Profit Enhanced Russell Graph Slack Based Measure

The profit enhanced Russell graph-slack based measure is computed by solving an enhanced Russell graph-slack based measure model for the technical efficiency.

The enhanced Russell graph measure $ERG$ (or slack-based measure) was designed as a new global efficiency measure to overcome the computational difficulties of the Profit Russell model. The novelty lays on the definition of a non-radial model that accounts for both inputs and outputs (graph or non-oriented) as the Russell proposal, but that is easier to compute through linear programming–as opposed to the standard Russell model. The ERG=SBM measure is formulated resorting to the same variables as the Russell graph measure: the 'thetas' ($\theta$'s) for the proportional individual reduction of each of the $M$ inputs, the 'phys' ($\phi$'s) for the proportional individual output increase of each of the $N$ outputs, and the `lambdas' ($\lambda$'s) defining the reference hyperplanes from the obeservations that constitute the production frontier. The technical efficiency model corresponds to:

\[\begin{split} T{{E}_{ERG=SBM(G)}}\left( {{x}_{o}},{{y}_{o}} \right) = \, & \underset{\pmb{\theta} , \pmb{\phi} ,\pmb{\lambda} }{\mathop{\min }} \, \frac{\frac{1}{M}\sum\limits_{m=1}^{M}{{{\theta }_{m}}}}{\frac{1}{N}\sum\limits_{n=1}^{N}{{{\phi }_{n}}}} \\ & s.t. \quad \sum\limits_{j=1}^{J}{{{\lambda }_{j}}{{x}_{mj}}}={{\theta }_{m}}{{x}_{om}}, \quad m=1,...,M \\ & \quad \quad \sum\limits_{j=1}^{J}{{{\lambda }_{j}}{{y}_{nj}}}={{\phi }_{n}}{{y}_{on}}, \quad n=1,...,N \\ & \quad \quad \sum\limits_{j=1}^{J}{{{\lambda }_{j}}}=1, \\ & \quad \quad {{\theta }_{m}}\le 1, \quad m=1,...,M \\ & \quad \quad {{\phi }_{n}}\ge 1, \quad n=1,...,N \\ & \quad \quad {{\lambda }_{j}}\ge 0, \quad j=1,...,J \\ \end{split}\]

Comparing this model with that defining the Russell graph measure the only difference is the objective function, originally formulated as $\frac{1}{M+N}\left( \sum\limits_{m=1}^{M}{{{\theta }_{m}}}+\sum\limits_{n=1}^{N}{\frac{1}{{{\phi }_{n}}}} \right)$. From the perspective of finding the solution of this model this difference is relevant because we have replaced the original nonlinear objective function with a linear fractional objective function, i.e., a fraction of two linear expressions, that is easier to solve.

For this purpose, after performing a change of variables that formulates the model in terms of slacks instead of the multiplicative reduction of each input and increase of each output, and introducing a variable $\beta$ corresponding to the inverse of the denominator of the resulting objective function–see Pastor, Aparicio and Zofío (2022, Ch. 7)–we obtain the final linear program that calculates the value of the ERG=SMB measure of technical efficiency:

\[\begin{split} T{{E}_{ERG=SBM(G)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}} \right)= & \underset{\textbf{t}_{{}}^{-},\textbf{t}_{{}}^{+},\pmb{\mu} ,\pmb{\beta} }{\mathop{\min }}\, \beta -\frac{1}{M}\sum\limits_{m=1}^{M}{\frac{t_{m}^{-}}{{{x}_{om}}}} \\ & s.t. \quad \beta +\frac{1}{N}\sum\limits_{n=1}^{N}{\frac{t_{n}^{+}}{{{y}_{on}}}=1} \\ & \quad \quad \sum\limits_{j=1}^{J}{{{\mu }_{j}}{{x}_{jm}}}=\beta {{x}_{om}}-t_{m}^{-}, \quad m=1,...,M \\ & \quad \quad \sum\limits_{j=1}^{J}{{{\mu }_{j}}{{y}_{jn}}}=\beta {{y}_{on}}+t_{n}^{+}, \quad n=1,...,N \\ & \quad \quad \sum\limits_{j=1}^{J}{{{\mu }_{j}}}=\beta , \\ & \quad \quad \beta \ge 0,\,\, \\ & \quad \quad t_{m}^{-}\ge 0,t_{n}^{+}\ge 0, \quad \forall m,n, \\ & \quad \quad{{\mu }_{j}}\ge 0, \quad j=1,...,J. \\ \end{split}\]

