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Data Envelopment Analysis

This article is based on a talk given by Professor Robert Dyson of Warwick Business School to the Mathematical Programming Study Group.

Measuring Efficiency

Data Envelopment Analysis (DEA) is concerned with comparing the efficiency of organisations such as local authority departments, British Telecom districts, schools, retailers and bank branches. It is applied where there are many fairly similar units each of which has multiple inputs and multiple outputs.

For instance, to assess the efficiency of petrol stations one might draw up this list of inputs:

  • number of pumps;
  • population in catchment area;
  • number of cars per head in catchment area;
  • competitors in the catchment area;
  • income of households;
  • state of repair of petrol station;
  • shelf space in convenience store;

and outputs:

  • petrol sales;
  • convenience store sales.

If one simply had a single input and a single output one would define a measure of efficiency as:

    efficiency = output / input

and normalise it to be less than or equal to 1.

The natural extension to multiple inputs and outputs is to use weighted sums of the inputs and outputs:

    efficiency = weighted outputs / weighted inputs

                   = Σi ui xi / Σj vj yj

The DEA Approach

If everyone could agree on a common set of weights {ui, vj} that would be the end of the story. But people cannot agree. This is where DEA comes in.

It allows units in the system to choose their own weights in the way which is most advantageous to themselves. If a unit is inefficient even with the set of weights which is most favourable to it, then there are serious grounds for investigating further.

Each unit is considered in turn and its most favourable weights are selected. The efficiency of all other units is computed using this set of weights. The result is that for each unit we obtain a series of relative efficiencies both using those weights most favourable to itself and those most favourable to other units.

Allowing this flexibility in setting the weights has both strengths and weaknesses. Permitting each unit to show itself in its most favourable light strengthens the evidence where units are found to be inefficient and at the same time makes such results more palatable. On the other hand, if too many inputs and outputs are considered, every unit may be efficient on its own terms. The underlying linear model of efficiency is also open to criticism, although this can be addressed to some extent by transforming the raw data (e.g. taking logarithms).

Aligning DEA with an Organisation's Aims

The implication is that DEA is a tool which needs to be applied with care and judgement. There needs to be a large enough number of similar units and this number must be much greater than the number of inputs and outputs chosen. Value judgements may be needed to constrain weights to ensure that the results are consistent with the purpose of the units.

At their least controversial, constraints on weights are simply common-sense. For instance, in a study of perinatal care there were separate measures of output for the numbers of very satsified mothers, satisfied mothers and not dissatisfied mothers. The weights on these were constrained so that the weight on very satisfied mothers was greater than or equal to that on satsified mothers which was greater than or equal to that on not dissatisfied mothers. It would be an odd health authority which proclaimed its efficiency by placing greater weight on those who acquiesced to its services than those who were pleased with them.

Similarly, few would challenge the need to restrict weights when faced with a finding that Liverpool's Rates Department appeared efficient only by loading its entire output weights onto the number of summons and distress warrants issued with zero weight on the revenue raised.

Performance Targets

When DEA assesses a unit as inefficient, it identifies those other units m* by which the unit has been found to be inefficient. It produces a set of weighting factors {lm*} such that the outputs of the inefficient unit could be produced using fewer inputs by a "target" unit comprising a weighted combination of the units m*. (An alternative formulation constrains the inputs and then shows the greater outputs which would be achieved by the composite efficient unit).

It is one of the strengths of DEA that it automatically produces targets where it finds units to be inefficient. However, the target which is generated automatically is not necessarily that to which the inefficient unit would aspire and DEA software (such as that developed at Warwick) enables the user to explore the entire efficient frontier which improves on the unit's current performance.

Figure 1: Setting Targets for an Inefficient Unit

In the diagram there are two outputs, y1 and y2 whose values are shown relative to a single (fixed) input. The efficient frontier is defined by points P1, P2, P3 and P4. Unit P5 is inefficient and its natural target is the point P* which lies where the ray traced from the origin intersects the efficient frontier (i.e. pro rata increases in the outputs). However, there is no reason why it should not choose any target on the efficient frontier between P' and P": this would amount to increasing one of the outputs more than the other. If it really wanted to, it could choose any point on the efficient frontier at all, but this would raise questions about why it was producing its current levels of the outputs.

Using DEA in Practice

One way in which DEA is used, for instance in studies commissioned by the Audit Commission, is to identify those units worth further investigation (either because they are very good or very bad). The more detailed study then leads on to a report describing best practice and making recommendations.

Somewhat similar to this is the use which a brewery made of DEA when required to divest itself of a large number of pubs. Rather than simply selling those which were least profitable, it used DEA to assess the performance of its pubs. Those which were efficient but unprofitable or marginally profitable were sold but those which were inefficient and unprofitable were investigated further to determine whether they could become profitable and should be retained.

When DEA is used to set targets one needs to beware of the possibility of undesirable behavioural responses. For instance, if one of the output measures for a university department is the number of papers published, academics may improve their measured efficiency by submitting each minor advance as a fresh paper to a different journal.

DEA therefore cannot be used in isolation but must be seen as a tool within the complete cycle of management.

Figure 2: Performance Measurement and Control

The stages of DEA: defining inputs and outputs; determining value systems; measuring performance; assessing it and setting targets, must all be related to the aims and values of the organisation. Within this context DEA becomes part of a mission-driven framework of performance measurement and improvement.

Related articles include Data Envelopment Analysis and its Use in Banking. To find other articles, refer to the MP in Action page.

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