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Data Envelopment Analysis

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Data envelopment analysis can measure the relative efficiency of a group of hospitals. The following data from a particular study involving seven teaching hospitals include three input measures and four output measures.

Input Measures
Hospital Full-Time Supply Bed-Days
Equivalent Expense Available
Nonphysicians (1000s) (1000s)
A 310.0 134.60 116.00
B 278.5 114.30 106.80
C 165.6 131.30 65.52
D 250.0 316.00 94.40
E 206.4 151.20 102.10
F 384.0 217.00 153.70
G 530.1 770.80 215.00

Output Measures
Hospital Patient-Days Patient-Days Nurses Interns
(65 or older) (Under 65) Trained Trained
(1000s) (1000s)
A 55.31 49.52 291 47
B 37.64 55.63 156 3
C 32.91 25.77 141 26
D 33.53 41.99 160 21
E 32.48 55.30 157 82
F 48.78 81.92 285 92
G 58.41 119.70 111 89

a.Formulate a linear programming model so that data envelopment analysis can be used to evaluate the performance of hospital D.
b.Solve the model.
c.Is hospital D relatively inefficient? What is the interpretation of the value of the objective function?
d.How much patient-days of each type are produced by the composite hospital?
e.Which hospitals would you recommend hospital D consider emulating to improve the efficiency of its operation?

(See attached file for full problem description)

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For theory: have a look at the reference website

a. Let Ei be the efficiency of hospital I, we have:

where wj the weight of output j
mj the value of output measure j
wk the weight of input k
mk the value of input measure k

We want to maximize the weighted output of hospital D, which is:
Maximise
Subject to the following constraints:

The first two constraints require the weighing factor to be positive. The third constraint requires the weighted input of hospital D to be 1. This is required because we want to normalize and linearise the program (see reference for more details).

There is also another constraint which ...

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