Efficiency and Productivity in Agriculture by DEA Peter Fandel Slovak 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 DEA Part 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

Productionefficiency Inputs Outputs labour; production; capital; sales; materialsprofit Productivity Partial productivity measures (output per worker employed, output per worker hour, Total factor productivity measures (all outputs, all inputs) Transformation

Productivity and Efficiency Single input and single output case FarmABCDEFGH Employees Sale Sale/Empl. (productivity) Efficiency – prod. relative to B Sales per employee of others Sales per employee of B 0 ≤≤ 1 Relative efficiency: productivity / max. productivity Source: Cooper, W.W. – Seiford, L.M., - Tone, K., 2002

Productivity and Efficiency Single input and single output case Source: Cooper, W.W. – Seiford, L.M., - Tone, K., 2002

Productivity and Efficiency Single input and single output case Improvement of input Improvement of output

Productivity and Efficiency Two inputs and one output case Production possibility set P(3,4;2,6) Efficiency of A = 0P / 0A = { { Source: Cooper, W.W. – Seiford, L.M., - Tone, K., 2002 d(0,D) = d(0,A) =

Productivity and Efficiency One input and two outputs case Efficiency of D = d(0,D) d(0,P) = 0.75 d(0,D) = d(0,P) = Production possibility set Source: Cooper, W.W. – Seiford, L.M., - Tone, K., 2002 d(0,P) d(0,D) = 1.33

Efficiency measurement when 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 measurement when more inputs and more outputs 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

Efficiency measurement when more inputs and more outputs When fixed weights are available: Source: Cooper, W.W. – Seiford, L.M., - Tone, K., 2002

Efficiency measurement when more inputs and more outputs 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

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 DEA Part 2 Software available

Introduction to DEA and DEA formulation Regression line y=0.662x Source: Cooper, W.W. – Seiford, L.M., - Tone, K., 2002

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 max : TE i = 0 ≤≤ 1 y r = quantity of output r; v r = weight attached to output r; y i = quantity of input i; v i = weight attached to input i for s outputs and m inputs

DEA formulation Fractional programming problem max θ = ≤ 1 (j = 1, …, n) u 1 y 10 +u 2 y 20 + … + u s y s0 v 1 x 10 +v 2 x 20 + … + v m x m0 u 1 y 1j +u 2 y 2j + … + u s y sj v 1 x 1j +v 2 x 2j + … + v m x mj v 1,v 2, …, v m ≥ 0 u 1,u 2, …, u m ≥ 0 subject to

DEA formulation (primal) Transformation to a linear programming problem max θ = (j = 1, …, n) u 1 y 10 +u 2 y 20 + … + u s y s0 subject to v 1 x 10 +v 2 x 20 + … + v m x m0 = 1 u 1 y 1j +u 2 y 2j + … + u s y sj ≤ v 1 x 1j +v 2 x 2j + … + v m x mj v 1,v 2, …, v m ≥ 0 u 1,u 2, …, u m ≥ 0

DEA formulation (primal) An example of linear programming problem for firm C max θ =160u 1 +55u 2 subject to 25v v 2 = 1 A: 100u 1 +90u 2 ≤ 19v v 2 C: 160u 1 +55u 2 ≤ 20v v 2 B: 150u 1 +50u 2 ≤ 27v v 2 E: 94u 1 +66u 2 ≤ 25v v 2 D: 180u 1 +72u 2 ≤ 22v v 2 F: 230u 1 +90u 2 ≤55v v 2 v 1,v 2 ≥ 0u 1,u 2 ≥ 0

Primal DEA results 1.A firm (DMU 0 ) is efficient if θ * = 1 and there exists at least one optimal solution (u *, v * ), with u * > 0 and v * > 0 2.Otherwise a firm (DMU 0 ) is inefficient 3.If a firm (DMU 0 ) 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 DMU 0 should be equiproportionally reduced, or what level of possible outputs is DMU 0 generating from its inputs.

