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188 CHAPTER 4 LINEAR PROGRAMMING APPLICATIONS

But with an additional constraint that:

275:7v 1 þ 348:5v 2 þ 104:1v 3 ¼ 1

The full formulation is then:

Max :

36:72u 1 þ 45:98u 2 þ 175u 3 þ 23u 4
s:t:

275:7v 1 þ 348:5v 2 þ 104:1v 3 þ 1
48:14u 1 þ 43:1u 2 þ 253u 3 þ 41u 4 285:2v 1 123:8v 2 106:72v 0
34:62u 1 þ 27:11u 2 þ 148u 3 þ 27u 4 162:3v 1 128:7v 2 64:21v 3 0

36:72u 1 þ 45:98u 2 þ 175u 3 þ 23u 4 275:7v 1 348:5v 2 104:1v 3 0
33:16u 1 þ 56:46u 2 þ 160u 3 þ 84u 4 210:4v 1 154:1v 2 104:04v 3 0

u 1 ; u 2 ; u 3 ; u 4 ; v 1 ; v 2 ; v 3 0

The LP problem can now be solved in the usual way. Figure 4.12 shows the results.
The objective function takes a value of 0.902. This is the efficiency score for County
hospital. Given that this is less than 1 tells us that County hospital is relatively
inefficient compared to at least one other hospital in the data set. Using the
optimum weights for County hospital, we can calculate the efficiency scores for
the other hospitals:

General hospital: 0.99
University hospital: 1.00
County hospital: 0.90
City hospital 1.00

In fact we see that all the other hospitals are more efficient than County with
both University and City hospitals having maximum efficiency. This implies that
The ‘envelopment’ part
of DEA comes from the County’s efficiency can be improved – more outputs and/or less inputs are
fact that the efficient possible given the performance of the other hospitals. Advanced analysis of the
units create an ‘envelope’ DEA results (which is beyond this text) would also enable managers to set
or frontier envelope, or numerical targets in terms of what input/output quantities should be for County.
frontier, around the data
set. We could now repeat the analysis for the other three hospitals in the data set to
assess their relative efficiency. For example, we could re-formulate the problem
to look at General hospital. The objective function would now use the numerical
parameters for General hospital from Table 4.17 and the equality constraint
would also use appropriate numerical parameters. Solving the new problem
would give the optimum input/output weights for General hospital and again
its own efficiency score.

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