The measurement of relative efficiency where there are multiple possibly incommensurate inputs and outputs was addressed by Farrell and developed by Farrell and Fieldhouse(2), focusing on the construction of a hypothetical efficient unit, as a weighted average of efficient units, to act as a comparator for an inefficient unit.
A common measure for relative efficiency is,
which introducing the usual notation can be written as
(Note efficiency is usually constrained to the range [0,1]).
The initial assumption is that this measure of efficiency requires a common set of weights to be applied across all units. This immediately raises the problem of how such an agreed common set of weights can be obtained. There can be two kinds of difficulties in obtaining a common set of weights. First of all it may simply be difficult to value the inputs or outputs. For example in the depot data the weights on the outputs presumably relate to the values or cost of producing the outputs but these costs or values may be difficult to measure. Alternatively different depots may choose to organise their operations differently so that the relative values of the different outputs may legitimately be different. This perhaps becomes clearer if an attempt has been made to compare the relative efficiency of schools with achievements at music and sport amongst the outputs. Some schools may legitimately value achievements in sport or music differently to other schools, and in general units may value inputs and outputs differently and thus require different weights. This measure of efficiency coupled with the assumption that a single common set of weights is required is thus unsatisfactory.