Russell Models

Russell Input Model

Based on the data matrix $(X,Y)$, we calculate the Russell measure of input efficiency (Färe & Lovell, 1978; and Färe et al., 1985) of each observation o by solving $n$ times the following linear programming problem:

\[\begin{aligned} & \underset{\theta_i ,\lambda_j }{\mathop{\min }}\,\quad \quad \quad \;\ \frac{1}{m} \sum_{i=1}^{m}{\theta_i} \\ & \text{subject}\ \text{to} \\ & \quad \quad \quad \quad \quad \ \sum_{j=1}^{n}{\lambda_j x_{ij} }\ \le \theta_i {x}_{io} \qquad i = 1,...,m \\ & \quad \quad \quad \quad \quad \ \sum_{j=1}^{n}{\lambda_j y_{rj} }\ \ge {y}_{ro} \qquad r = 1,...,s \\ & \quad \quad \quad \quad \quad \ \theta_i \le 1 \qquad i = 1,...,m \\ & \quad \quad \quad \quad \quad \ \lambda_j \ge 0 \qquad j = 1,...,n. \end{aligned}\]

The measurement of technical efficiency assuming variable returns to scale, VRS, adds the following condition:

\[\sum\nolimits_{j=1}^{n}\lambda_j=1\]

In this example we compute the Russell input DEA model under constant returns to scale:

using DataEnvelopmentAnalysis

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];

dearussellio = dearussell(X, Y, orient = :Input, rts = :CRS)

Russell DEA Model 
DMUs = 8; Inputs = 2; Outputs = 1
Orientation = Input; Returns to Scale = CRS
──────────────────────────────────────────
   efficiency     effX1     effX2  slackY1
──────────────────────────────────────────
1    1.0       1.0       1.0           0.0
2    1.0       1.0       1.0           0.0
3    1.0       1.0       1.0           0.0
4    0.583333  0.5       0.666667      0.0
5    0.4       0.4       0.4           0.0
6    0.833333  0.666667  1.0           0.0
7    0.65      0.5       0.8           0.0
8    0.5625    0.625     0.5           0.0
──────────────────────────────────────────

To compute the variable returns to scale model, we simply set the rts parameter to :VRS:

Estimated efficiency scores are returned with the efficiency function:

efficiency(dearussellio)

8-element Vector{Float64}:
 1.0
 1.0
 1.0
 0.5833333333333334
 0.4
 0.8333333333333334
 0.65
 0.5625

efficiency(dearussellio, :X)

8×2 Matrix{Float64}:
 1.0       1.0
 1.0       1.0
 1.0       1.0
 0.5       0.666667
 0.4       0.4
 0.666667  1.0
 0.5       0.8
 0.625     0.5

Russell Output Model

It is possible to calculate the Russell measure of output efficiency of each observation by solving the following linear program:

\[\begin{aligned} & \underset{\phi_r ,\lambda_j }{\mathop{\max }}\,\quad \quad \quad \;\ \frac{1}{s} \sum_{r=1}^{s}{\phi_r} \\ & \text{subject}\ \text{to} \\ & \quad \quad \quad \quad \quad \ \sum_{j=1}^{n}{\lambda_j x_{ij} }\ \le {x}_{io} \qquad i = 1,...,m \\ & \quad \quad \quad \quad \quad \ \sum_{j=1}^{n}{\lambda_j y_{rj} }\ \ge \phi_r {y}_{ro} \qquad r = 1,...,s \\ & \quad \quad \quad \quad \quad \ \phi_r \ge 1 \qquad r = 1,...,s \\ & \quad \quad \quad \quad \quad \ \lambda_j \ge 0 \qquad j = 1,...,n. \end{aligned}\]

with the following condition when assuming variable returns to scale:

\[\sum\nolimits_{j=1}^{n}\lambda_j=1\]

In this example we compute the Russell output DEA model under constant returns to scale:

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] ;

dearusselloo = dearussell(X, Y, orient = :Output, rts = :CRS)

