英语论文网

ossibly do that. Bank A is therefore deemed to beecient. 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, thenwe create a bank that processes 600 checks and 85 loan applications with just 10 tellers. Thisdominates B (we would much rather have the virtual bank we created than bank B). Bank B is
therefore inecient.
Another way to see this is that we can scale down the inputs to B (the tellers) and still have atleast 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'seciency rating is .63. Banks A and C have an eciency rating of 1.
12.2 Graphical Example
The single input two-output or two input-one output problems are easy to analyze graphically.
The previous numerical example is now solved graphically. (An assumption of constant returns to
scale is made and explained in detail later.) The analysis of the eciency for bank B looks like the
following:
O
20
50
150
200 400 1000 Checks
Loans
C
B
A
V
12.3. USING LINEAR PROGRAMMING 149
If it is assumed that convex combinations of banks are allowed, then the line segment connectingbanks A and C shows the possibilities of virtual outputs that can be formed from these two banks.
Similar segments can be drawn between A and B along with B and C. Since the segment AC liesbeyond the segments AB and BC, this means that a convex combination of A and C will create
the most outputs for a given set of inputs.
This line is called the eciency frontier. The eciency frontier de nes the maximum combina-
tions of outputs that can be produced for a given set of inputs.
Since bank B lies below the eciency frontier, it is inecient. Its eciency can be determinedby comparing it to a virtual bank formed from bank A and bank C. The virtual player, called V,
is approximately 54% of bank A and 46% of bank C. (This can be determined by an applicationof the lever law. Pull out a ruler and measure the lengths of AV, CV, and AC. The percentage of
bank C is then AV/AC and the percentage of bank A is CV/AC.)The eciency of bank B is then calculated by nding the fraction of inputs that bank V would
need to produce as many outputs as bank B. This is easily calculated by looking at the line from
the origin, O, to V. The eciency of player B is OB/OV which is approximately 63%. This gure
also shows that banks A and C are ecient since they lie on the eciency frontier. In other words,
any virtual bank formed for analyzing banks A and C will lie on banks A and C respectively.
Therefore since the eciency is calculated as the ratio of OA/OV or OA/OV, banks A and C will
have eciency scores equal to 1.0.
The graphical method is useful in this simple two dimensional example but gets much harder in
higher dimensions. The normal method of evaluating the eciency of bank B is by using an linear
programming formulation of DEA.
Since this problem uses a constant input value of 10 for all of the banks, it avoids the com-
plications caused by allowing di erent returns to scale. Returns to scale refers to increasing or
decreasing eciency based on size. For example, a manufacturer can achieve certain economies of