The additive model measures technical efficiency based solely on input excesses and output shortfalls, and characterizes efficiency in terms of the input and output slacks: $\mathbf{s}^-\mathbb{\in R}^m$ and $\mathbf{s}^+$$\mathbb{\in R}^s$, respectively. . The package implements the weighted additive formulation of Cooper and Pastor (1995) and Pastor, Lovell and Aparicio (2011), whose associated linear program is:

\begin{aligned} & \underset{\mathbf{\lambda },\,{{\mathbf{s}}^{-}},\,{{\mathbf{s}}^{+}}}{\mathop{\max }}\,\quad \quad \quad \quad \omega =\mathbf{\rho_{x}^{-}}{{\mathbf{s}}^{\mathbf{-}}}+\mathbf{\rho_{y}^{+}}{{\mathbf{s}}^{+}} \\ & \text{subject}\ \text{to} \\ & \quad \quad \quad \quad \quad \quad X\mathbf{\lambda }+{{\mathbf{s}}^{\mathbf{-}}}= \ {{\mathbf{x}}_{o}} \\ & \quad \quad \quad \quad \quad \quad Y\mathbf{\lambda }-{{\mathbf{s}}^{+}}=\ {{\mathbf{y}}_{o}} \\ & \quad \quad \quad \quad \quad \quad \mathbf{e\lambda=1} \\ & \quad \quad \quad \quad \quad \quad \mathbf{\lambda }\ge \mathbf{0},\ {{\mathbf{s}}^{\mathbf{-}}}\ge 0,{{\mathbf{s}}^{+}}\ge 0, \end{aligned}

where $(\mathbf{\rho_{x}^{-}, \mathbf{\rho_{y}^{+}}}) \mathbb{\in R}^m_{+}\times \mathbb{R}_+^{s}$ are the inputs and outputs weight vectors whose elements can vary across DMUs.

In this example we compute the additive DEA model with all weights equal to one:

using DataEnvelopmentAnalysis

X = [5 13; 16 12; 16 26; 17 15; 18 14; 23 6; 25 10; 27 22; 37 14; 42 25; 5 17];

Y = [12; 14; 25; 26; 8; 9; 27; 30; 31; 26; 12];

deaadd(X, Y)
Weighted Additive DEA Model
DMUs = 11; Inputs = 2; Outputs = 1
Orientation = Graph; Returns to Scale = VRS
Weights = Ones
────────────────────────────────────────────────────
efficiency       slackX1  slackX2      slackY1
────────────────────────────────────────────────────
1    0.0           0.0              0.0  0.0
2    7.33333       4.33333          0.0  3.0
3    0.0           0.0              0.0  0.0
4   -8.03397e-16  -8.03397e-16      0.0  0.0
5   18.0          13.0              1.0  4.0
6    6.48305e-16   2.70127e-16      0.0  3.78178e-16
7    0.0           0.0              0.0  0.0
8    0.0           0.0              0.0  0.0
9    0.0           0.0              0.0  0.0
10  35.0          25.0             10.0  0.0
11   4.0           0.0              4.0  4.78849e-16
────────────────────────────────────────────────────

The same model is computed with:

deaadd(X, Y, :Ones)
Weighted Additive DEA Model
DMUs = 11; Inputs = 2; Outputs = 1
Orientation = Graph; Returns to Scale = VRS
Weights = Ones
────────────────────────────────────────────────────
efficiency       slackX1  slackX2      slackY1
────────────────────────────────────────────────────
1    0.0           0.0              0.0  0.0
2    7.33333       4.33333          0.0  3.0
3    0.0           0.0              0.0  0.0
4   -8.03397e-16  -8.03397e-16      0.0  0.0
5   18.0          13.0              1.0  4.0
6    6.48305e-16   2.70127e-16      0.0  3.78178e-16
7    0.0           0.0              0.0  0.0
8    0.0           0.0              0.0  0.0
9    0.0           0.0              0.0  0.0
10  35.0          25.0             10.0  0.0
11   4.0           0.0              4.0  4.78849e-16
────────────────────────────────────────────────────

