DEA
Data Envelopment Analysis
Nonparametric efficiency measurement by data envelopment analysis. Provides radial (Charnes-Cooper-Rhodes and Banker-Charnes-Cooper) technical efficiency under constant, variable, non-increasing and non-decreasing returns to scale, the slacks-based measure of Tone (2001), the additive model of Charnes and others (1985), and the directional distance function of Chambers, Chung and Fare (1996), all through one interface and one result object. Efficiency estimates are accompanied by peers, slacks, returns-to-scale classification, scale efficiency and the optimal multipliers, and by bias-corrected estimates and confidence intervals from the smoothed homogeneous bootstrap of Simar and Wilson (1998). Where prices are known, cost, revenue and Nerlovian profit efficiency separate the technical component from the allocative one; where they are not, cross-efficiency with the secondary goals of Doyle and Green (1994) ranks units that a self-appraisal leaves tied. This package succeeds the archived 'DEA' package of Diaz-Martinez and Fernandez-Menendez (2008).
DEA
Data envelopment analysis in R. Five nonparametric efficiency estimators behind one interface and one result object; cost, revenue and profit efficiency where prices are known, and cross-efficiency where they are not; and the sampling theory that DEA practice usually leaves out: bias correction, confidence intervals, and an honest statement of how fast any of it can converge.
By David H. Bernstein, Zuleyka Díaz-Martínez and José Fernández-Menéndez.
A companion to sfa, which does the
parametric half of the same problem.
On the name. This package succeeds
DEA0.1-2, published in 2008 by Zuleyka Díaz-Martínez and José Fernández-Menéndez and archived shortly afterwards; both are authors here. It covers the same models, but no code is carried over — the 2008 package bundled a GLPK simplex in C, this one has no compiled code at all, and the API is entirely different. SeeNEWS.mdfor a table mapping the old function names onto the new arguments.
Installation
Not yet on CRAN — a first submission is pending. From GitHub:
# install.packages("remotes")
remotes::install_github("davidhbernstein/DEA")
Once it is accepted, install.packages("DEA").
Quick start
library(DEA)
## The original DEA application: 70 school sites, 5 inputs, 3 outputs
## (Charnes, Cooper and Rhodes 1981), bundled with the package.
data(charnes1981)
x <- charnes1981[, paste0("x", 1:5)]
y <- charnes1981[, paste0("y", 1:3)]
fit <- dea(x, y, rts = "crs", orientation = "in")
fit
#> --- Data envelopment analysis ---
#> model: radial (Debreu-Farrell)
#> technology: CRS, input orientation
#> DMUs: 70 inputs: 5 outputs: 3
#>
#> theta (input-oriented, 1 = on the frontier)
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.7883 0.9002 0.9404 0.9378 1.0000 1.0000
#>
#> efficient DMUs: 19 of 70
head(efficiency(fit)) # theta, in (0, 1]
head(peers(fit)) # who each DMU is benchmarked against
head(slacks(fit)) # what the radial cut leaves behind
dea_rts(x, y) # scale efficiency and returns to scale
x and y can be matrices, data frames, bare vectors, or — with data —
column names or one-sided formulas:
dea(~ x1 + x2 + x3 + x4 + x5, ~ y1 + y2 + y3,
data = charnes1981, rts = "crs")
For checking an estimator rather than using one, dea_sim() draws a technology
whose true efficiency is known in closed form:
sim <- dea_sim(n = 200, p = 2, q = 1, returns = 0.9, seed = 1)
fit <- dea(sim$x, sim$y, rts = "vrs", orientation = "out")
mean(fit$eff - sim$phi) # the bias, against a truth written down first
plot(fit)
What is here
| function | what it estimates |
|---|---|
dea() |
radial Debreu–Farrell efficiency: CCR, BCC, non-increasing and non-decreasing returns, and the free disposal hull; either orientation; second-stage slacks; Andersen–Petersen super-efficiency |
dea_sbm() |
the slacks-based measure of Tone (2001), non-oriented or oriented |
dea_ddf() |
the directional distance function of Chambers, Chung and Färe (1996), which handles zero and negative data |
dea_add() |
the additive model of Charnes et al. (1985), unweighted or as the Range Adjusted Measure |
dea_rts() |
scale efficiency and the returns-to-scale classification |
dea_cost(), dea_revenue() |
cost and revenue efficiency, each factoring exactly into technical × allocative |
dea_profit() |
Nerlovian profit inefficiency, which decomposes by addition rather than multiplication |
dea_cross() |
cross-efficiency, with the Doyle–Green benevolent and aggressive secondary goals |
multipliers() |
the optimal weights v, u, u0 from the multiplier form of any radial fit |
dea_boot() |
bias correction and confidence intervals, by the Simar–Wilson (1998) smoothed homogeneous bootstrap |
dea_sim() |
a technology with a closed-form answer, for testing an estimator against a truth |
dea_rate() |
the rate at which any of this can converge |
charnes1981 |
the Program Follow Through data the CCR model was introduced on |
All the estimators return an object of class "dea", so print(),
summary(), plot(), efficiency(), peers(), slacks(), multipliers()
and fitted() work the same way across them. The price models and
cross-efficiency return their own classes, with the same methods where the same
question makes sense.
