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

deacostadditive(X, Y, W, model)