The current paper is focused on assessment of farms from marginal regions, in Less Favourable Area (LFA) by Data Envelopment Analysis (DEA). Farming in marginal regions does not have to be profitable, but it is necessary for social or ecological reasons. Therefore, negative values of outputs are present in the considered set of farms.

A group of 55 farms of similar characteristics is considered. As inputs, total assets, agricultural land area, man effort and fiscal assets are taken into account, and, as outputs, yields in sum and income from operations before tax.

DEA models described by Farrel (1957) and Charnes *et al.* (1978) assume non-negative values of inputs and outputs. The dataset of the current paper contains negative values and therefore this condition can not be met. For this reason, the generic directional distance model proposed by Chambers *et al.* (1996, 1998) are used to handle these negative data.

Consider a set of units *k*=1, 2, …, *p*, with input levels *x* _{ij}, *i*=1, 2, …, *m* and output levels *y* _{jk}, *j*=1, 2, …, *n* and unit *o* ∊ *k* which is to be assessed. Vector g_{xi}(g_{yj}) represents possible changes of input (output). The generic directional distance model is as follows:

This model (1) is valid for the case of variable returns to scale (VRS) and with input and output vectors in **R**^{m+n}. Target values of inputs (outputs) were obtained as product X (Y) and λ.

This method provides efficiency scores similar to radial efficiencies traditionally used in DEA without previous transformation of negative data. The second advantage of this model is the ability to project inefficient units onto the efficiency frontier with a selected direction. This feature is applicable to the application in the current paper, because the model provides each unit with its own path to efficiency representing its improvement potentials.

Efficiency scores are defined by Eqn (2).

Where ϕ_{o} is efficiency for assessed unit o,

• *x**_{io} is target value of *i*-th input projected on efficiency frontier,

• *y**_{jo} is target value of *j*-th input projected on efficiency frontier,

• *R* _{jo}=max_{k}{ *y* _{jk}}−*y* _{jo}, *j*=1, 2, …, *n*, and

• *R* _{io}=*x* _{io}−min_{k}{ *x* _{ik} }, *i*=1, 2, …, *m*.

Efficient units B, D, F, G and inefficient units A, C, E are presented. Classical DEA model using radial projection recommends, for point E, raising output 1 to 4387 and output 2 to 6047. However, it is not always possible to follow this recommendation. For example, for point E output 2 cannot be changed, therefore directional vector g=(1, 0) was used. With this direction vector, output 2 remains the same and output 1 should rise to 7·5. This projection is depicted with the dotted line. Direction vectors allow restriction of changes of inputs or outputs. For unit A, a projection on efficient frontier is conducted according to direction vector g=(0, 1). Projected point for unit A would be in another place on the efficiency frontier, but the model takes into account efficiency frontier from point B to point G (not vertical and horizontal part of that). Therefore, projection of A can go through point B.

Farrell, M. J. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society, Series A, General 120, 253–281.

Chambers, R. G., Chung, Y. & Färe, R. (1996). Benefit and functions. Journal of Economic Theory 70, 407–419.

Chambers, R. G., Chung, Y. & Färe, R. (1998). Profit, directional distance functions and Nerlovian efficiency. Journal of Optimization Theory and Applications 98, 351–364.

Charnes, A., Cooper, W. W. & Rhodes, E. (1978). Measuring efficiency of decision-making units. European Journal of Operational Research 2, 429–444.