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 DEA 0.1-2, 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. See NEWS.md for 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

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 Benchmarking for every radial technology and orientation, and against DJL for 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).