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Resource allocation and target setting based on virtual profit improvement

We thank four reviewers of this journal for their most constructive comments

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  • One application of Data Envelopment Analysis (DEA) is the resource allocation and target setting among homogeneous Decision Making Units (DMUs). In this paper, we assume that all units are under the supervision and control of a central decision making unit, for instance chain stores, banks, schools, etc. The aim is to allocate available resources among units in a way that the so-called organisational overall "virtual profit" is maximized. Our method is highly flexible in decision making to achieve the goals of the Decision Maker (DM). The resulting production plans maintain the following characteristics: (1) the virtual profit of each unit is calculated with a common set of weights; (2) the selected weights for calculating the virtual profit prevent the virtual profit of the system from getting worse; (3) the virtual profits of less profitable units are improved as much as possible. The proposed method is illustrated with a simple numerical example and a real life application.

    Mathematics Subject Classification: Primary: 97M40, 90B50; Secondary: 90C90, 90C05.

    Citation:

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  • Figure 1.  Farell's frontier before resource allocation

    Figure 2.  Farell's frontier after resource allocation in cases I and II

    Table 1.  Data set and results of numerical example

    Case I Case II
    $ DMU $ $ I_1 $ $ I_2 $ $ O $ $ I_1 $ $ I_2 $ $ O $ $ I_1 $ $ I_2 $ $ O $
    A 8 24 4 6.4 24 4.1 6.4 21.32 4.2
    B 27 27 9 32.4 27 8.55 32.4 32.4 9
    C 56 8 8 54.3 8 7.6 67.2 9.6 8
    D 8 14 2 6.4 14 2.1 6.4 11.2 2.1
    E 42 24 6 33.6 24 6.3 33.6 20.28 6.3
    F 32 8 4 25.6 8 4.2 30.18 9.6 4.2
    G 30 3 3 24 3 3.15 26.82 3.6 3.15
    Central 203 108 36 182.7 108 36 203 108 36.95
     | Show Table
    DownLoad: CSV

    Table 2.  Results of numerical example

    Case I Case II
    $ DMU $ $ p_j $ $ p_j^\prime $ $ \theta_j $ $ \theta_j^\prime $ $ p_j $ $ p_j^\prime $ $ \theta_j $ $ \theta_j^\prime $
    A -0.0457 -0.0239 1 1 -0.0457 -0.0091 1 1
    B 0 0 1 1 0 0 1 1
    C -0.0047 0 1 1 -0.0047 0 1 1
    D -0.042 -0.0287 0.63 0.7 -0.042 -0.0151 0.59 0.73
    E -0.0869 -0.0459 0.59 0.77 -0.0869 -0.0151 0.59 0.91
    F -0.0307 -0.0037 0.75 0.97 -0.0307 -0.0099 0.69 0.9
    G -0.0239 -0.0028 0.85 1 -0.0239 0 0.85 1
    Central -0.234 -0.1049 0.77 0.9 -0.234 -0.0491 0.77 0.95
     | Show Table
    DownLoad: CSV

