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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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    Solution Preview

    Please see attached files.

    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:
    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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