The additive DEA model can be computed under constant returns to scale setting the rts parameter to :CRS:

deaadd(X, Y, :Ones, rts = :CRS)
Weighted Additive DEA Model
DMUs = 11; Inputs = 2; Outputs = 1
Orientation = Graph; Returns to Scale = CRS
Weights = Ones
──────────────────────────────────────────────────
efficiency       slackX1    slackX2  slackY1
──────────────────────────────────────────────────
1    0.0           0.0           0.0           0.0
2   10.7692        6.84615       3.92308       0.0
3   10.8378        0.0          10.8378        0.0
4   -8.03397e-16  -8.03397e-16   0.0           0.0
5   22.1538       12.7692        9.38462       0.0
6   17.9231       17.1154        0.807692      0.0
7    0.0           0.0           0.0           0.0
8   12.0769        7.38462       4.69231       0.0
9   11.6138       11.6138        0.0           0.0
10  35.0          25.0          10.0           0.0
11   4.0           0.0           4.0           0.0
──────────────────────────────────────────────────

The package can compute a wide class of different DEA models known as general efficiency measures (GEMs):

• The measure of inefficiency proportions (MIP).
• The normalized weighted additive DEA model.
• The range adjusted measure (RAM).
• The bounded adjusted measure (BAM).

## Measure of Inefficiency Proportions (MIP)

The measure of inefficiency proportions (MIP), Charnes et al. (1987) and Cooper et al. (1999), use the weights:

$$$(\mathbf{\rho_{x}^{-}, \mathbf{\rho_{y}^{+}}})=(1/{\mathbf{x}}_{o},1/{{\mathbf{y}}_{o}})$$$
deaadd(X, Y, :MIP)
Weighted Additive DEA Model
DMUs = 11; Inputs = 2; Outputs = 1
Orientation = Graph; Returns to Scale = VRS
Weights = MIP
─────────────────────────────────────────────────────
efficiency       slackX1  slackX2       slackY1
─────────────────────────────────────────────────────
1    0.0           0.0              0.0   0.0
2    0.507519      0.0              0.0   7.10526
3    0.0           0.0              0.0   0.0
4   -4.72586e-17  -8.03397e-16      0.0   0.0
5    2.20395       0.0              0.0  17.6316
6    1.31279e-16   8.10382e-16      0.0   8.64407e-16
7    0.0           0.0              0.0   0.0
8    0.0           0.0              0.0   0.0
9    0.0           0.0              0.0   0.0
10   1.04322      17.0             15.0   1.0
11   0.235294      0.0              4.0   0.0
─────────────────────────────────────────────────────

The normalized weighted additive DEA model, Lovell and Pastor (1995), use the weights:

$$$(\mathbf{\rho_{x}^{-}, \mathbf{\rho_{y}^{+}}})=(1/{\mathbf{σ^-}},1/{{\mathbf{σ^+}}})$$$

where $\mathbf{σ^-}$and $\mathbf{σ^+}$ are the standard deviations of inputs and outputs respectively.

deaadd(X, Y, :Normalized)
Weighted Additive DEA Model
DMUs = 11; Inputs = 2; Outputs = 1
Orientation = Graph; Returns to Scale = VRS
Weights = Normalized
──────────────────────────────────────────────────────────
efficiency       slackX1       slackX2       slackY1
──────────────────────────────────────────────────────────
1    0.0           0.0           0.0           0.0
2    0.804925      0.0           0.65          6.25
3    0.0           0.0           0.0           0.0
4   -9.79609e-17   0.0          -6.09909e-16   0.0
5    2.01497       0.0           2.95         13.75
6    4.81529e-16   2.49057e-15   0.0           2.37658e-15
7    0.0           0.0           0.0           0.0
8    0.0           0.0           0.0           0.0
9    0.0           0.0           0.0           0.0
10   3.98989      17.0          15.0           1.0
11   0.642462      0.0           4.0           0.0
──────────────────────────────────────────────────────────