Primal optimum for the firm C Peers for the firm C: firm B and firm D

Primal optimum for all firms

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 =160u 1 +55u 2 θ : 25v v 2 = 1 λ 1 : 100u 1 +90u 2 -19v v 2 ≤ 0 λ 3 : 160u 1 +55u 2 -20v v 2 ≤ 0 λ 2 : 150u 1 +50u 2 -27v v 2 ≤ 0 λ 5 : 94u 1 +66u 2 -25v v 2 ≤ 0 λ 4 : 180u 1 +72u 2 -22v v 2 ≤ 0 λ 6 : 230u 1 +90u 2 -55v v 2 ≤ 0 v 1,v 2 ≥ 0u 1,u 2 ≥ 0

DEA formulation (dual) Dual DEA problem for the firm C min θ subject to 100 λ λ λ λ λ λ 6 ≥ λ λ λ λ λ λ 6 ≥ 55 25θ - 25 λ λ λ λ λ λ 6 ≥ 0 160θ λ λ λ λ λ λ 6 ≥ 0 λ 1, λ 2, λ 3, λ 4, λ 5, λ 6 ≥ 0 θ - free

DEA formulation (dual) general formulation min θ subject to y r1 λ 1 + y r2 λ 2 + … + y rn λ n ≥ y r0, r = 1, 2,…, s - θx i0 + x i1 λ 1 + x i2 λ 2 + … + x im λ n ≤ 0, i = 1,2, …, m λ 1, λ 2, …, λ n ≥ 0 θ - free

Dual DEA results 1.A firm (DMU 0 ) is efficient if θ * = 1, all λ j =0, except λ 0 =1 2.A firm (DMU 0 ) is inefficient if θ* < 1. 3.If a firm (DMU 0 ) 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 assumed 6.Output oriented measure φ = 1/ θ

Dual DEA results all firms Firms A, B, D are efficient Firms C, E, F are inefficient Target values of inputs and outputs

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 DEA Part 2 Software available

Input and output efficiency Input oriented measures keep output fixed input oriented technical efficiency (TEi) by how much can input quantities be proportionally reduced holding output constant Output oriented measures keep input fixed output oriented technical efficiency (TEo) by how much can output quantities be proportionally expanded holding input constant

Input and output efficiency

Input and output orientated DEA Input oriented DEA min θ s.t. Yλ ≥ y 0 - θx 0 +Xλ ≤ 0 λ ≥ 0 0 ≤ θ ≤ 1 Output oriented DEA max φ s.t. - φy 0 + Yλ ≥ 0 Xλ ≤ x 0 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

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 DEA Part 2 Software available

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

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

Treatment of slacks Input orientated DEA min θ– ε ∙1's + - ε ∙1's - s.t. Yλ - s + = y 0 - θx 0 +Xλ + s - = 0 λ ≥ 0 0 ≤ θ ≤ 1 Output orientated DEA max φ s.t. - φy 0 + Yλ - s + = 0 Xλ + s - = x 0 λ ≥ 0 1 ≤ φ < +∞ DMU is efficient if and only if θ = 1 and all slacks s + = 0, s - = 0

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 DEA Part 2 Software available

y PCPC x 0 Q R P A PVPV CRS Frontier NiRS Frontier VRS Frontier B TE CRS = AP C /AP, TE VRS = AP V /AP, ER = AP C /AP V Returns to scale Source: Cooper, W.W. – Seiford, L.M., - Tone, K., 2002

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 DEA Part 2 Software available

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 weights Disadvantages: 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 EMS: Efficiency Measurement System version 1.3 – University of Dortmund, by Holger Scheel – Available freely at – 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

References 1.Cooper, W.W. – Seiford, L.M., - Tone, K Data Envelopment Analysis. A Comprehensive Text with Models, Applications, References and DEA-Solver Software 2.Jacobs Rovena, An Introduction to Measuring Efficiency and Productivity in Public Sector, Workshop material, Data Envelopment Analysis workshop, Centre for Health Economics, University of York, January 2005