Additive Models
Weighted Additive Model
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
─────────────────────────────────────────────────────Normalized Weighted Additive Model
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
──────────────────────────────────────────────────────────Range Adjusted Measure (RAM)
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
──────────────────────────────────────────────────────────Bounded Adjusted Measure (BAM)
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
─────────────────────────────────────────────────deaadd Function Documentation
DataEnvelopmentAnalysis.deaadd — Functiondeaadd(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:CRSor variable returns to scale:VRS.rhoX: matrix of weights of inputs. Only ifmodel=:Custom.rhoY: matrix of weights of outputs. Only ifmodel=: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.