DEA Tutorial
Abstract
Data envelopment analysis (DEA) is a linear programming based technique for measuring the relative performance of organisational units where the presence of multiple inputs and outputs makes comparisons difficult. This tutorial paper introduces the technique and uses an example to show how relative efficiencies can be determined and targets for inefficient units set. The paper also considers a number of practical issues of concern in applying the technique.
Introduction
There is an increasing concern with measuring and comparing the efficiency of organisational units such as local authority departments, schools, hospitals, shops, bank branches and similar instances where there is a relatively homogeneous set of units.
The usual measure of efficiency, i.e.:
[pic]
is often inadequate due to the existence of multiple inputs and outputs related to different resources, activities and environmental factors. This problem can be illustrated for depots of a large retailing organisation which distributes goods to supermarkets. In this case the inputs for an efficiency measure are taken to be the value of the stock and the recurrent costs mainly in the form of wages, as those are resources supporting the operation of the depots. Stock is reasonably an input to an efficiency measure as an efficient depot will attempt to give a good service at lower stock levels, saving on capital and space. The outputs correspond to activities of the depots and are measured by the number of issues representing deliveries to supermarkets, the number of receipts in bulk from suppliers, and the number of requisitions on suppliers where they are out of stock or approaching stock out. (A further possible output would be a measure of the service level). The data on these measures is shown in Appendix l.
With two inputs and three outputs the difficulty of comparing the efficiency of depots becomes apparent. Some statements concerning the relative efficiency of depots can be made, for example comparing depots 10 and 11. These both have the same stock levels and the same costs but depot 10 has activity levels which are all as great or greater than those of depot 11. Clearly if these inputs and outputs are representative then depot 10 is more efficient than depot 11. However the majority of comparisons are difficult to make. For example comparing depots 14 and 15, depot 14 has higher inputs and has higher outputs on two measures but lower on one measure. Here different patterns of activity levels are supported by different amounts of resources making efficiency comparisons difficult and this is generally the case.
A formula for relative efficiency incorporating multiple inputs and outputs is introduced now, and the DEA model which allows relative efficiency measures to be determined is developed. This is followed by a discussion of the information made available by solving the model and some issues of practical concern in applying the technique.
Relative efficiency measurement
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,
[pic]
which introducing the usual notation can be written as
[pic]
(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.
The Data Envelopment Analysis
Charnes, Cooper and Rhodes(3) recognised the difficulty in seeking a common set of weights to determine relative efficiency. They recognised the legitimacy of the proposal that units might value inputs and outputs differently and therefore adopt different weights, and proposed that each unit should be allowed to adopt a set of weights which shows it in the most favourable light in comparison to the other units. Under these circumstances, efficiency of a target unit j0 can be obtained as a solution to the following problem:
Maximise the efficiency of unit j0, subject to the efficiency of all units being < =1.
The variables of the above problem are the weights and the solution produces the weights most favourable to unit j0 and also produces a measure of efficiency.
The algebraic model is as follows:
[pic]
For the depot data, the efficiency of depot 1 is obtained by solving the following model:
[pic]
The u's and v's are variables of the problem and are constrained to be greater than or equal to some small positive quantity [pic]in order to avoid any input or output being totally ignored in determining the efficiency. The solution to the above model gives a value h0, the efficiency of depot 1, and the weights leading to that efficiency. If h0 = 1 then depot I is efficient relative to the others but if ho turns out to be less than l then some other depot(s) is more efficient than depot l, even when the weights are chosen to maximise depot l 's efficiency.
This flexibility in the choice of weights is both a weakness and a strength of this approach. It is a weakness because the judicious choice of weights by a unit possibly unrelated to the value of any input or output may allow a unit to appear efficient but there may be concern that this is more to do with the choice of weights than any inherent efficiency. This flexibility is also a strength, however, for if a unit turns out to be inefficient even when the most favourable weights have been incorporated in its efficiency measure then this is a strong statement and in particular the argument that the weights are incorrect is not tenable.