    Table 3.  Data set and results of numerical example

    Case I Case II
    $ DMU $ $ I_1 $ $ I_2 $ $ O_1 $ $ O_2 $ $ I_1 $ $ I_2 $ $ O_1 $ $ O_2 $ $ I_1 $ $ I_2 $ $ O_1 $ $ O_2 $
    1 79.1 4.99 115.3 1.71 87.01 4.99 121.06 1.8 71.19 4.49 121.06 1.8
    2 60.1 3.3 75.2 1.81 66.11 3.3 78.96 1.9 54.09 2.97 78.96 1.9
    3 126.7 8.12 225.5 10.39 139.37 8.12 214.22 9.87 139.37 8.93 225.5 10.91
    4 153.9 6.7 185.6 10.42 169.29 6.7 194.88 10.94 138.51 7.37 194.88 10.94
    5 65.7 4.74 84.5 2.36 72.27 4.74 88.73 2.48 59.13 4.27 88.73 2.48
    6 76.8 4.08 103.3 4.35 84.48 4.08 108.46 4.57 69.12 4.24 108.46 4.57
    7 50.2 2.53 78.8 0.16 55.22 2.53 82.74 0.17 55.22 2.78 82.74 0.17
    8 44.8 2.47 59.3 1.3 49.28 2.47 62.27 1.37 40.32 2.72 62.27 1.37
    9 48.1 2.32 65.7 1.49 52.91 2.32 68.99 1.56 43.29 2.55 68.99 1.56
    10 89.7 4.91 163.2 6.26 98.67 4.91 155.04 5.95 98.67 5.4 163.2 6.26
    11 56.9 2.24 70.7 2.8 62.59 2.24 74.23 2.94 51.21 2.46 74.24 2.94
    12 112.6 5.42 142.6 2.75 123.86 5.42 149.73 2.89 101.34 4.88 149.73 2.89
    13 106.9 6.28 127.8 2.7 117.59 6.28 134.19 2.84 96.21 5.65 134.19 2.84
    14 54.9 3.14 62.4 1.42 60.39 3.14 65.52 1.49 60.39 3.45 65.52 1.49
    15 48.8 4.43 55.2 1.38 53.68 4.43 57.96 1.45 53.68 3.99 57.96 1.45
    16 59.2 3.98 95.9 0.74 65.12 3.98 100.7 0.78 65.12 4.38 100.7 0.78
    17 74.5 5.32 121.6 3.06 81.95 5.32 127.68 3.21 67.05 5.85 127.68 3.21
    18 94.6 3.69 107 2.98 104.06 3.69 112.35 3.13 102.17 3.32 112.35 3.13
    19 47 3 65.4 0.62 51.7 3 68.67 0.65 42.3 2.7 68.67 0.65
    20 54.6 3.87 71 0.01 60.06 3.87 74.55 0.01 57.72 3.48 74.55 0.01
    21 90.1 3.31 81.2 5.12 99.11 3.31 85.26 5.38 99.11 2.98 85.26 5.38
    22 95.2 4.25 128.3 3.89 104.72 4.25 134.72 4.08 104.72 3.83 134.72 4.08
    23 80.1 3.79 135 4.73 88.11 3.79 135.39 4.97 87.47 4.17 141.75 4.97
    24 68.7 2.99 98.9 1.86 75.57 2.99 103.85 1.95 75.57 2.69 103.85 1.95
    25 62.3 3.1 66.7 7.41 68.53 3.1 63.37 7.04 68.53 3.41 66.7 7.41
    Central 1901.5 102.97 2586.1 81.72 2091.65 102.97 2663.52 83.42 1901.5 102.96 2692.66 85.14
     | Show Table
    DownLoad: CSV

    Table 4.  Results of numerical example

    Case I Case II
    $ DMU $ $ p_j $ $ p_j^\prime $ $ \theta_j $ $ \theta_j^\prime $ $ p_j $ $ p_j^\prime $ $ \theta_j $ $ \theta_j^\prime $
    1 -0.0237 -0.0205 0.359 0.418 -0.0246 -0.0136 0.343 0.575
    2 -0.0176 -0.0154 0.431 0.494 -0.0178 -0.0098 0.423 0.62
    3 0 0 1 1 -0.0015 0 1 1
    4 -0.0196 -0.0109 0.849 0.923 -0.0181 -0.0006 0.849 0.992
    5 -0.0213 -0.0188 0.5 0.566 -0.023 -0.0138 0.436 0.618
    6 -0.0127 -0.0085 0.712 0.809 -0.0128 -0.0036 0.709 0.915
    7 -0.0146 -0.0128 0.078 0.095 -0.0143 -0.0154 0.078 0.085
    8 -0.0125 -0.0107 0.431 0.502 -0.0126 -0.0092 0.427 0.566
    9 -0.0113 -0.0093 0.516 0.57 -0.011 -0.0068 0.516 0.659
    10 0 0 1 1 0 0 1 1
    11 -0.0097 -0.0069 0.805 0.86 -0.0088 -0.0027 0.805 0.902
    12 -0.0321 -0.0281 0.401 0.451 -0.0315 -0.0169 0.401 0.59
    13 -0.0362 -0.0328 0.367 0.418 -0.0371 -0.0237 0.346 0.502
    14 -0.019 -0.0174 0.366 0.414 -0.0194 -0.0214 0.346 0.37
    15 -0.0225 -0.0211 0.393 0.444 -0.0248 -0.0236 0.306 0.353
    16 -0.018 -0.0156 0.243 0.3 -0.019 -0.0207 0.232 0.247
    17 -0.0159 -0.0118 0.628 0.757 -0.0175 -0.011 0.583 1
    18 -0.0247 -0.0214 0.562 0.617 -0.0233 -0.0196 0.562 0.739
    19 -0.0164 -0.0149 0.229 0.267 -0.017 -0.011 0.215 0.332
    20 -0.0243 -0.0232 0.003 0.004 -0.0255 -0.0237 0.003 0.004
    21 -0.02 -0.0164 0.729 0.799 -0.0185 -0.0153 0.729 1
    22 -0.0182 -0.0137 0.677 0.735 -0.0173 -0.0125 0.677 0.914
    23 -0.0038 0 1 1 -0.0031 0 1 1
    24 -0.0146 -0.0117 0.573 0.617 -0.0138 -0.0107 0.573 1
    25 0 0 1 1 0 0 1 1
    central -0.4088 -0.3419 0.591 0.658 -0.4123 -0.2856 0.588 0.714
     | Show Table
    DownLoad: CSV
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