Radial Models

Radial Input Oriented Model

Based on the data matrix $(X,Y)$, we calculate the input oriented efficiency of each observation o by solving $n$ times the following linear programming problem – known as the Charnes, Cooper, and Rhodes (1978), CCR, model:

\[\begin{aligned} & \underset{\theta ,\mathbf{\lambda }}{\mathop{\min }}\,\quad \quad \quad \;\ \theta \\ & \text{subject}\ \text{to} \\ & \quad \quad \quad \quad \quad \ X\mathbf{\lambda } \le \theta {{\mathbf{x}}_{o}} \\ & \quad \quad \quad \quad \quad \;Y\mathbf{\lambda }\ \ge {{\mathbf{y}}_{o}} \\ & \quad \quad \quad \quad \quad \ \mathbf{\lambda }\ge \mathbf{0}. \end{aligned}\]

The measurement of technical efficiency assuming variable returns to scale, VRS, as introduced by Banker, Charnes and Cooper (1984) – known as the Banker, Charnes and Cooper, BCC, model – adds the following condition:

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

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

using DataEnvelopmentAnalysis

X = [5 13; 16 12; 16 26; 17 15; 18 14; 23 6; 25 10; 27 22; 37 14; 42 25; 5 17];

Y = [12; 14; 25; 26; 8; 9; 27; 30; 31; 26; 12];

dea(X, Y, orient = :Input, rts = :CRS)

Radial DEA Model 
DMUs = 11; Inputs = 2; Outputs = 1
Orientation = Input; Returns to Scale = CRS
──────────────────────────────────────────────────
    efficiency       slackX1      slackX2  slackY1
──────────────────────────────────────────────────
1     1.0        0.0          0.0              0.0
2     0.62229   -4.41868e-15  0.0              0.0
3     0.819856   0.0          8.17926e-15      0.0
4     1.0       -8.03397e-16  0.0              0.0
5     0.310371   1.80764e-15  0.0              0.0
6     0.555556   4.44444      0.0              0.0
7     1.0        0.0          0.0              0.0
8     0.757669   1.60679e-15  0.0              0.0
9     0.820106   1.64021      0.0              0.0
10    0.490566   9.68683e-15  0.0              0.0
11    1.0        0.0          4.0              0.0
──────────────────────────────────────────────────

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

dea(X, Y, orient = :Input, rts = :VRS)

Radial DEA Model 
DMUs = 11; Inputs = 2; Outputs = 1
Orientation = Input; Returns to Scale = VRS
───────────────────────────────────────────────────────
    efficiency       slackX1       slackX2      slackY1
───────────────────────────────────────────────────────
1     1.0        0.0           0.0          0.0
2     0.869986   0.0           0.0          0.0
3     1.0        0.0           2.56789e-13  0.0
4     1.0       -8.03397e-16   0.0          0.0
5     0.71164    0.0           0.0          2.69841
6     1.0        2.70127e-16   0.0          3.78178e-16
7     1.0        0.0           0.0          0.0
8     1.0        0.0          -1.27018e-14  0.0
9     1.0        0.0           0.0          0.0
10    0.493121   3.90444e-15   0.0          0.0
11    1.0        0.0           4.0          4.78849e-16
───────────────────────────────────────────────────────

Estimated efficiency scores are returned with the efficiency function:

deaiovrs = dea(X, Y, orient = :Input, rts = :VRS);

efficiency(deaiovrs)

11-element Vector{Float64}:
 1.0
 0.8699861687413553
 1.0000000000000002
 1.0
 0.7116402116402116
 1.0
 1.0
 0.9999999999999999
 1.0
 0.4931209268645909
 1.0

The optimal peers, $λ$, are returned with the peers function and are returned as a DEAPeers object:

peers(deaiovrs)

DEA Peers
1: 1 ( 1.0 ) 
2: 1 ( 0.5255878284923927 ) 6 ( 0.2842323651452281 ) 7 ( 0.1901798063623792 ) 
3: 3 ( 1.0000000000000002 ) 
4: 4 ( 1.0 ) 
5: 1 ( 0.5661375661375662 ) 6 ( 0.4338624338624339 ) 
6: 6 ( 1.0 ) 
7: 7 ( 1.0 ) 
8: 8 ( 0.9999999999999999 ) 
9: 9 ( 1.0 ) 
10: 1 ( 0.03711078928312814 ) 4 ( 0.4433381607530775 ) 7 ( 0.5195510499637944 ) 
11: 11 ( 1.0 ) 

Input and output slacks are returned with the slacks function:

slacks(deaiovrs, :X)

11×2 Matrix{Float64}:
  0.0           0.0
  0.0           0.0
  0.0           2.56789e-13
 -8.03397e-16   0.0
  0.0           0.0
  2.70127e-16   0.0
  0.0           0.0
  0.0          -1.27018e-14
  0.0           0.0
  3.90444e-15   0.0
  0.0           4.0

slacks(deaiovrs, :Y)

