## Input-oriented

A term used in conjunction with the ratio models to indicate that an inefficient unit is made efficient through the proportional reduction of its inputs while its outputs proportions are held constant. While the CCR model yields the same efficiencies regardless of whether it is input or output-oriented, this is not so with the BCC model.

## Input

Any factor used as a resource by the DMU for producing something of value.

## Facet

In higher dimensions, the segments forming the efficiency frontier are known as facets.

## Dual weights – dual multipliers

While in the primal DEA model weights are associated with factors, each dual weight (so called because they are obtained as a result of  solving the dual DEA model ) is associated with a DMU. The value of the dual weight ? indicates the importance attributed to unit j in defining tlie input-output mix of the composite unit (Oral and Yolalan 1990).

## Dual Model

While the dual (CCR model) yields the same efficiency score as the primal model, it provides another way at looking at the same problem. Mathematically the dual may look more complex, but is much faster to solve as it has only as many constraints as there are factors. To  see if a unit is efficient or not, the dual model internally tries to construct a hypothetical composite unit out of existing units that will outperform the given unit. If it can, the unit in question is inefficient, otherwise it is efficient.

## DMU (Decision making unit)

In their original paper on OEA (Chanes et al 1978), the authors used this term to emphasise the fact that the focus was not on profit generating enterprises, but rather on decision making entities. Later on, DEA came to be applied to the private sector as well.

## Discriminatory power

measure of the extent that the OEA model has succeeded in discriminating between the DMUs. The discriminatory power is directly related to the number of inefficient units. Since the efficiency measure is relative and at least one unit will always be efficient, OEA cannot discriminate between the whole field.

## Discretionary factor

A unit is said to operate at decreasing returns to scale (DRS) if a proportionate increase in all of its inputs results in a less than proportionate increase in its outputs. If for a given DMU, the sum of the dual weights in the dual model (M2) is greater than 1, then that unit can be said to operate at DRS, assuming it is technically efficient. See Banker and Morey (1986) or Banker and Thrall (1992) for an in-depth analysis.
– Banker R D and Morey R (1986) ‘The use of categorical variables in data envelopment analysis’, Mgmt. Sci., 32, pp 1613-1627.
– Banker R D and Thrahl R M (1992) ‘Estimation of returns to scale using data envelopment analysis’, Eur. J. of OpI. Res., 62, pp 74-84.

## Cross efficiency matrix

This tool for interpreting the results consists of creating a table where the number of rows (j) and columns (j) equals the number of units in the analysis. For each cell (ij), the efficiency of unit jis computed with weights that are optimal to unit j. The higher the values in a given column j, the more likely it is that the unit jis an example of truly efficient operating practices (Doyle and Green 1994).
-Doyle J and Green R (1994) ‘Efficiency and cross-efficiency in DEA: derivations, meanings and uses’, J. of the Opi. Res. Soc., 45, pp 567-578.

## Convexity constraint

Ensures that the region specified by a set of points (units) forms a convex set. The BCC model is obtained by simply adding a convexity constraint Sigma (Lambdas)=1 to the dual of the CCR model, which means that each composite unit is a convex combination of its reference units. For a dull discussion on convexity, see a standard non-linear programming text such as Bazaraa et aI (1993). For details about BCC model See Banker et al (1984).

-Banker, R.D., R.F. Charnes, & W.W. Cooper (1984) “Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis, Management Science vol. 30, pp. 1078–1092.
– Bazaraa M, Sherahi H and Shetty C M, Nonlinear programming, John Wiley & Sons, 1993, or later edition.

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