cross_efficiency function

Cross efficiency analysis

Computes arbitrary, benevolent and aggressive formulations of cross-efficiency under any returns-to-scale. Doyle and Green (1994) present three alternatives ways of formulating the secondary goal (wich will minimize or maximize the other DMUs' cross-efficiencies in some way). Methods II and III are implemented in deaR with any returns-to-scale. The maverick index is also calculated.

cross_efficiency(datadea,
                 dmu_eval = NULL,
                 dmu_ref = NULL,
                 epsilon = 0, 
                 orientation = c("io", "oo"),
                 rts = c("crs", "vrs", "nirs", "ndrs", "grs"),
                 L = 1,
                 U = 1,
                 selfapp = TRUE,
                 correction = FALSE,
                 M2 = TRUE,
                 M3 = TRUE)

Arguments

• datadea: An object of class dea or deadata. If it is of class dea it must have been obtained with some of the multiplier DEA models.

• dmu_eval: A numeric vector. Only the multipliers of DMUs in dmu_eval

  are computed. If NULL (default), all DMUs are considered.

• dmu_ref: A numeric vector containing which DMUs are the evaluation reference set. If NULL (default), all DMUs are considered.

• epsilon: Numeric, multipliers must be >= epsilon.

• orientation: A string, equal to "io" (input-oriented) or "oo" (output-oriented).

• rts: A string, determining the type of returns to scale, equal to "crs" (constant), "vrs" (variable), "nirs" (non-increasing), "ndrs" (non-decreasing) or "grs" (generalized).

• L: Lower bound for the generalized returns to scale (grs).

• U: Upper bound for the generalized returns to scale (grs).

• selfapp: Logical. If it is TRUE, self-appraisal is included in the average scores of A and e.

• correction: Logical. If it is TRUE, a correction is applied in the "vrs" input-oriented model in order to avoid negative cross-efficiencies, according to Lim & Zhu (2015).

• M2: Logical. If it is TRUE, it computes Method II for aggresive/benevolent estimations.

• M3: Logical. If it is TRUE, it computes Method III for aggresive/benevolent estimations.

Note

(1) We can obtain negative cross-efficiency in the input-oriented DEA model under no constant returns-to-scale. However, the same does not happen in the case of the output-oriented VRS DEA model. For this reason, the proposal of Lim and Zhu (2015) is implemented in deaR to calculate the input-oriented cross-efficiency model under no constant returns-to-scale.

(2) The multiplier model can have alternate optimal solutions (see note 1 in model_multiplier). So, depending on the optimal weights selected we can obtain different cross-efficiency scores.

Examples

# Example 1.
# Arbitrary formulation. Input-oriented model under constant returns-to-scale.
data("Golany_Roll_1989")
data_example <- make_deadata(datadea = Golany_Roll_1989, 
                             inputs = 2:4, 
                             outputs = 5:6)
result <- cross_efficiency(data_example, 
                           orientation = "io", 
                           rts = "crs", 
                           selfapp = TRUE)
result$Arbitrary$cross_eff
result$Arbitrary$e

# Example 2.
# Benevolent formulation (method II). Input-oriented.
data("Golany_Roll_1989")
data_example <- make_deadata(datadea = Golany_Roll_1989, 
                             inputs = 2:4, 
                             outputs = 5:6)
result <- cross_efficiency(data_example, 
                           orientation = "io", 
                           selfapp = TRUE)
result$M2_ben$cross_eff
result$M2_ben$e

# Example 3.
# Benevolent formulation (method III). Input-oriented.
data("Golany_Roll_1989")
data_example <- make_deadata(datadea = Golany_Roll_1989, 
                             inputs = 2:4, 
                             outputs = 5:6)
result <- cross_efficiency(data_example, 
                           orientation = "io", 
                           selfapp = TRUE)
result$M3_ben$cross_eff
result$M3_ben$e
  
# Example 4.
# Arbitrary formulation. Output-oriented.
data("Golany_Roll_1989")
data_example <- make_deadata(datadea = Golany_Roll_1989,
                             inputs = 2:4, 
                             outputs = 5:6)
result <- cross_efficiency(data_example, 
                           orientation = "oo", 
                           selfapp = TRUE)
result$Arbitrary$cross_eff
result$Arbitrary$e

# Example 5.
# Arbitrary formulation. Input-oriented model under vrs returns-to-scale.
data("Lim_Zhu_2015")
data_example <- make_deadata(Lim_Zhu_2015,
                             ni = 1, 
                             no = 5)
cross <- cross_efficiency(data_example,
                          epsilon = 0,
                          orientation = "io",
                          rts = "vrs",
                          selfapp = TRUE,
                          M2 = FALSE,
                          M3 = FALSE)
cross$Arbitrary$e

References

Sexton, T.R., Silkman, R.H.; Hogan, A.J. (1986). Data envelopment analysis: critique and extensions. In: Silkman RH (ed) Measuring efficiency: an assessment of data envelopment analysis, vol 32. Jossey-Bass, San Francisco, pp 73–104. tools:::Rd_expr_doi("10.1002/ev.1441")

Doyle, J.; Green, R. (1994). “Efficiency and cross efficiency in DEA: derivations, meanings and the uses”, Journal of Operational Research Society, 45(5), 567–578. tools:::Rd_expr_doi("10.2307/2584392")

Cook, W.D.; Zhu, J. (2015). DEA Cross Efficiency. In: Zhu, J. (ed) Data Envelopment Analysis. A Handbook of Models and Methods. International Series in Operations Research & Management Science, vol 221. Springer, Boston, MA, 23-43. tools:::Rd_expr_doi("10.1007/978-1-4899-7553-9_2")

Lim, S.; Zhu, J. (2015). "DEA Cross-Efficiency Under Variable Returns to Scale". Journal of Operational Research Society, 66(3), p. 476-487. tools:::Rd_expr_doi("10.1057/jors.2014.13")

See Also

model_multiplier, cross_efficiency_fuzzy

Author(s)

Vicente Coll-Serrano (vicente.coll@uv.es ). Quantitative Methods for Measuring Culture (MC2). Applied Economics.

Vicente Bolós (vicente.bolos@uv.es ). Department of Business Mathematics

Rafael Benítez (rafael.suarez@uv.es ). Department of Business Mathematics

University of Valencia (Spain)