# Revenue Reverse Directional Distance Function model

The cost reverse directional distance function model is computed by solving an input-oriented Reverse DDF model for the technical inefficiency.

As their graph counterpart presented in the Profit Reverse Directional Distance Function model, the output-oriented $RDDF$ transforms any additive measure of output technical inefficiency, $EM(O)$, such as the Russell output measure, into a single scalar measure corresponding to an output-oriented $DDF$, $\beta_O$. Therefore, given the set of $J$ firms under study, ${F}_{J}$, and their projections on the frontier, denoted by ${\hat{F}_{J}}$, the $RDDF$ assigns a new $DDF$ score to the original $EM(O)$, while keeping the same projections ${\hat{F}_{J}}$. We denote this score by $\beta_{RDDF(EM(O), F_J, \hat{F}_{J})}$. The advantage of the RDDF is that it relates the additive and multiplicative measures of technical inefficiency because the ouput-oriented DDFs is equivalent to the Farrell's output radial measures shown in the Revenue Radial model, i.e., $\beta_O^{*}=\phi^*-1$.

To calculate this distance functions for firm $\left( {{\mathbf{x}_o,\mathbf{y}_{o}}} \right)$ we need to determine the directional vector $\mathbf{g}= ({\textbf{0}_N},{\mathbf{g_{x}^+}})$ connecting the firm to its output-oriented projection, $\left( {{{\textbf{x}}}_{o}},{{{\hat{\textbf{y}}}}_{o}} \right)\in {{\hat{F}}_{J}}$. Afterwards we calculate the value of the $RDDF(O)$. However, when calculating the new scores we need to differentiate between firms that are deemed technically efficient with $EM(O)=0$ and those that are technically inefficient. The measure $EM(O)$ splits the sample into two disjoint subsets: ${{F}_{E}}=\left\{ \left( {{x}_{j}},{{y}_{j}} \right)\in {{F}_{J}}:T{{I}_{E{{M(O)}}}}\left( {{x}_{j}},{{y}_{j}} \right)=0 \right\}$ and ${{F}_{J\sim E}}=\left\{ \left( {{x}_{j}},{{y}_{j}} \right)\in {{F}_{J}}:T{{I}_{E{{M(O)}}}}\left( {{x}_{j}},{{y}_{j}} \right)>0 \right\}$. Then, for the output orientation we define the directional vector joining the firm under evaluation and its projection as follows

• If $\left( {{\textbf{x}}_{j}},{{\textbf{y}}_{j}} \right)\in {{F}_{J\sim E}}$, define
$$$\left({\textbf{g}^{-}_{\textbf{x}_{j}}},{\textbf{g}^{+}_{\textbf{y}_{j}}} \right)= \left( \textbf{0}_M, \frac{{{{\mathbf{\hat{y}}}}_{j}}-{{\mathbf{y}}_{j}}}{T{{I}_{EM\left( G \right)}}\left( {{\mathbf{x}}_{j}},{{\mathbf{y}}_{j}} \right)} \right) \, \, and \, \,\beta_{RDDF(O)}^{*}=T{{I}_{E{{M}(O)}}}\left( {{\textbf{x}}_{j}},{{\textbf{y}}_{j}} \right)>0.$$$
• If $\left( {{\textbf{x}}_{j}},{{\textbf{y}}_{j}} \right)\in {{F}_{E}}$, define
$$$\beta _{RDDF(I)}^{*}=T{{I}_{EM\left( O \right)}}\left( {{\textbf{x}}_{j}},{{\textbf{y}}_{j}} \right)=0 \, \, and \, \,\left( \textbf{0}_N, {\textbf{g}^{+}_{\textbf{y}_{j}}} \right)= \left(\textbf{0}_M,{{{\vec{\textbf{k}}}}_{jN}} \right)\,\in \mathbb{R}_{++}^{M+N},$$$

where ${{\vec{\textbf{k}}}_{jN}}\in \mathbb{R}_{++}^{N}$ is a vector whose units of measurement are identical to those of the firm under evaluation $\left( {{\textbf{x}}_{j}},{{\textbf{y}}_{j}} \right)\in {{F}_{E}}$–making revenue inefficiency units' invariant. For consistency we search for a value that yields a normalization factor whose value is equal to that associated to the underlying efficiency measure, i.e, $\sum\limits_{n=1}^{N}{p_{n}^{{}}k_{jn}^{+}}= NF_{EM(O)}$. This choice of ${{\vec{\textbf{k}}}_{jN}}$ is numerically relevant because it makes the values of normalized cost inefficiency based on the $RDDF$ and the original $EM(O)$ equivalent, and therefore their normalized allocative efficiencies can be compared to each other–their technical inefficiency values being null.