## On the frontier, but buying the wrong mix for the prices you face?
ce <- dea_cost(x, y, w = c(1.4, 0.9, 2.1, 1.2, 1.0), rts = "crs")
sum(ce$technical > 1 - 1e-9) # 19 sites are technically efficient
sum(ce$eff == 1) # 2 of them are also allocatively efficient
## 19 sites tie at 1 and cannot be ranked. Cross-efficiency ranks all 70.
ben <- dea_cross(x, y, secondary = "benevolent")
agg <- dea_cross(x, y, secondary = "aggressive")
range(ben$eff - agg$eff) # how much the answer depends on whose weights
Cross-efficiency is reported through a pair of secondary goals on purpose. An efficient DMU has a whole face of optimal weight vectors, all giving it the same score of 1 but giving everyone else different ones, so a single cross-efficiency number is an artefact of the vertex a solver stopped at. The benevolent and aggressive answers bracket it: a ranking that survives from one to the other is in the data, and one that does not was in the solver.
Four things this package insists on
A DEA score is an estimate. The frontier is spanned by the observed DMUs,
so it lies inside the true one and every score is biased toward 1. On a
200-unit simulated sample the mean bias is around 0.05 on a (0, 1] scale — not
a rounding error. dea_boot() estimates it:
b <- dea_boot(fit, B = 2000, seed = 1)
head(b$table) # eff, bias, se, bias_corrected, ci_lower, ci_upper
It also reports correct_worthwhile, which applies Simar and Wilson's own
warning: the correction removes a bias and adds the variance of the estimate of
that bias, so it is a net loss where the bias is small relative to the noise.
The package reports that judgement per DMU rather than making it silently.
Convergence is slow, and slower with every variable added. DEA is not
root-n consistent. dea_rate() gives the slope that log mean squared error
can attain against log sample size:
dea_rate(1, 1, "vrs") # -1.333
dea_rate(4, 4, "vrs") # -0.444
At four inputs and four outputs, a hundredfold increase in sample size buys
about a factor of 8 in mean squared error where a parametric estimator would
buy 100. dea() warns when n < 3(p+q), where most DMUs are efficient by
dimension alone rather than by performance.
Radial efficiency is not Pareto–Koopmans efficiency. A DMU can score 1 and
still be dominated, because no common factor cuts every input even though one
input alone could come down. The second-stage slack program is on by default,
print() reports both counts when they differ, and $efficient is the
stricter judgement. dea_sbm() folds the distinction into a single number.
Peer sets are not unique. These linear programs frequently have multiple
optima, so peers() returns one valid benchmark set rather than the
benchmark set. Scores are unique; the weights achieving them generally are not,
and no DEA package can make them so. summary() says this out loud.
Correctness
Efficiency scores are checked three ways, in the package's own test suite:
- against hand-worked answers on a five-DMU example, so at least one check depends on no other package agreeing;
- against
Benchmarkingfor every radial technology and orientation, and againstDJLfor all three slacks-based orientations — agreement to around 1e-12, which is linear-programming tolerance; - against
dea_sim()'s closed-form truth, which is the only check that catches a misunderstanding rather than a coding error. Two packages making the same mistake agree perfectly.
Two further pipelines live in the development repository rather than the
package: a convergence study that measures each estimator's rate against the
theoretical target from dea_rate(), and a horserace benchmarking this package
against the other DEA packages on CRAN for speed, agreement and edge-case
behaviour.
References
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, 1078–1092.
Chambers, R. G., Chung, Y. and Färe, R. (1996). Benefit and distance functions. Journal of Economic Theory 70, 407–419.
Charnes, A., Cooper, W. W. and Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research 2, 429–444.
Kneip, A., Park, B. U. and Simar, L. (1998). A note on the convergence of nonparametric DEA estimators for production efficiency scores. Econometric Theory 14, 783–793.
Simar, L. and Wilson, P. W. (1998). Sensitivity analysis of efficiency scores: how to bootstrap in nonparametric frontier models. Management Science 44, 49–61.
Tone, K. (2001). A slacks-based measure of efficiency in data envelopment analysis. European Journal of Operational Research 130, 498–509.
License
GPL (>= 2).