Russell DEA Model 
DMUs = 8; Inputs = 1; Outputs = 2
Orientation = Output; Returns to Scale = CRS
────────────────────────────────────────
   efficiency    effY1    effY2  slackX1
────────────────────────────────────────
1     1.0      1.0      1.0          0.0
2     1.0      1.0      1.0          0.0
3     1.0      1.0      1.0          0.0
4     1.86667  2.33333  1.4          0.0
5     2.33333  2.33333  2.33333      0.0
6     1.5      1.0      2.0          0.0
7     1.45833  1.16667  1.75         0.0
8     3.05556  5.11111  1.0          0.0
────────────────────────────────────────

Estimated efficiency scores are returned with the efficiency function:

efficiency(dearusselloo)

8-element Vector{Float64}:
 1.0
 1.0
 1.0
 1.8666666666666665
 2.333333333333333
 1.5
 1.4583333333333335
 3.0555555555555554

efficiency(dearusselloo, :Y)

8×2 Matrix{Float64}:
 1.0      1.0
 1.0      1.0
 1.0      1.0
 2.33333  1.4
 2.33333  2.33333
 1.0      2.0
 1.16667  1.75
 5.11111  1.0

Russell Graph Model

It is possible to calculate the Russell graph measure of technical efficiency of each observation by solving the following linear program:

\[\begin{aligned} & \underset{\theta_i, \phi_r ,\lambda_j }{\mathop{\min }}\,\quad \quad \quad \;\ \frac{1}{m + s} (\sum_{i=1}^{m}{\theta_i} + \sum_{r=1}^{s}{\frac{1}{\phi_r}}) \\ & \text{subject}\ \text{to} \\ & \quad \quad \quad \quad \quad \ \sum_{j=1}^{n}{\lambda_j x_{ij} }\ \le \theta_i {x}_{io} \qquad i = 1,...,m \\ & \quad \quad \quad \quad \quad \ \sum_{j=1}^{n}{\lambda_j y_{rj} }\ \ge \phi_r {y}_{ro} \qquad r = 1,...,s \\ & \quad \quad \quad \quad \quad \ \theta_i \le 1 \qquad i = 1,...,m \\ & \quad \quad \quad \quad \quad \ \phi_r \ge 1 \qquad r = 1,...,s \\ & \quad \quad \quad \quad \quad \ \lambda_j \ge 0 \qquad j = 1,...,n. \end{aligned}\]

with the following condition when assuming variable returns to scale:

\[\sum\nolimits_{j=1}^{n}\lambda_j=1\]

In this example we compute the Russell graph DEA model under variable returns to scale:

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

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

dearussellgr = dearussell(X, Y, orient = :Graph, rts = :VRS)

Russell DEA Model 
DMUs = 8; Inputs = 1; Outputs = 1
Orientation = Graph; Returns to Scale = VRS
────────────────────────────────
   efficiency     effX1    effY1
────────────────────────────────
1    1.0       1.0       1.0
2    1.0       1.0       1.0
3    1.0       1.0       1.0
4    1.0       1.0       1.0
5    0.633333  0.666667  1.66667
6    0.723214  0.571429  1.14286
7    0.928571  0.857143  1.0
8    0.447795  0.424989  2.12495
────────────────────────────────

Estimated efficiency scores are returned with the efficiency function:

efficiency(dearussellgr)

8-element Vector{Float64}:
 0.9999996492279167
 0.9999999749520232
 0.9999999913580742
 0.9999999703644951
 0.6333333226351024
 0.7232142937165499
 0.9285714048857328
 0.4477946881737884

efficiency(dearussellgr, :X)

8×1 Matrix{Float64}:
 1.0000000084319445
 0.9999999519036574
 0.9999999790875087
 0.9999999328388507
 0.6666666303008444
 0.57142851550475
 0.8571428024480117
 0.42498937010375143

efficiency(dearussellgr, :Y)

8×1 Matrix{Float64}:
 1.000000709976615
 1.0000000019996111
 0.9999999963713605
 0.9999999921098606
 1.6666666250851108
 1.1428570489099183
 0.9999999926765463
 2.124946848134728

dearussell Function Documentation

dearussell(X, Y)