# 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. ```r 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 ```r # 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)

cross_efficiency function