DEA thus may be appropriate where units can properly value inputs or outputs differently, or where there is a high uncertainty or disagreement over the value of some input or outputs.
The DEA model M1 is a fractional linear program. To solve the model it is first necessary to convert-it into linear form so that the methods of linear programming can be applied. The linearisation process is relatively straightforward. The linear version of the constraints of Ml is shown in model M3. For the objective function it is necessary to observe that in maximising a fraction or ratio it is the relative magnitude of the numerator and denominator that are of interest and not their individual values. It is thus possible to achieve the same effect by setting the denominator equal to a constant and maximising the numerator. The resultant linear program is as follows:
[pic]
Solutions to the DEA model
The efficiency of the target unit in a set can be obtained by solving M3. The solution to this LP provides a measure of the relative efficiency of the target unit and the weights leading to that efficiency. These weights are the most favourable ones from the point of view of the target unit. To obtain the efficiencies of the entire set of units it is necessary to solve a linear program focusing on each unit in turn. Clearly as the objective function is varying from problem to problem the weights obtained for each target unit may be different. The efficiencies obtained for the full set of depots are shown in Table l.
Table 1
|Depot|Efficienc|
| |y |
|19 |1.00 |
|15 |1.00 |
|14 |1.00 |
|12 |1.00 |
|9 |0.96 |
|5 |0.95 |
|2 |0.94 |
|16 |0.91 |
|10 |0.89 |
|20 |0.84 |
|13 |0.83 |
|6 |0.83 |
|1 |0.82 |
|3 |0.82 |
|7 |0.71 |
|4 |0.65 |
|11 |0.63 |
|17 |0.55 |
|8 |0.52 |
|18 |0.42 |

A concern with the DEA model is that if all units can adopt their most favourable weights, they may all appear efficient. With the depot data we can see that there is in fact a considerable difference in efficiencies. At one extreme depot 18 has an efficiency of only 0.42. This can broadly be interpreted as saying that depot 18 should have been able to support its activity levels with only 42% of its resources. In fact 12 of the depots have an efficiency below 0.9 indicating a fair degree of discrimination. This degree of discrimination is similar in scale to that reported by Thanassoulis, Dyson and Foster(4) and in other published studies.
In solving each linear program the solution technique will attempt to make the efficiency of the target unit as large as possible. This search procedure will terminate when either the efficiency of the target unit or the efficiency of one or more other units hits the upper limit of 1. Thus for an inefficient unit at least one other unit will be efficient with the target unit's set of weights. These efficient units are known as the peer group for the inefficient unit. Table 2 shows the peers for depot 18.
Table 2 Peer Units for Depot 18
|Depot 18 |Input or Output|Depot 12 |Depot 15 |Depot 19 |
|4.0 |- stock |2.0 |2.0 |3.0 |
|6.0 |- wages |4.0 |3.0 |4.0 |
|25.0 |+ issues |45.0 |20.0 |45.0 |
|38.0 |+ receipts |40.0 |50.0 |67.0 |
|20.0 |+ reqs |44.0 |15.0 |32.0 |

(Note - indicates an input and + an output).
It is sometimes useful to scale the data on the peer units so that a better comparison of the inefficient unit with the peer units can be made. In Table 3 the peers have been scaled on one of the inputs, so that each peer unit uses no more of an input than the inefficient unit. In this example inefficiency is clearly demonstrated as all the outputs of the peer units are greater than those of depot 18. (These comparisons assume constant returns to scale).
Table 3 Scaled Peer Units for Depot 18
|Depot 18 |Input or Output|Depot 12 |Depot 15 |Depot 19 |
|4.0 |scale |1.500 |2.000 |1.33 |
|6.0 |- stock |3.0 |4.0 |4.0 |
|25.0 |+ issues |67.5 |40.0 |60.0 |
|38.0 |+ receipts |60.0 |100.0 |89.3 |
|20.0 |+ reqs |66.0 |30.0 |42.7 |

Finally for each inefficient unit the linear programming solution will provide a set of target inputs and outputs. The targets correspond to either a pro rata decrease in inputs or increase in outputs. For a unit at an extreme of the data set a pro rata decrease on inputs may be insufficient and will need to be coupled with an increase on one or more of the outputs or further decreases in certain inputs. This will be illustrated in the subsequent section. For the inefficient depot 18 the targets are shown in Table 4.