The range adjusted measure (RAM), Cooper et al. (1999), use the weights::

$$$(\mathbf{\rho^{-}, \mathbf{\rho^{+}}})=(1/(m+s)R^-,(1/(m+s)R^+)$$$

where $R^-$and $R^+$are the inputs and outputs variables' ranges.

deaadd(X, Y, :RAM)
Weighted Additive DEA Model
DMUs = 11; Inputs = 2; Outputs = 1
Orientation = Graph; Returns to Scale = VRS
Weights = RAM
──────────────────────────────────────────────────────────
efficiency       slackX1       slackX2       slackY1
──────────────────────────────────────────────────────────
1    0.0           0.0           0.0           0.0
2    0.102975      0.0           0.0           7.10526
3    0.0           0.0           0.0           0.0
4   -1.01651e-17   0.0          -6.09909e-16   0.0
5    0.25553       0.0           0.0          17.6316
6    5.68808e-17   2.49057e-15   0.0           2.37658e-15
7    0.0           0.0           0.0           0.0
8    0.0           0.0           0.0           0.0
9    0.0           0.0           0.0           0.0
10   0.417646     17.0          15.0           1.0
11   0.0666667     0.0           4.0           0.0
──────────────────────────────────────────────────────────

The bounded adjusted measure (BAM), Cooper et al. (2011), use the weights:::

$$$(\mathbf{\rho_{x}^{-}, \mathbf{\rho_{y}^{+}}})=(1/(m+s)({\mathbf{x}}_{o}-{\mathbf{\underline{x}}}),(1/(m+s)({\mathbf{\overline{y}}} - {\mathbf{y}}_{o})$$$

where $\mathbf{\underline{x}}$ and $\mathbf{\overline{y}}$ are the minimum and maximum observed values of inputs and outputs respectively.

deaadd(X, Y, :BAM)
Weighted Additive DEA Model
DMUs = 11; Inputs = 2; Outputs = 1
Orientation = Graph; Returns to Scale = VRS
Weights = BAM
─────────────────────────────────────────────────
efficiency   slackX1  slackX2       slackY1
─────────────────────────────────────────────────
1    0.0           0.0          0.0   0.0
2    0.199894      6.59649      0.0   0.0
3    0.0           0.0          0.0   0.0
4   -3.78838e-17   0.0          0.0  -5.68256e-16
5    0.432971     13.0          1.0   4.0
6    1.11554e-17   0.0          0.0   7.36254e-16
7    0.0           0.0          0.0   0.0
8    0.0           0.0          0.0   0.0
9    0.0           0.0          0.0   0.0
10   0.571361      5.0         11.0   5.0
11   0.121212      0.0          4.0   0.0
─────────────────────────────────────────────────

DataEnvelopmentAnalysis.deaaddFunction
deaadd(X, Y, model)

Compute related data envelopment analysis weighted additive models for inputs X and outputs Y.

Model specification:

• :Ones: standard additive DEA model.
• :MIP: Measure of Inefficiency Proportions. (Charnes et al., 1987; Cooper et al., 1999)
• :Normalized: Normalized weighted additive DEA model. (Lovell and Pastor, 1995)
• :RAM: Range Adjusted Measure. (Cooper et al., 1999)
• :BAM: Bounded Adjusted Measure. (Cooper et al, 2011)
• :Custom: User supplied weights.

Optional Arguments

• orient=:Graph: choose between graph oriented :Graph, input oriented :Input, or output oriented model :Output.
• rts=:VRS: choose between constant returns to scale :CRS or variable returns to scale :VRS.
• rhoX: matrix of weights of inputs. Only if model=:Custom.
• rhoY: matrix of weights of outputs. Only if model=:Custom.
• Xref=X: Identifies the reference set of inputs against which the units are evaluated.
• Yref=Y: Identifies the reference set of outputs against which the units are evaluated.
• disposX=:Strong: chooses strong disposability of inputs. For weak disposability choose :Weak.
• disposY=:Strong: chooses strong disposability of outputs. For weak disposability choose :Weak.
• names: a vector of strings with the names of the decision making units.
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