Editing Data envelopment analysis (section)

==Techniques==
In a one-input, one-output scenario, [[efficiency]] is merely the ratio of output over input that can be produced, while comparing several entities/DMUs based on it is trivial. However, when adding more inputs or outputs the efficiency computation becomes more complex. Charnes, Cooper, and Rhodes (1978)<ref name=":0" /> in their basic DEA model (the CCR) define the objective function to find <math>DMU_j's</math> efficiency <math>(\theta_j)</math> as:

:<math>\max \quad \theta_j = \frac{\sum\limits_{m=1}^{M}y_m^j u_m^j}{\sum\limits_{n=1}^{N}x_n^j v_n^j},</math>

where the <math>DMU_j's</math> known <math>M</math> outputs <math>y_1^j,...,y_m^j</math> are multiplied by their respective weights <math>u_1^j,...,u_m^j</math> and divided by the <math>N</math> inputs <math>x_1^j,...,x_n^j</math> multiplied by their respective weights <math>v_1^j,...,v_n^j</math>.

The efficiency score <math>\theta_j</math> is sought to be maximized, under the constraints that using those weights on each <math>DMU_k \quad k=1,...,K</math>, no efficiency score exceeds one:

:<math>\frac{\sum\limits_{m=1}^{M}y_m^k u_m^j}{\sum\limits_{n=1}^{N}x_n^k v_n^j} \leq 1 \qquad k = 1,...,K,</math>

and all inputs, outputs and weights have to be non-negative. To allow for linear optimization, one typically constrains either the sum of outputs or the sum of inputs to equal a fixed value (typically 1. See later for an example).

Because this [[optimization]] problem's dimensionality is equal to the sum of its inputs and outputs, selecting the smallest number of inputs/outputs that collectively, accurately capture the process one attempts to characterize is crucial. And because the production frontier envelopment is done empirically, several guidelines exist on the minimum required number of DMUs for good discriminatory power of the analysis, given homogeneity of the sample. This minimum number of DMUs varies between twice the sum of inputs and outputs (<math>2 (M + N)</math>) and twice the product of inputs and outputs (<math>2 M N</math>).

Some advantages of the DEA approach are:
* no need to explicitly specify a mathematical form for the production function
* capable of handling multiple inputs and outputs
* capable of being used with any input-output measurement, although ordinal variables remain tricky
* the sources of inefficiency can be analysed and quantified for every evaluated unit
* using the dual of the optimization problem identifies which DMU is evaluating itself against which other DMUs

Some of the disadvantages of DEA are:
* results are sensitive to the selection of inputs and outputs
* high-efficiency values can be obtained by being truly efficient or having a niche combination of inputs/outputs
* the number of efficient firms on the frontier increases with the number of inputs and output variables
* a DMU's efficiency scores may be obtained by using non-unique combinations of weights on the input and/or output factors