Table 4 Targets for Depot 18 (efficiency 0.42)
|Variable |Actual|Target |
|- stock |4.0 |1.7 |
|- wages |6.0 |2.5 |
|+ issues |25.0 |25.3 |
|+ receipts|38.0 |38.0 |
|+ reqs |20.0 |20.0 |

The above targets simply consist of a reduction in inputs of 42% of their current levels plus a small increase in issues. It will often be the case that some inputs or outputs are uncontrollable and therefore targets associated with them are meaningless. Banker and Morey(5) address the problem where some inputs or outputs are exogenously fixed, and Thanassoulis and Dyson(6) show how a variety of different targets can be obtained for an inefficient unit.
The solution to the DEA model thus provides a relative efficiency measure for each unit in the set, a subset of peer units for each inefficient unit, and a set of targets for each inefficient unit.
Graphical representation
Figure l shows a set of units P1, P2,... P6 with each unit consuming the same amount of a single resource and producing different amounts of outputs, y1 and y2 as shown. For a given amount of resource input, units providing greater amounts of the outputs will be the efficient ones. Applying the DEA approach to this set of units will identify units P1, P2, P3 and P4 as efficient and they provide an envelope round the entire data set units P5 and P6 are within this envelope and are inefficient. The data envelope has been notionally extended to the axes by the lines P1y2' and P4y1' to enclose the data set.
For unit P5 the peer group consists of the units P1 and P2 and a set of targets for P5 is provided at P5'. These targets are obtained by a pro rata increase in the outputs of unit P5. Clearly there are other possible targets for P5 and for example if the output level Y2 could not be increased for P5 then a target P5" could be set which would rely entirely on increasing output y1. For unit P6 the pro rata increase leads to the set of targets P6'. However P6' is clearly dominated by P4 which produces the same amount of output y1 but more output y2. In this case the pro rata increase needs to be
[pic]
supplemented by a further increase in the output of y2 to provide an efficient target. Returning to unit P5 the set of targets P5 can be obtained from a weighted average of the peer units P1 and P2. Thus P5 can be thought of as a composite unit made up of a weighted average of the peer units and this composite unit provides a target for the inefficient unit.
Issues in DEA
The concern with the DEA approach is that by a judicious choice of weights a high proportion of units in the set will turn out to be efficient and DEA will thus have little discriminatory power. The first thing to note is that a unit which has the highest ratio of one of the outputs to one of the inputs will be efficient, or have an efficiency very close to one by putting as much weight as possible on that ratio and the minimum weight ([pic]) on the other inputs and outputs. In a typical analysis each such ratio may be associated with a different unit and the number of such ratios will be the product of the number of inputs and the number of outputs. Hence in an analysis with six inputs and six outputs, there is a clear possibility of 36 efficient units. In general if there are t outputs and m inputs we would expect the order of tm efficient units, suggesting that the number of units in the set should be substantially greater than tm, in, order for there to be suitable discrimination between the units.
This highlights one of the concerns regarding DEA analysis. A unit can appear efficient simply because of its pattern of inputs and outputs and not because of any inherent efficiency. An approach to resolving this issue is to constrain the weights in some way. Determining a minimum weight for any input and output would ensure that each input and output played some part in the determination of the efficiency measure. Similarly, a maximum limit could be placed on weights to avoid any input or output being over-represented. Clearly these limits should not be heavily constraining as this would tend towards each unit being measured using a common set of weights. Hence a compromise is sought between weights flexibility on the one hand and a common set throughout the system on the other.
