Super-efficiency
During our first residency, we talked about how to rank or distinguish the performance of best-practice DMUs. I mentioned a model called "super-efficiency".
What is a super-efficiency model?
DEA model discussed in Chapter 1 looks like the following (suppose DMU1 is under evaluation and we have 20 DMUs)
Max weighted outputs for DMU1
subject to (constraints)
Weighted outputs < weighted inputs for DMU1
Weighted outputs < weighted inputs for DMU2
Weighted outputs < weighted inputs for DMU3
...
Weighted outputs < weighted inputs for DMU20
Weighted inputs = 1 (for DMU1)
The super-efficiency model for DMU1 is as follows
Max weighted outputs for DMU1
subject to (constraints)
Weighted outputs < weighted inputs for DMU1(remove this constraint in Solver)
Weighted outputs < weighted inputs for DMU2
Weighted outputs < weighted inputs for DMU3
...
Weighted outputs < weighted inputs for DMU20
Weighted inputs = 1 (for DMU1)
As you can see, the super efficiency model removes the DMU under consideration from the constraints. If DMU2 is under evaluation, we will have the following super efficiency model
Max weighted outputs for DMU2
subject to (constraints)
Weighted outputs < weighted inputs for DMU1
Weighted outputs < weighted inputs for DMU2(remove this constraint in Solver)
Weighted outputs < weighted inputs for DMU3
...
Weighted outputs < weighted inputs for DMU20
Weighted inputs = 1 (for DMU2)
Now, for the model discussed in Chapter 6, the super efficiency model is different. Let us present the model in chapter 6 in the following manner.
Suppose the (unknown) benchmarking shares are (j = 1, 2, ...,17 as in 17 airlines). These represent cells J2:J18. Now, let the score (or efficiency score) in cell E22 be . Then the excel version of the model in chapter 6 can be expressed in an algebraic form
Minimize (objective cell)
subject to (constraints)
*input1 (for DMU1) + *input1 (for DMU2) +...+ *input1 (for DMU17) < *input 1 (for DMU1)
*input2 (for DMU1) + *input2 (for DMU2) +...+ *input2 (for DMU17) < *input 2 (for DMU1)
*input3 (for DMU1) + *input3 (for DMU2) +...+ *input3 (for DMU17) < *input 3 (for DMU1)
*input4 (for DMU1) + *input4 (for DMU2) +...+ *input4 (for DMU17) < *input 4 (for DMU1)
*output1 (for DMU1) + *output1 (for DMU2) +...+ *output1 (for DMU17) > output1 (for DMU1)
*output2 (for DMU1) + *output2 (for DMU2) +...+ *output2 (for DMU17) > output2 (for DMU1)
+ +...+ = 1
If DMU2 is under evaluation, then we need to replace the data on the right-hand-side of the constraints with DMU2.
The super efficiency model for DMU1 is
Minimize (objective cell)
subject to (constraints)
*input1 (for DMU1) + *input1 (for DMU2) +...+ *input1 (for DMU17) < *input 1 (for DMU1)
*input2 (for DMU1) + *input2 (for DMU2) +...+ *input2 (for DMU17) < *input 2 (for DMU1)
*input3 (for DMU1) + *input3 (for DMU2) +...+ *input3 (for DMU17) < *input 3 (for DMU1)
*input4 (for DMU1) + *input4 (for DMU2) +...+ *input4 (for DMU17) < *input 4 (for DMU1)
*output1 (for DMU1) + *output1 (for DMU2) +...+ *output1 (for DMU17) > output1 (for DMU1)
*output2 (for DMU1) + *output2 (for DMU2) +...+ *output2 (for DMU17) > output2 (for DMU1)
+ +...+ = 1
That is, we obtain a super efficiency model, by removing the DMU under evaluation, or setting the benchmarking share = 0
For the homework, please use the your data set for Chapter 1 Homework and solve for the super efficiency scores in Excel. You need to solve for the two models described above.
For the model in chapter 6, it is relatively easy to automate the solving procedure using VBA. For the model in chapter 1, it may be difficult to use VBA - if so, please change one DMU to another manually.
Attachment:- VBA.rar