From the solution to this program we can recover the following measure of technical inefficiency:

\[\begin{split} & T{{I}_{ERG=SBM(G)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}} \right)=1-T{{E}_{ERG=SBM(G)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}} \right)= \\ & \quad 1-\frac{1-\frac{1}{M}\sum\limits_{m=1}^{M}{\frac{s_{m}^{-*}}{{{x}_{om}}}}}{1+\frac{1}{N}\sum\limits_{n=1}^{N}{\frac{s_{n}^{+*}}{y_{on}^{{}}}}}=\frac{\frac{1}{N}\sum\limits_{n=1}^{N}{\frac{s_{n}^{+*}}{y_{on}^{{}}}}+\frac{1}{M}\sum\limits_{m=1}^{M}{\frac{s_{m}^{-*}}{{{x}_{om}}}}}{1+\frac{1}{N}\sum\limits_{n=1}^{N}{\frac{s_{n}^{+*}}{y_{no}^{{}}}}}. \end{split}\]

Resorting to the duality between this expression and the profit function we can establish the associated decomposition of profit inefficiency: $\Pi I_{ERG=SBM(G)}^{{}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\tilde{\textbf{w}},\tilde{\textbf{p}} \right)=T{{I}_{ERG=SBM(G)}}\left( {{\textbf{x}}_{o}},{{\textbf{\textbf{y}}}_{o}} \right)+A{{I}_{ERG=SBM(G)}}\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\tilde{\textbf{w}},\tilde{\textbf{p}} \right)$; i.e.,

\[\begin{split} & \underbrace{\frac{\Pi \left( \mathbf{w},\mathbf{p} \right)-\left( \sum\limits_{n=1}^{N}{{{p}_{n}}{{y}_{on}}}-\sum\limits_{m=1}^{M}{{{w}_{m}}{{x}_{om}}} \right)}{{{\delta }_{\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\textbf{p},\textbf{w} \right)}}\left( 1+\frac{1}{N}\sum\limits_{n=1}^{N}{\frac{s_{n}^{+*}}{y_{on}^{{}}}} \right)}}_{\text{Norm}\text{.}\ \text{Profit}\ \text{Inefficency}}= \\ & \quad =\underbrace{\left( \frac{\frac{1}{N}\sum\limits_{n=1}^{N}{\frac{s_{n}^{+*}}{y_{on}^{{}}}+}\frac{1}{M}\sum\limits_{m=1}^{M}{\frac{s_{m}^{-*}}{x_{om}^{{}}}}}{\left( 1+\frac{1}{N}\sum\limits_{n=1}^{N}{\frac{s_{n}^{+*}}{y_{on}^{{}}}} \right)} \right)}_{\text{Graph Techncial Inefficency}}+\underbrace{A{{I}_{ERG=SBM(G)}}\left( {{\mathbf{x}}_{o}},{{\mathbf{y}}_{o}},\tilde{\textbf{p}},\tilde{\textbf{w}} \right)}_{\text{Norm}\text{.}\ \text{Allocative}\ \text{Inefficency}}\ge 0, \end{split}\]

where ${{\delta }_{\left( {{\textbf{x}}_{o}},{{\textbf{y}}_{o}},\textbf{p},\textbf{w} \right)}}=\min \left\{ N{{p}_{n}}y_{on}^{{}},n=1,...,N,M{{w}_{m}}x_{om}^{{}},m=1,...,M \right\}$ in the normalization factor of the ERG=SBM.

Reference

Chapter 7 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 profit efficiency Enhanced Russell Graph Slack Based measure:

using BenchmarkingEconomicEfficiency

X = [2; 4; 8; 12; 6; 14; 14; 9.412];

Y = [1; 5; 8; 9; 3; 7; 9; 2.353];

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

P = [2; 2; 2; 2; 2; 2; 2; 2];

profiterg = deaprofiterg(X, Y, W, P)

Enhanced Russell Graph Slack Based Measure Profit DEA Model 
DMUs = 8; Inputs = 1; Outputs = 1
Returns to Scale = VRS
──────────────────────────────────
     Profit  Technical  Allocative
──────────────────────────────────
1  4.0        0.0         4.0
2  0.5        0.0         0.5
3  0.0        0.0         0.0
4  0.166667   0.0         0.166667
5  0.8        0.6         0.2
6  0.571429   0.52381     0.047619
7  0.285714   0.142857    0.142857
8  1.2706     0.8         0.4706
──────────────────────────────────

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

efficiency(profiterg, :Economic)

8-element Vector{Float64}:
 4.0
 0.5
 0.0
 0.16666666666666666
 0.8
 0.5714285714285714
 0.2857142857142857
 1.2705999999999997

efficiency(profiterg, :Technical)

8-element Vector{Float64}:
 0.0
 0.0
 0.0
 0.0
 0.6000000000000001
 0.5238095238095238
 0.14285714285714257
 0.8

efficiency(profiterg, :Allocative)

8-element Vector{Float64}:
 4.0
 0.5
 0.0
 0.16666666666666666
 0.19999999999999996
 0.04761904761904756
 0.14285714285714313
 0.4705999999999997

deaprofiterg Function Documentation

deaprofiterg(X, Y, W, P)