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Data Envelopment Analysis and Malmquist Productivity Index

Source: R/DEA.R
DEA.Rd

Computes standard and super-efficiency Data Envelopment Analysis (DEA) models (CCR, BCC, and slacks-based), including efficiency scores, slacks, lambdas, targets, returns to scale, and references, with support for undesirable outputs. Additionally, calculates the Malmquist productivity index to measure productivity changes over time, including efficiency change (EC), technical change (TC), and other decomposition components.

Usage

basic_DEA(
  data,
  inputs,
  outputs,
  ud_outputs = NULL,
  orientation = "io",
  rts = "vrs"
)

super_DEA(
  data,
  inputs,
  outputs,
  ud_outputs = NULL,
  orientation = "io",
  rts = "vrs"
)

basic_SBM(
  data,
  inputs,
  outputs,
  ud_outputs = NULL,
  orientation = "io",
  rts = "vrs"
)

super_SBM(data, inputs, outputs, orientation = "io", rts = "vrs")

malmquist(
  data,
  period,
  inputs,
  outputs,
  orientation = "oo",
  rts = "vrs",
  type1 = "glob",
  type2 = "rd"
)

Arguments

data

For DEA functions (basic_DEA, super_DEA, basic_SBM, super_SBM): A data frame where the first column contains DMU (Decision Making Unit) names/identifiers, and subsequent columns are input/output variables. For malmquist: A long-format data frame with a period column, DMU identifiers, and input/output variables.

inputs

A numeric vector of column indices or a character vector of column names indicating input variables.

outputs

A numeric vector of column indices or a character vector of column names indicating (desirable) output variables.

ud_outputs

Optional. A numeric vector indicating the position of undesirable outputs within the outputs parameter. Defaults to NULL. Not applicable for super_SBM or malmquist.

orientation

Character string. Model orientation: "io" (input-oriented, default for DEA functions) or "oo" (output-oriented, default for malmquist).

rts

Character string. Returns to scale assumption: "vrs" (variable returns to scale, default) or "crs" (constant returns to scale).

period

For malmquist only: A numeric or character index/name indicating the column in data containing time periods.

type1

For malmquist only: Reference technology for Malmquist index: "cont" (contemporary), "seq" (sequential), or "glob" (global, default).

type2

For malmquist only: Decomposition method for Malmquist index: "fgnz" (Färe et al., 1994), "rd" (Ray and Desli, 1997, default), "gl" (generalized), or "bias" (biased).

Value

For DEA functions (basic_DEA, super_DEA, basic_SBM, super_SBM): A list containing six elements:

  • efficiencies: A named numeric vector of efficiency scores for each DMU, standardized to (0, 1] for both input- and output-oriented models.

  • slacks: A data frame or matrix containing slack values for inputs and outputs (including undesirable outputs, if specified).

  • lambdas: A matrix or data frame of intensity variables (\(\lambda\)), representing the contribution of reference DMUs to the efficiency frontier (self-excluded in super-efficiency models).

  • targets: A data frame or matrix of efficient target values for inputs and outputs, adjusted for undesirable outputs in DDF models.

  • returns: A character vector indicating returns-to-scale status for each DMU: "crs" (constant), "irs" (increasing), or "drs" (decreasing).

  • references: A matrix or data frame listing reference DMUs (peers) contributing to the efficiency frontier (\(\lambda > 0\)).

For malmquist: A data frame containing:

  • Period: Time period transitions (e.g., "t~t+1").

  • DMU: Decision Making Unit identifiers.

  • mi: Malmquist productivity index, measuring total productivity change.

  • ec: Efficiency change (EC), computed directly or as mi / tc if not available from the model.

  • tc: Technical change (TC), measuring frontier shift.

  • pech: Pure efficiency change (if applicable, based on decomposition method).

  • sech: Scale efficiency change (if applicable, based on decomposition method).

