# Cost Additive model

The cost additive model is computed by solving an input-oriented additive DEA model for the technical inefficiency.

The decomposition of cost inefficiency based on the weighted additive distance function measures input-oriented technical inefficiency based solely on input excesses, given by the following slack variables: $\mathbf{s}^-$$\mathbb{\in R}^M$. When the firm under evaluation ($\mathbf{x}_o$, $\mathbf{y}_o$) belongs to the production technology (as it is the case in cross-sectional studies), the DEA graph WADF model for measuring technical inefficiency is equivalent to the standard weighted additive model, which corresponds to the following DEA program:

$$$\begin{split} & T{{I}_{WADF\left( I \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},{{\rho }^{-}} \right)= \underset{\begin{smallmatrix} \pmb{s}^{-}, \pmb{\lambda} \end{smallmatrix}}{\mathop{\max }}\, \sum\limits_{m=1}^{M}{\rho _{m}^{-}s_{m}^{-}} \\ & s.t. \quad \sum\limits_{j=1}^{J}{{{\lambda }_{j}}{{x}_{jm}}}+s_{m}^{-}\le {{x}_{om}}, \quad m=1,...,M \\ & \quad \quad -\sum\limits_{j=1}^{J}{{{\lambda }_{j}}{{y}_{jn}}}\le -{{y}_{on}}, \quad n=1,...,N \\ & \quad \quad \sum\limits_{j=1}^{J}{{{\lambda }_{j}}}=1, \\ & \quad \quad s_{m}^{-}\ge 0, \quad m=1,...,M \\ & \quad \quad {{\lambda }_{j}}\ge 0, \quad j=1,...,J. \\ \end{split}$$$

For ($\mathbf{x}_o$, $\mathbf{y}_o$), this program seeks the maximum feasible reduction in inputs while remaining in $L(\textbf{y}_o)$. An observation is technically efficient if the optimal solution ($\mathbf{s}^{-*}, \mathbf{\lambda}^{*}$) is $\mathbf{s}^{-*}=0$, with $T{{I}_{WA\text{(}O\text{)}}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},{{\rho }^{-}}\right)=0$. Otherwise individual input reductions are feasible, and the larger the sum of the slacks, the larger the inefficiency.

The components of ${\rho}_{\textbf{x}}^{-}=\rho_{1}^{-},...,\rho_{M}^{-} \in R_{++}^{M}$ represent the relative importance of the unit inputs and are called input weights. Therefore, assigning unitary values, the previous program collapse to the standard input-oriented additive model. As with the Profit Additive model we can change the value of the weights to obtain specific DEA models of the family known as general efficiency measures (GEMs). The relevance of these transformations is that they make the additive measures independent of the units of measurement, which is a desirable property. In the accompanying documentation for this function presented below we present the different options that are available in BenchmarkingEconomicEfficiency.jl.

Following Pastor, Aparicio and Zofio (2022, Ch. 6) we can decompose normalized cost inefficiency into the technical inefficiency component and the residual allocative measure of cost inefficiency, i.e., $C{{I}_{WADF\left( I \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\rho _{{}}^{-},{\tilde{\textbf{w}}} \right)$ = $T{{I}_{WADF\left( I \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}} \right)$ + $A{{I}_{WADF\left( I \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\rho _{{}}^{-}, {\tilde{\textbf{w}}} \right)$:

$$$\underbrace{\frac{\sum\limits_{m=1}^{M}{{{w}_{m}}{{x}_{om}}} - C \left( \mathbf{y}_o,\mathbf{w} \right)}{\min \left\{ \frac{{{w}_{1}}}{\rho _{1}^{-}},...,\frac{{{w}_{M}}}{\rho _{M}^{-}} \right\}}}_{\text{Norm. Cost Inefficiency}}= \underbrace{ \sum\limits_{m=1}^{M}{\rho _{m}^{-}s_{m}^{-}}}_{\text{Input Technical Inefficiency}}+\underbrace{A{{I}_{WADF\left( I \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\rho _{{}}^{-},\tilde{\textbf{w}} \right)}_{\text{Norm. Allocative Inefficiency}}\ge 0, \\$$$

Reference

Chapter 6 in Pastor, J.T., Aparicio, J. and Zofío, J.L. (2022) Benchmarking Economic Efficiency: Technical and Allocative Fundamentals, International Series in Operations Research and Management Science, Vol. 315, Springer, Cham.

Example

In this example we compute the cost efficiency additive measure:

using BenchmarkingEconomicEfficiency

X = [2 2; 1 4; 4 1; 4 3; 5 5; 6 1; 2 5; 1.6 8];

Y = [1; 1; 1; 1; 1; 1; 1; 1];

W = [1 1; 1 1; 1 1; 1 1; 1 1; 1 1; 1 1; 1 1];

costadd = deacostadd(X, Y, W, :Ones)
Cost Additive DEA Model
DMUs = 8; Inputs = 2; Outputs = 1
Orientation = Input; Returns to Scale = VRS
Weights = Ones
──────────────────────────────
Cost  Technical  Allocative
──────────────────────────────
1   0.0        0.0         0.0
2   1.0        0.0         1.0
3   1.0        0.0         1.0
4   3.0        3.0         0.0
5   6.0        6.0         0.0
6   3.0        2.0         1.0
7   3.0        3.0         0.0
8   5.6        5.2         0.4
──────────────────────────────

Estimated economic, technical and allocative efficiency scores are returned with the efficiency function:

efficiency(costadd, :Economic)
8-element Vector{Float64}:
0.0
1.0
1.0
3.0
6.0
3.0
3.0
5.6
efficiency(costadd, :Technical)
8-element Vector{Float64}:
0.0
0.0
0.0
3.0
6.0
2.0000000000000004
3.0
5.200000000000001
efficiency(costadd, :Allocative)
8-element Vector{Float64}:
0.0
1.0
1.0
0.0
0.0
0.9999999999999996
0.0
0.3999999999999986

### deacostadd Function Documentation

BenchmarkingEconomicEfficiency.deacostaddFunction
deacostadditive(X, Y, W, model)

Compute cost efficiency using additive data envelopment analysis for inputs X, outputs Y and price of inputs W.

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

• rts=:VRS: chooses variable returns to scale. For constant returns to scale choose :CRS.
• rhoX: matrix of weights of inputs. Only if model=:Custom.
• disposal=:Strong: chooses strong disposal of outputs. For weak disposal choose :Weak.
• monetary=false: decomposition in normalized terms. Monetary terms if true.
• names: a vector of strings with the names of the decision making units.
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