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In an output-oriented model, an inefficient unit is made efficient through the proportional increase of its outputs, while the inputs proportions remain unchanged.

Reference set

The set of efficient units from which an inefficient unit’s inefficiency has been determined. Originally, the term was used to denote the set of all units in the analysis (i.e. the field) and the set of efficient units was known as a reference subset, a term still used by some.


An input or output factor. Since these are always known beforehand, their values are actually constants.

Overall efficiency

Same as aggregate efficiency

Reference unit

Any unit that is part of a reference set

Variable returns to scale

If it is suspected that an increase in inputs does not result in a proportional change in the outputs, a model which allows variable returns to scale (VRS) such as the BCC model should be considered.

Pareto-efficiency/Pareto-Koopmans efficiency:

Simply stated, a unit is Pareto-efficient when an attempt to improve on any of its inputs or outputs will adversely affect some other inputs or outputs. Formally, Chames et al (1981) consider a DMU to be 100% efficient only when ‘none of its inputs can be decreased without either (i) decreasing some of its outputs, or (ii) increasing some of its other inputs, and none of its outputs can be increased without either (i) increasing one or more of its inputs or (ii) decreasing some of its other outputs’. Since the condition for Pareto-efficiency is that a DMU’s efficiency score is 1, efficiency and Pareto-efficiency are synonymous. [Chames A, Cooper W W and Rhodes E (1981), ‘Evaluating program and managerial efficiency: an application of data envelopment analysis to program follow through’, Mgmt. Sci., 6, pp 668-697.]

Scale efficiency

A unit is said to be scale efficient when its size of operations is optimal so that any modifications on its size will render the unit less efficient. The value for scale efficiency is obtained by dividing the aggregate efficiency by the technical efficiency.

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.

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