Details

This package provides a unified interface for computing efficiency scores and productivity changes using the deaR package. It includes five functions: basic_DEA, super_DEA, basic_SBM, super_SBM, and malmquist, each tailored to specific DEA or productivity analysis models.

  • DEA Models:

    • basic_DEA: Implements standard radial DEA models (CCR for CRS, BCC for VRS) as described by Charnes et al. (1978) and Banker et al. (1984), optimizing radial efficiency (input contraction or output expansion).

    • super_DEA: Computes super-efficiency radial DEA, excluding the evaluated DMU from the reference set to allow efficiency scores beyond 1 (output-oriented) or below 1 (input-oriented) for efficient DMUs (Andersen & Petersen, 1993).

    • basic_SBM: Implements standard Slacks-Based Measure (SBM) models (Tone, 2001), optimizing input and output slacks for a non-radial efficiency measure.

    • super_SBM: Combines SBM with super-efficiency properties, excluding the evaluated DMU from the reference set. Note: super_SBM does not support undesirable outputs.

  • Malmquist Productivity Index:

    • malmquist: Calculates the Malmquist productivity index to measure productivity changes over time, decomposing it into efficiency change (EC) and technical change (TC), with optional further decomposition into pure efficiency change (PECH) and scale efficiency change (SECH) based on type2 (Färe et al., 1994; Ray & Desli, 1997). If EC is unavailable (e.g., under rts = "vrs" and type1 = "glob"), it is computed as ec = mi / tc to ensure consistent output.

Orientation:

  • Input-oriented ("io"): Minimizes inputs while maintaining outputs. Efficiency scores are in \((0, 1]\) (\(\theta \leq 1\) for radial models, \(\rho\) or \(\delta \leq 1\) for SBM models).

  • Output-oriented ("oo"): Maximizes outputs for given inputs. Efficiency scores are in \((0, 1]\). Radial models output \(\eta \geq 1\), converted to \(1/\eta\); SBM models output \(1/\rho^*\) or \(1/\delta\).

Returns to Scale (RTS):

  • CRS ("crs"): Assumes constant returns to scale, suitable for long-run analysis.

  • VRS ("vrs"): Allows variable returns to scale, with increasing ("irs") or decreasing ("drs") returns determined by the sum of intensity variables (\(\lambda\)).

Undesirable Outputs:

  • Supported in basic_DEA, super_DEA, and basic_SBM using directional distance functions (DDF) with direction vector \((g_y, -g_b)\) to increase desirable outputs and decrease undesirable outputs (Färe & Grosskopf, 2004). Not supported in super_SBM or malmquist.

Malmquist-Specific Parameters:

  • type1: Defines the reference technology for the Malmquist index:

    • "cont": Contemporary technology, using each period's frontier.

    • "seq": Sequential technology, incorporating all prior periods.

    • "glob": Global technology, using a single frontier across all periods.

  • type2: Specifies the decomposition method:

    • "fgnz": Färe et al. (1994) decomposition.

    • "rd": Ray and Desli (1997) decomposition.

    • "gl": Generalized decomposition.

    • "bias": Bias-corrected decomposition.

Handling NA Values:

  • Super-efficiency models (super_DEA, super_SBM) may return NA for efficient DMUs due to infeasible linear programming solutions, especially under VRS (Andersen & Petersen, 1993). Users can replace NA with standard efficiency scores or exclude affected DMUs.

  • For malmquist, if ec is NULL (e.g., under rts = "vrs" and type1 = "glob"), it is computed as mi / tc to ensure a complete result.

The package leverages deaR for robust computation, handling zero values internally and ensuring compatibility with input/output specifications. Efficiency scores are standardized to \((0, 1]\) for DEA models, and Malmquist results are formatted as a data frame for easy analysis.

