Example

Consider analyzing the efficiencies of 3 DMUs where 2 inputs and 3 outputs are used. The data is as follows:

The linear programs for evaluating the 3 DMUs are given by:

• LP for evaluating DMU 1:

  min THETA
  st
  5L1+8L2+7L3 - 5THETA < 0
  14L1+15L2+12L3 - 14THETA < 0
  9L1+5L2+4L3 > 9
  4L1+7L2+9L3 > 4
  16L1+10L2+13L3 > 16

• LP for evaluating DMU 2:

  min THETA
  st
  5L1+8L2+7L3 - 8THETA < 0
  14L1+15L2+12L3 - 15THETA < 0
  9L1+5L2+4L3 > 5
  4L1+7L2+9L3 > 7
  16L1+10L2+13L3 > 10
  end

• LP for evaluating DMU 3:

  min THETA
  st
  5L1+8L2+7L3 - 7THETA < 0
  14L1+15L2+12L3 - 12THETA < 0
  9L1+5L2+4L3 > 4
  4L1+7L2+9L3 > 9
  16L1+10L2+13L3 > 13
  end

The solution to each of these is as follows:

• DMU 1.

          OBJECTIVE FUNCTION VALUE
  
          1)     1.0000000    
  
    VARIABLE        VALUE          REDUCED COST
       THETA         1.000000          0.000000
          L1         1.000000          0.000000
          L2         0.000000          0.411491
          L3         0.000000          0.000000
  
  
         ROW   SLACK OR SURPLUS     DUAL PRICES
          2)         0.000000          0.000000
          3)         0.000000          0.071429
          4)         0.000000          0.000000
          5)         0.000000         -0.007764
          6)         0.000000         -0.060559

• DMU 2.

          OBJECTIVE FUNCTION VALUE
  
          1)    0.77333331    
  
    VARIABLE        VALUE          REDUCED COST
       THETA         0.773333          0.000000
          L1         0.261538          0.000000
          L2         0.000000          0.226667
          L3         0.661538          0.000000
  
  
         ROW   SLACK OR SURPLUS     DUAL PRICES
          2)         0.248205          0.000000
          3)         0.000000          0.066667
          4)         0.000000         -0.080000
          5)         0.000000         -0.053333
          6)         2.784615          0.000000

• DMU 3.

          OBJECTIVE FUNCTION VALUE
  
          1)     1.0000000    
  
    VARIABLE        VALUE          REDUCED COST
       THETA         1.000000          0.000000
          L1         0.000000          0.000000
          L2         0.000000          0.480072
          L3         1.000000          0.000000
  
  
         ROW   SLACK OR SURPLUS     DUAL PRICES
          2)         0.000000          0.000000
          3)         0.000000          0.083333
          4)         0.000000          0.000000
          5)         0.000000         -0.009058
          6)         0.000000         -0.070652

Note that DMUs 1 and 3 are overall efficient and DMU 2 is inefficient with an efficiency rating of 0.773333.

Hence the efficient levels of inputs and outputs for DMU 2 are given by:

• Efficient levels of Inputs:

  

• Efficient levels of Outputs:

  

Note that the outputs are at least as much as the outputs currently produced by DMU 2 and inputs are at most as big as the 0.773333 times the inputs of DMU 2. This can be used in two different ways: The inefficient DMU should target to cut down inputs to equal at most the efficient levels. Alternatively, an equivalent statement can be made by finding a set of efficient levels of inputs and outputs by dividing the levels obtained by the efficiency of DMU 2. This focus can then be used to set targets primarily for outputs rather than reduction of inputs.