## 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.

## Convex hull

The convex hull for a set of units (represented as points) is defined as the smallest convex polygon that encompasses all the points and is a further restriction on a convex cone. The envelopment surface specified by the BCC model results in a convex hull. See also: convex one.

## Convex cone – Conical hull

A convex cone has its vertex as its origin and includes all nonnegtive linear combinations of its points. The envelopment surface obtained from the CCR model has the shape of a convex cone. If the situation in Figure 1 were drawn 3-dimensionally, the efficiency frontier would consist of a set of interlaced facets formed by hyperplanes emanating from the origin and forming a conical structure. The efficient units would lie on top of the facets, while the inefficient ones would be covered under the cone. For a clear visual representation, refer to Ah and Seiford (1993).
-Ah A and Seiford L (1993), ‘The mathematical programming approach to efficiency analysis’, in: Fried H, Knox C A K and Schmidt S (editors), The measurement of productive efficiency: techniques and applications, Oxford University Press.

## Constant returns to scale

unit operates under constant returns to scale if an increase in inputs results in a proportionate increase in the output levels. If the inputs values for a unit are all doubled, then the unit must produce twice as much outputs. In a single input and output case, the efficiency frontier reduces to a straight line. See also: variable returns to scale.

## Decreasing returns to scale

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.