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Resource allocation: A common set of weights model

  • * Corresponding author: Sedighe Asghariniya

    * Corresponding author: Sedighe Asghariniya 
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  • Allocation problem is an important issue in management. Data envelopment analysis (DEA) is a non-parametric method for assessing a set of decision making units (DMUs). It has proven to be a useful technique to solve allocation problems. In recent years, many papers have been published in this regard and many researchers have tried to find a suitable allocation model based on DEA. Common set of weights (CSWs) is a DEA model which, in contrast with traditional DEA models, does not allow individual weights for each decision making unit. In this manner, all DMUs are assessed through choosing a same set of weights. In this article, we will use the weighted-sum method to solve the multi-objective CSW problem. Then, via introducing a set of special weights, we will connect the CSW model to a non-linear (fractional) CSW model. After linearization, the proposed model is used for allocating resources. To illustrate our model, some examples are also provided.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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  • Figure 1.  The graph of the weighted-sum example

    Figure 2.  PPS in a two dimensions case

    Figure 3.  PPS$ _{CCR} $ for the peresentedDMUs in Table 1

    Figure 4.  virtual plane of the example

    Table 1.  Information related to example

    DMU A B C D Agregated
    input1 1 2 6 3 12
    input2 5 2 1 3 11
    output 1 1 1 1 4
     | Show Table
    DownLoad: CSV

    Table 2.  Optimal solution of model (5) and virtual inputs and outputs connected with it for the example

    $ u^* $ $ v^* $ $ ({v^ * }{x_1},{u^ * }{y_1}) $ $ ({v^ * }{x_2},{u^ * }{y_2}) $ $ ({v^ * }{x_3},{u^ * }{y_3}) $ $ ({v^ * }{x_4},{u^ * }{y_4}) $ $ ({v^ * }{\bar x},{u^ * }{\bar y}) $
    $ \frac{8}{47} $ $ (\frac{8}{47},\frac{1}{47}) $ $ (\frac{8}{47},\frac{8}{47}) $ $ (\frac{8}{47},\frac{8}{47}) $ $ (\frac{10}{47},\frac{8}{47}) $ $ (\frac{12}{47},\frac{8}{47}) $ $ (\frac{32}{47},\frac{47}{47}) $
     | Show Table
    DownLoad: CSV

    Table 3.  Data set of [14]

    DMU Input1 Input2 Input3 Output1 Output2 $ EFF_CCR $
    1 350 39 9 67 751 0.75663
    2 298 26 8 73 611 0.92300
    3 422 31 7 75 584 0.74384
    4 281 16 9 70 665 1.00000
    5 301 16 6 75 445 1.00000
    6 360 29 17 83 1070 0.96112
    7 540 18 10 72 457 0.85863
    8 276 33 5 78 590 1.00000
    9 323 25 5 75 1074 1.00000
    10 444 64 6 74 1072 0.83102
    11 323 25 5 25 350 0.33325
    12 444 64 6 104 1199 1.00000
     | Show Table
    DownLoad: CSV

    Table 4.  Allocated cost to DMUs obtained by different methods

    DMU Allocated resource Efficiency DMU Allocated resource Efficiency
    1 8.47412 1.00000 7 4.84712 1.00000
    2 6.90521 1.00000 8 6.68270 1.00000
    3 6.45563 1.00000 9 12.32408 1.00000
    4 7.56372 1.00000 10 12.13035 1.00000
    5 4.96678 1.00000 11 3.76675 1.00000
    6 12.22981 1.00000 12 13.65374 1.00000
     | Show Table
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    Table 5.  Different allocation in selected methods

    Eff. invariance Output orientation Input orientation
    no no no no no no yes no yes
    our Besley Du et al. Li et al. Hossein Zadeh Si et al. Cook and Lin and Yang and
    DMU approach [4] [23] [34] Lotfi et al. [40] [49] Kress [14] Chen [38] Zhang [53]
    1 8.47 6.78 5.79 5.54 8.20 7.65 14.52 9.83 7.54
    2 6.91 7.21 7.95 7.53 7.46 8.41 6.74 7.53 8.65
    3 6.46 6.83 6.54 7.35 4.28 8.62 9.32 9.93 7.52
    4 7.56 8.47 11.10 7.87 9.30 8.11 5.6 5.20 9.05
    5 4.97 7.08 8.69 6.38 4.81 8.69 5.79 5.20 9.07
    6 12.23 10.06 13.49 11.50 15.37 9.57 8.15 9.10 8.81
    7 4.85 5.09 7.10 5.90 0 8.33 8.86 5.85 8.17
    8 6.68 7.74 6.83 7.77 7.34 9.96 6.26 8.96 9.06
    9 12.32 15.11 16.68 11.90 16.33 8.65 7.31 8.07 10.46
    10 12.13 10.08 5.42 11.38 11.60 8.35 10.08 9.96 8.01
    11 3.77 1.58 0 2.74 0 2.80 7.31 8.07 4.55
    12 13.65 13.97 10.41 14.14 15.31 11.85 10.08 12.56 9.12
    Gap 9.88 13.53 16.68 11.40 16.33 9.05 8.92 7.36 5.91
     | Show Table
    DownLoad: CSV
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