Examples

# Sample DEA data
data_dea = data.frame(
  DMU = paste0("DMU", 1:5),
  input1 = c(10, 20, 15, 25, 30),
  input2 = c(5, 8, 7, 10, 12),
  output = c(100, 150, 120, 180, 200),
  ud_output = c(10, 15, 12, 20, 25)
)

# Standard DEA
result = basic_DEA(data_dea, inputs = 2:3, outputs = 4)
#> Warning: We note that we call 'targets' to the 'efficient projections' in the
#>           strongly efficient frontier.
result$efficiencies
#>      DMU1      DMU2      DMU3      DMU4      DMU5 
#> 1.0000000 1.0000000 0.9166667 1.0000000 1.0000000 

# DEA with undesirable outputs
result = basic_DEA(data_dea, inputs = 2:3, outputs = 4:5, ud_outputs = 2)
#> Warning: Undesirable (bad) outputs with no output-oriented model.
#> Warning: We note that we call 'targets' to the 'efficient projections' in the
#>           strongly efficient frontier.
result$efficiencies
#>      DMU1      DMU2      DMU3      DMU4      DMU5 
#> 1.0000000 1.0000000 0.9333333 1.0000000 1.0000000 

# Super-efficiency DEA
result = super_DEA(data_dea, inputs = 2:3, outputs = 4)
#> Warning: We note that we call 'targets' to the 'efficient projections' in the
#>           strongly efficient frontier.
result$efficiencies
#>      DMU1      DMU2      DMU3      DMU4      DMU5 
#> 1.5000000 1.0156250 0.9166667 1.0400000        NA 

# Standard SBM
result = basic_SBM(data_dea, inputs = 2:3, outputs = 4)
#> Warning: We note that we call 'targets' to the 'efficient projections' in the
#>           strongly efficient frontier.
result$efficiencies
#>      DMU1      DMU2      DMU3      DMU4      DMU5 
#> 1.0000000 1.0000000 0.9047619 1.0000000 1.0000000 

# Super-efficiency SBM
result = super_SBM(data_dea, inputs = 2:3, outputs = 4)
#> Warning: For oriented models and non constant returns to scale, feasibility is not
#>              guaranteed. Proceed with caution, some DMUs results may be missing!
#> Warning: We note that we call 'targets' to the 'efficient projections' in the
#>           strongly efficient frontier.
result$efficiencies
#>     DMU1     DMU2     DMU3     DMU4     DMU5 
#> 1.450000 1.007812 1.000000 1.040000       NA 

# Sample Malmquist data (long format)
data_malm = data.frame(
  DMU = rep(paste0("DMU", 1:5), 3),
  Period = rep(1:3, each = 5),
  input1 = c(10, 20, 15, 25, 30, 12, 22, 17, 27, 32, 14, 24, 19, 29, 34),
  input2 = c(5, 8, 7, 10, 12, 6, 9, 8, 11, 13, 7, 10, 9, 12, 14),
  output = c(100, 150, 120, 180, 200, 110, 160, 130, 190, 210, 120, 170, 140, 200, 220)
)
malmquist(data_malm, period = 2, inputs = 3:4, outputs = 5)
#> # A tibble: 10 × 7
#>    Period DMU      mi    ec    tc  pech  sech
#>    <chr>  <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
#>  1 1~2    DMU1  0.917 0.922 0.994  1    0.922
#>  2 1~2    DMU2  0.948 0.972 0.976  1    0.972
#>  3 1~2    DMU3  0.948 0.953 0.995  1.00 0.949
#>  4 1~2    DMU4  0.949 0.949 1      1    0.949
#>  5 1~2    DMU5  0.923 0.923 1      1    0.923
#>  6 2~3    DMU1  0.936 0.940 0.995  1    0.940
#>  7 2~3    DMU2  0.976 0.979 0.998  1    0.979
#>  8 2~3    DMU3  0.958 0.962 0.996  1.00 0.958
#>  9 2~3    DMU4  0.955 0.955 1      1    0.955
#> 10 2~3    DMU5  0.929 0.929 1      1    0.929