A DMU is said to operate at increasing returns to scale (IRS) if a proportionate increase in all of its inputs results in a greater than proportionate increase in its outputs. If for a given unit the sum of its dual weights in the dual model turns out to be less than 1, then that unit can be said to operate at IRS, assuming it is technically efficient.

## Input

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

## 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.

## Efficiency/Productive efficiency

A measure of a unit’s ability to produce outputs from its set of inputs. Since the efficiency of a given DMU is measured with respect to other DMUs in the field, the obtained efficiency is always relative. From the standpoint of the efficiency frontier, an enveloped unit’s efficiency is related to its radial distance from the frontier.

## Input/output mix

The proportions in which a unit’s input/output values are combined.

## Efficiency score

Efficiency is measured on a scale of 0 to 1, where a value of I indicates the unit is relatively efficient, and a value less than 1 indicates the unit is inefficient. The efficiency score of a unit will vary according to the factors and DMUs included in the analysis. In the early days of DEA, there was some concern as to whether it was justified to rank inefficient units according to their efficiency scores on the grounds that the scores may have been calculated from distinct reference sets.

## Envelopment form

A DEA model formulation involving the concept of a composite unit. The term ‘envelopment’ stresses the fact that a composite unit is a combination of efficient units enveloping an inefficient unit.

## Environmental factor

While in the strictest sense all factors depend on their environment, an input or output factor that purposefully attempts to make up for the inherent differences in the DMIJs being analysed is generally referred to as an environmental factor. The larger the field, the more probable that some units have intrinsic advantages over others. Environmental inputs typically take the age and the nature of its resources (je educational level of the staff) into account, while environmental outputs usually focus on the quality, rather than the quantity, of the outputs.

## Epsilon

A very small positive constant (generally taken to be 1.0E-6) whose purpose in the primal model, Epsilon prevents any factor from being abused as a free commodity and avoid a unit being falsely classified as efficient. In the dual model, epsilon’s role is to make sure that the minimisation of the intensity factor, takes precedence over the maximisation of the sum of the slacks. The value of epsilon may cause computational inaccuracies, a solution is provided in Ali and Seiford (1993). Ali 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.

## Exogeneously fixed factor

An input or output that is beyond the control of the DMU manager. A fast-food restaurant may have an input such as years of operations, that it cannot control and hence the unit cannot improve its efficiency by reducing that input.