To illustrate how DEA works, let's take an example of three banks. Each bank has exactly 10 tellers (the only input), and we measure a bank based on two outputs: Checks cashed and Loan applications. The data for these banks is as follows:
Now, the key to DEA is to determine whether we can create a virtual bank that is better than one or more of the real banks. Any such dominated bank will be an inefficient bank.
Consider trying to create a virtual bank that is better than Bank A. Such a bank would use no more inputs than A (10 tellers), and produce at least as much output (1000 checks and 20 loans). Clearly, no combination of banks B and C can possibly do that. Bank A is therefore deemed to be efficient. Bank C is in the same situation.
However, consider bank B. If we take half of Bank A and combine it with half of Bank C, then we create a bank that processes 600 checks and 85 loan applications with just 10 tellers. This dominates B (we would much rather have the virtual bank we created than bank B). Bank B is therefore inefficient.
Another way to see this is that we can scale down the inputs to B (the tellers) and still have at least as much output. If we assume (and we do), that inputs are linearly scalable, then we estimate that we can get by with 6.3 tellers. We do that by taking .34 times bank A plus .29 times bank B. The result uses 6.3 tellers and produces at least as much as bank B does. We say that bank B's efficiency rating is .63. Banks A and C have an efficiency rating of 1.