The primal model allows the DMU being measured to determine the set of optimal weights for each of its factors (outputs are denoted by y. and inputs by x in the following model) so as to maximise its efficiency. The solution consists of a set of weights (for outputs and y for inputs) chosen so that the efficiency of any other unit with these weights won’t exceed 1, the value at which a unit is relatively efficient. Which model is denoted primal and which dual is arbitrary, some authors prefer to call this model the primal model, as it conveys better the basic idea behind DEA. The convention followed here is the same as that used by the developers in their original paper, Chames et al (1978).

The efficiency scores of the DEA ratio models are independent of the units in which the factors are measured. The input and output values can thus be scaled through multiplication by a constant as proven in Chames and Cooper (1985). [Charnes A, Cooper W W (1985) ‘Preface to topics in data envelopment analysis’, in Thompson R G and Thrall R M (editors), The annals of operations research]

Given a set of inputs that produce outputs, the production function defines an optimum relationship for producing the maximal amount of output from the given inputs. The DEA equivalent of the production function is the efficiency frontier which is based on empirical data (inputs and outputs). See Chames et al (1981) for details and more references. [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.]

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

For a single input-output case, the ratio of a unit’s output to its input. Productivity varies according to changes that occur in the production technology, the efficiency of the production process (which can be measured through DEA) and the production environment (Lovell 1993). [Lovel C A K (1993), ‘Productive frontiers and productive efficiency’, in: Fried H, Knox C A K and Schmidt S (editors) The measurement of productive efficiency: techniques and applications, Oxford University Press.]

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.

The requirement that the relationship between inputs and outputs not be erratic. Increasing the value of any input while keeping other factors constant should not decrease any output but should instead lead to an increase in the value of at least one output.

Refers to an inefficient DMU’s composite unit to emphasise that geometrically it involves the projection of the inefficient DMU onto the efficiency frontier (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.]

Virtual input is obtained for each input by taking the product of the input’s value and its corresponding optimal weight as given by the solution to the primal model. Virtual outputs are obtained analogously. A virtual input or output describes the importance attached to the given factor. The virtual inputs always add up to the maximum efficiency score (ie 1) for the unit being analysed, while the sum of the virtual outputs will equal that unit’s efficiency.

MPSS is a unit (point) on the efficiency frontier that maximises the average productivity for its given input-output mix and after which decreasing returns to scale set in. See Banker & Kemerer (1989) on how to compute the MPSS. [Banker R and Kemerer C (1989) ‘Scale economies in new software development’, IEEE Trans. On Softw. Eng.., 15, pp 1199-1205.]