Simple DEA example

Efficiency ratio
Efficiency is the ratio between the outputs produced with the amount of inputs used. The 'volume of sales per employee', the 'GDP per capita', or the 'average number of passenger per flight' are common example of efficiency ratio.

The following example introduces several stores. Their performance will measured using efficiency ratios as per illustrated above. Because in DEA's jargons entities are called Decision Making Units (DMU in short — this is to reflect the fact entities can make decision which affect their performance), the stores will often be referred to as 'DMU'.

A stores example
Following the same approach to efficiency, food stores' efficiency can be measured by comparing the volume of sales against the number of employees. We can illustrate the principle with the following data:



Plotting this data with employees on the x axis (the horizontal axis), and the sales on the y axis (the vertical axis) will give the following graph:



The table and graph allows to find the best performers. In this case, store E shows the best performance with an efficiency of 1. The line that spans from the origin to store E is the efficiency frontier. The efficiency frontier illustrates the best observed performance and is said in mathematics to 'envelop' the data; hence DEA's name.

It is interesting to note the difference between statistical approaches, which tend to look at average tendencies (like the regression line above), and DEA, which evaluates an entity's performance by comparing it to the efficiency frontier directly determined from the data.

Becoming efficient
All the DMU that are not on the efficiency frontier are inefficient. An inefficient unit must consequently reach the efficiency frontier in order to become efficient. There are three possibilities:


 * Reduce the inputs while keeping the outputs constant (this is an input oriented approach),
 * Increase the outputs while keeping the inputs constant (this is an output orientated approach).
 * Both increasing outputs while reducing the inputs (this can be done with non-oriented versions of models such as the SBM model — it is unfortunately out of the scope of this introduction to discuss this further).

In effect, the further away from the frontier an entity is, the worse its performance.

This can be illustrated by considering DMUC from the previous graph:



This illustration shows how DMUC can either reach the frontier by reducing its inputs (whilst keeping its output levels constant) and reach 'P', or by increasing its output (whilst keeping its input levels constant) and reach Q.

Returns to scale
It is important to note that the frontier pictures in the graphs above is assumed to stretch to infinity; i.e. that the performance levels of DMUE's (the only efficient store) are possible regardless of the number of employee the store has. This is called Constant returns to scale (Constant RTS). Although the constant RTS assumption is sometimes true for a local range of production, it sometimes need to be relaxed. This is possible with variable returns to scale (there also exist other types of RTS) which are illustrated in the graph below:



Note that under variable returns to scale, DMUB is now also efficient.

Although DMUH is on the efficiency frontier, it is not efficient. This is caused by DMUE which produces a similar amount of 'sales' (i.e. 5) but uses 3 less units of 'employees' to do so. In order for H to be efficient, it will need to reduce its sales force by 3 employees in order to reach E's coordinates. DMUH could also become efficient by increasing its sales. However there is no way to know for sure if this is possible as such production levels have not been observed (i.e. above the current efficiency frontier).

How does DEA work?
As explained earlier, DEA uses a total factor productivity ratio to measure performance (i.e. a unique ratio with all the inputs and outputs). DEA attributes a virtual weight tp each of these input and output.

Entities' performance is then calculated using a linear optimisation process which tries to maximise each entity's ratio by finding the best set of weight for this particular entity.

The optimisation process is contrained by existing data so that each entity is compared against the best observed performance.

This simple example should have introduced most of DEA's basic concepts. Although the example only used a single input and a single output, DEA is more useful when performance needs to take multiple inputs and outputs into account. More information specific to different DEA models (e.g. CCR, BCC, SBM) will be made available on this site correspondingly to the library releases.