Cross-efficiency and benevolent weights

Functions for the cross-efficiency analysis of a DEA problem: the cross-efficiency matrix itself (compute_cross_efficiency), the benevolent multiplier weights used to build it (compute_cross_efficiency_weights), and a helper to put those weights on a common scale (standardize_weights).

Usage

compute_cross_efficiency(
  x,
  approach = c("benevolent", "aggressive"),
  epsilon = 0
)

compute_cross_efficiency_weights(
  x,
  approach = c("benevolent", "aggressive"),
  epsilon = 0,
  normalize = TRUE
)

standardize_weights(weights)

Arguments

x: A dea_data object, or a data frame coerced by as_dea_data.
approach: Secondary goal: "benevolent" (maximise others' weighted output) or "aggressive" (minimise it).
epsilon: Lower bound applied to every multiplier weight. The default 0 reproduces the plain model (weights may be zero); set a small positive value (e.g. 1e-6) to enforce strictly positive, non-Archimedean weights.
normalize: Logical; if TRUE (default) inputs and outputs are max-normalised column-wise before solving, so the returned weights are on a comparable scale (useful for plotting).
weights: A list with input_weights and output_weights matrices, such as the result of compute_cross_efficiency_weights.

Value

compute_cross_efficiency returns an \(n \times n\) numeric matrix whose entry [k, j] is the efficiency of DMU j evaluated with the weights of DMU k (rows are rating units, columns are rated units). Row and column names are the DMU labels.

compute_cross_efficiency_weights returns a list with matrices input_weights and output_weights (one row per DMU, labelled).

standardize_weights returns a list of the same shape as weights, row-standardised and rounded to 5 digits.

Functions

compute_cross_efficiency(): Compute the cross-efficiency matrix.
compute_cross_efficiency_weights(): Secondary-goal multiplier weights for every DMU.
standardize_weights(): Row-standardise a set of weights so that, within each DMU, the input weights sum to one and the output weights sum to one.

Secondary-goal model

Multiplier (weight) solutions in DEA are not unique, so cross-efficiency requires a secondary goal to choose among a unit's optimal weights. For unit \(o\) the model fixes the unit's own efficiency at its CRS score \(\theta_o\) and then, among all weight vectors that keep it there, optimises the total weighted output of the other units – maximising it for the "benevolent" approach, minimising it for the "aggressive" one. The unit's own efficiencies \(\theta_o\) and the fall-back weights come from a single dea solve.

References

Doyle, J., & Green, R. (1994). Efficiency and cross-efficiency in DEA: Derivations, meanings and uses. Journal of the Operational Research Society, 45(5), 567–578. doi:10.1057/jors.1994.84

Examples

df <- data.frame(
  dmu   = c("A", "B", "C", "D"),
  i_x1  = c(4, 7, 8, 4),
  i_x2  = c(3, 3, 1, 2),
  o_y   = c(1, 1, 1, 1)
)
ce <- compute_cross_efficiency(df, approach = "benevolent")
round(ce, 3)
#>      A     B   C D
#> A 1.00 0.571 0.5 1
#> B 0.75 0.632 1.0 1
#> C 0.75 0.632 1.0 1
#> D 0.75 0.632 1.0 1
colMeans(ce)   # each unit's average cross-efficiency
#>         A         B         C         D 
#> 0.8125000 0.6165414 0.8750000 1.0000000