We now present the expressions corresponding to the revenue inefficiency measure and its decomposition associated with the $RDDF$. This results in $R{{I}_{{RDDF\left( EM\left( O \right),{{F}_{J}},{{{\hat{F}}}_{J}} \right)}}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\mathbf{g_{y}^+},{\tilde{\textbf{p}}} \right)$ = $T{{I}_{{RDDF\left( EM\left( O \right),{{F}_{J}},{{{\hat{F}}}_{J}} \right)}}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o},\mathbf{g_{y}^+}} \right)$ + $A{{I}_{RDDF\left( EM\left( O \right),{{F}_{J}},{{{\hat{F}}}_{J}} \right)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\mathbf{g_{y}^+}, {\tilde{\textbf{p}}} \right)$. Hence we obtain the following expression:

$$$\underbrace{\frac{R \left( \mathbf{x}_o,\mathbf{p} \right) - \sum\limits_{n=1}^{N}{{{p}_{n}}{{y}_{om}}} }{\sum\limits_{n=1}^{N}{{{p}_{n}}{{g}^{-}_{on}}}}}_{\text{Norm. Revenue Inefficiency}}=\underbrace{\beta _{RDDF\left( EM\left( O \right),{{F}_{J}},{{{\hat{F}}}_{J}} \right)}^{*}}_{\text{Output Technical Inefficiency}}+\underbrace{A{{I}_{RDDF\left( EM\left( O \right),{{F}_{J}},{{{\hat{F}}}_{J}} \right)}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\textbf{g}_{{{\textbf{y}}_{o}}}^{{+}}, \tilde{\textbf{p}} \right)}}_{\text{Norm. Allocative Inefficiency}} \ge 0,$$$

where the efficiency scores $\beta _{RDDF\left( EM\left( O \right),{{F}_{J}},{{{\hat{F}}}_{J}} \right)}^{*}$ for technically inefficient firms is obtained by solving the Output-oriented Directional Distance Function model with the corresponding directional vectors. .

BenchmarkingEconomicEfficiency.jl with Julia offers the possibility of decomposing revenue inefficiency based on the $RDDF$ considering as original measure the Russell output-oriented model.

Reference

Chapter 12 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 revenue efficiency Reverse directional distance function measure for the Russell technical inefficiency:

using BenchmarkingEconomicEfficiency

X = [1; 1; 1; 1; 1; 1; 1; 1];

Y = [7 7; 4 8; 8 4; 3 5; 3 3; 8 2; 6 4; 1.5 5];

P = [1 1; 1 1; 1 1; 1 1; 1 1; 1 1; 1 1; 1 1];

revenuerddf = dearevenuerddf(X, Y, P, :ERG)
Revenue Reverse DDF DEA Model
DMUs = 8; Inputs = 1; Outputs = 2
Returns to Scale: VRS
Associated efficiency measure = ERG
────────────────────────────────────
Revenue  Technical    Allocative
────────────────────────────────────
1  0.0        0.0        0.0
2  0.25       0.0        0.25
3  0.25       0.0        0.25
4  0.464286   0.464286   0.0
5  0.571429   0.571429   0.0
6  0.666667   0.333333   0.333333
7  0.314286   0.314286  -5.55112e-17
8  0.818182   0.672727   0.145455
────────────────────────────────────

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

efficiency(revenuerddf, :Economic)
8-element Vector{Float64}:
0.0
0.25
0.25
0.4642857142857143
0.5714285714285715
0.6666666666666667
0.3142857142857144
0.8181818181818182
efficiency(revenuerddf, :Technical)
8-element Vector{Float64}:
0.0
0.0
0.0
0.4642857142857143
0.5714285714285715
0.33333333333333337
0.31428571428571445
0.6727272727272727
efficiency(revenuerddf, :Allocative)
8-element Vector{Float64}:
0.0
0.25
0.25
0.0
0.0
0.33333333333333337
-5.551115123125783e-17
0.1454545454545455

### dearevenuerddf Function Documentation

BenchmarkingEconomicEfficiency.dearevenuerddfFunction
dearevenuerddf(X, Y, P, measure)

Compute profit efficiency using data envelopment analysis Reverse DDF model for inputs X, outputs Y, price of outputs P, and efficiency measure.

Measure specification:

• :ERG: Enhanced Russell Graph Slack Based Measure.

Optional Arguments

• rts=:VRS: choose between constant returns to scale :CRS or variable returns to scale :VRS.
• atol=1e-6: tolerance for DMU to be considered efficient.
• monetary=false: decomposition in normalized terms. Monetary terms if true.
• names: a vector of strings with the names of the decision making units.