11×1 Matrix{Float64}:
 0.0
 0.0
 0.0
 0.0
 2.6984126984126924
 3.7817815923971297e-16
 0.0
 0.0
 0.0
 0.0
 4.788485288453362e-16

Input and output optimal targets are returned with the targets function:

targets(deaiovrs, :X)

11×2 Matrix{Float64}:
  5.0     13.0
 13.9198  10.4398
 16.0     26.0
 17.0     15.0
 12.8095   9.96296
 23.0      6.0
 25.0     10.0
 27.0     22.0
 37.0     14.0
 20.7111  12.328
  5.0     13.0

targets(deaiovrs, :Y)

11×1 Matrix{Float64}:
 12.0
 14.0
 25.0
 26.0
 10.698412698412692
  9.0
 27.0
 30.0
 31.0
 26.0
 12.0

Radial Output Oriented Model

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

\[\begin{aligned} & \underset{\phi ,\mathbf{\lambda }}{\mathop{\max }}\,\quad \quad \quad \quad \phi \\ & \text{subject}\ \text{to} \\ & \quad \quad \quad \quad \quad \ X\lambda\le {{\mathbf{x}}_{o}} \\ & \quad \quad \quad \quad \quad \ Y\mathbf{\lambda }\ \ge \phi {{\mathbf{y}}_{o}} \\ & \quad \quad \quad \quad \quad \ \mathbf{\lambda }\ge \mathbf{0}.\ & \quad \end{aligned}\]

with the following condition when assuming variable returns to scale:

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

In this example we compute the radial output oriented DEA model under variable returns to scale:

dea(X, Y, orient = :Output, rts = :VRS)

Radial DEA Model 
DMUs = 11; Inputs = 2; Outputs = 1
Orientation = Output; Returns to Scale = VRS
──────────────────────────────────────────────────
    efficiency       slackX1  slackX2      slackY1
──────────────────────────────────────────────────
1      1.0       0.0              0.0  0.0
2      1.50752   5.78599e-15      0.0  0.0
3      1.0       0.0              0.0  0.0
4      1.0      -8.03397e-16      0.0  0.0
5      3.20395  -3.38377e-15      0.0  0.0
6      1.0       2.70127e-16      0.0  3.78178e-16
7      1.0       0.0              0.0  0.0
8      1.0       0.0              0.0  0.0
9      1.0       0.0              0.0  0.0
10     1.19231   5.0             11.0  0.0
11     1.0       0.0              4.0  4.78849e-16
──────────────────────────────────────────────────

Radial Model in Multiplier Form

The dual to the input oriented and output oriented radial DEA models in envelopment form presented above is the multiplier form.

This example computes the radial input-oriented DEA model in multiplier form under variable returns to scale:

deaiovrsm = deam(X, Y, rts = :VRS)

Radial DEA Model (Multiplier form)
DMUs = 11; Inputs = 2; Outputs = 1
Orientation = Input; Returns to Scale = VRS
─────────────────────────────────────────────────
    efficiency         v1           v2         u1
─────────────────────────────────────────────────
1     1.0       0.0901826  0.0422374    0.0833333
2     0.869986  0.0197095  0.0570539    0.0148686
3     1.0       0.0612245  0.000784929  0.0525903
4     1.0       0.0416228  0.0194942    0.0384615
5     0.71164   0.0185185  0.047619     0.0
6     1.0       0.0        0.166667     0.037037
7     1.0       0.0261969  0.0345077    0.037037
8     1.0       0.0227671  0.0175131    0.0875657
9     1.0       0.0        0.0714286    0.0714286
10    0.493121  0.013034   0.0181028    0.0137581
11    1.0       0.2        2.61229e-17  0.0833333
─────────────────────────────────────────────────

Input and output virtual multipliers (shadow prices) are returned with the multipliers function:

multipliers(deaiovrsm, :X)

11×2 Matrix{Float64}:
 0.0901826  0.0422374
 0.0197095  0.0570539
 0.0612245  0.000784929
 0.0416228  0.0194942
 0.0185185  0.047619
 0.0        0.166667
 0.0261969  0.0345077
 0.0227671  0.0175131
 0.0        0.0714286
 0.013034   0.0181028
 0.2        2.61229e-17

multipliers(deaiovrsm, :Y)

11×1 Matrix{Float64}:
 0.08333333333333336
 0.014868603042876911
 0.05259026687598115
 0.038461538461538464
 0.0
 0.03703703703703698
 0.03703703703703704
 0.08756567425569173
 0.07142857142857138
 0.013758146270818254
 0.0833333333333333

The value measuring the returns to scale is returned with the rts function:

rts(deaiovrsm)

11-element Vector{Float64}:
  0.0
 -0.6618257261410788
  0.3147566718995288
  0.0
 -0.7116402116402116
 -0.6666666666666667
  0.0
  1.6269702276707518
  1.2142857142857133
 -0.13540912382331627
  0.0

dea Function Documentation

dea(X, Y)

deam(X, Y)