model_additive function

Additive DEA model.

Solve the additive model of Charnes et. al (1985). With the current version of deaR, it is possible to solve input-oriented, output-oriented, and non-oriented additive model under constant and non-constant returns to scale.

Besides, the user can set weights for the input slacks and/or output slacks. So, it is also possible to solve weighted additive models. For example: Measure of Inefficiency Proportions (MIP), Range Adjusted Measure (RAM), etc.

model_additive(datadea,
               dmu_eval = NULL,
               dmu_ref = NULL,
               orientation = NULL,
               weight_slack_i = 1,
               weight_slack_o = 1,
               rts = c("crs", "vrs", "nirs", "ndrs", "grs"),
               L = 1,
               U = 1,
               compute_target = TRUE,
               returnlp = FALSE,
               ...)

Arguments

• datadea: A deadata object with n DMUs, m inputs and s outputs.
• dmu_eval: A numeric vector containing which DMUs have to be evaluated. 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.
• orientation: This parameter is either NULL (default) or a string, equal to "io" (input-oriented) or "oo" (output-oriented). It is used to modify the weight slacks. If input-oriented, weight_slack_o are taken 0. If output-oriented, weight_slack_i are taken 0.
• weight_slack_i: A value, vector of length m, or matrix m x ne (where ne is the length of dmu_eval) with the weights of the input slacks. If 0, output-oriented.
• weight_slack_o: A value, vector of length s, or matrix s x ne (where ne is the length of dmu_eval) with the weights of the output slacks. If 0, input-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).
• compute_target: Logical. If it is TRUE, it computes targets.
• returnlp: Logical. If it is TRUE, it returns the linear problems (objective function and constraints).
• ...: Ignored, for compatibility issues.

Note

In this model, the efficiency score is the sum of the slacks. Therefore, a DMU is efficient when the objective value (objval) is zero.

Examples

# Example 1. 
# Replication of results in Charnes et. al (1994, p. 27)
x <- c(2, 3, 6, 9, 5, 4, 10) 
y <- c(2, 5, 7, 8, 3, 1, 7)
data_example <- data.frame(dmus = letters[1:7], x, y)
data_example <- make_deadata(data_example, 
                             ni = 1, 
                             no = 1)
result <- model_additive(data_example, 
                         rts = "vrs")
efficiencies(result)
slacks(result)
lambdas(result)

# Example 2.
# Measure of Inefficiency Proportions (MIP).
x <- c(2, 3, 6, 9, 5, 4, 10) 
y <- c(2, 5, 7, 8, 3, 1, 7)
data_example <- data.frame(dmus = letters[1:7], x, y)
data_example <- make_deadata(data_example,
                             ni = 1,
                             no = 1)
result2 <- model_additive(data_example,
                          rts = "vrs",
                          weight_slack_i = 1 / data_example[["input"]],
                          weight_slack_o = 1 / data_example[["output"]])
slacks(result2)

# Example 3.
# Range Adjusted Measure of Inefficiencies (RAM).
x <- c(2, 3, 6, 9, 5, 4, 10) 
y <- c(2, 5, 7, 8, 3, 1, 7)
data_example <- data.frame(dmus = letters[1:7], x, y)
data_example <- make_deadata(data_example,
                             ni = 1,
                             no = 1)
range_i <- apply(data_example[["input"]], 1, max) -
           apply(data_example[["input"]], 1, min)
range_o <- apply(data_example[["output"]], 1, max) -
           apply(data_example[["output"]], 1, min)
w_range_i <- 1 / (range_i * (dim(data_example[["input"]])[1] +
                             dim(data_example[["output"]])[1]))
w_range_o <- 1 / (range_o * (dim(data_example[["input"]])[1] +
                             dim(data_example[["output"]])[1]))
result3 <- model_additive(data_example,
                          rts = "vrs",
                          weight_slack_i = w_range_i,
                          weight_slack_o = w_range_o)
slacks(result3)

References

Charnes, A.; Cooper, W.W.; Golany, B.; Seiford, L.; Stuz, J. (1985) "Foundations of Data Envelopment Analysis for Pareto-Koopmans Efficient Empirical Production Functions", Journal of Econometrics, 30(1-2), 91-107. tools:::Rd_expr_doi("10.1016/0304-4076(85)90133-2")

Charnes, A.; Cooper, W.W.; Lewin, A.Y.; Seiford, L.M. (1994). Data Envelopment Analysis: Theory, Methology, and Application. Boston: Kluwer Academic Publishers. tools:::Rd_expr_doi("10.1007/978-94-011-0637-5")

Cooper, W.W.; Park, K.S.; Pastor, J.T. (1999). "RAM: A Range Adjusted Measure of Inefficiencies for Use with Additive Models, and Relations to Other Models and Measures in DEA". Journal of Productivity Analysis, 11, p. 5-42. tools:::Rd_expr_doi("10.1023/A:1007701304281")

See Also

model_addsupereff

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)