An arbitrary application of weights restrictions would be difficult to justify, but recognising that the weights imply value or a cost associated with an input or an output then investigation of the particular context may lead to justifiable restrictions on the weights. This issue has been addressed for the case of a single input by Dyson and Thanassoulis(7) in the context of a Local Authority Department. Here there was a single cost input so that the weights on the outputs related to the cost expended supporting a unit of the output. Given this economic interpretation on the weights of the output a range of possible weights can be justified. For example, if one output measures the number of households that rates must be collected from, or the number of individuals liable to a community charge, then the weight represents the cost of collection from one household or individual. Clearly there is some minimum cost associated with such a collection task and this minimum cost could be estimated and become a lower bound on the weight associated with that output. Constraining weights in this way will of course enhance the discriminatory power of DEA.
A key aspect of DEA is incorporating environmental factors into the model as either inputs or outputs. Resources available to units are classed as inputs whilst activity levels or performance measures are represented by outputs. One approach to incorporating environmental factors is to consider whether they are effectively additional resources to the unit in which case they can be incorporated as inputs, or whether they are resource users in which case they may be better included as outputs. For example in comparing efficiency of schools research has indicated that in general parents of higher educational attainment provide greater support to their children and therefore are effectively an additional resource to the schools and should be classed as an input. An appropriate measure might be simply the number of parents in the higher social classes. In a retailing environment, the strength of competition would be an environmental factor, but in this case the greater that strength the more resources a unit will need to compete. Hence competition is a resource user and a measure of competition should be included as an output. Where possible direct measures of environmental factors should be used, but sometimes surrogate measures are required. For example if social deprivation is seen as an environmental factor which is a resource user then this might be represented by, for example, the number of children qualifying for free lunches in the context of schools, or the number of individuals whose first language is not English in the context of say Social Services Departments.
If the set of organisational units is part of the profit-making sector of the economy (e.g. bank branches) it could be argued that profitability is the only performance measure of relevance. The main argument against this is that environmental factors can affect the attainment of profitability. A profitable retail outlet may be being managed efficiently or it may be just enjoying favourable environmental factors, whilst an unprofitable one may be being badly managed or simply experiencing unfavourable pressures from the environment. If poor profitability is not a sufficient measure of performance in the profit-making sector, neither can it be ignored. In making decisions about units, both efficiency and profitability are relevant and one approach would be to use DEA to determine efficiency and to separately determine profitability. Units could then be assessed on an efficiency/profitability matrix which has similarities in structure to the product portfolio matrix. This is displayed in figure 2.
Organisational units whose profitability and efficiency located them in the star quadrant are the flagship units and should provide examples of good operating practice and are probably also in a favourable environment. The sleepers are profitable but this is more to do with favourable environmental conditions than good management They are candidates for an efficiency drive leading to even greater profits. The question marks have potential for a greater efficiency and possibly greater profits. Attempts should be made to increase their efficiency and this may lead to greater profitability. The dogs are efficiently operated units but low on profitability due to an unfavourable environment. In the extreme case it may be sensible to divest of these and relocate the staff to other units. This approach thus sees efficiency and profitability as two key performance measures, each of which can help with the management of the overall system of units.
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Conclusions
DEA is a novel approach to relative efficiency measurement where there are multiple incommensurate inputs and outputs. If a suitable set of measures can be defined DEA provides an efficiency measure not relying on the application of a common weighting of the inputs and outputs. Additionally the method identifies peer units and targets for inefficient units. A number of issues arising from the application of DEA have also been addressed. A full bibliography of DEA including applications has been compiled by Seiford(8)
References
1. FARRELL M.J. (1957) The measurement of productive efficiency, J.R. Statis. Soc. Series A 120, 253-281.
2. FARRELL M.J. and FIELDHOUSE M. (1962) Estimating efficient production functions under increasing returns to scale, J.R. Statis. Soc. Series A 125, 252-267.
3. CHARNES A., COOPER W.W. and RHODES E. (1978) Measuring the efficiency of decision making units, Eur. J. Opl. Res 2, 429-444.
4. THANASSOULIS E., DYSON, R.G. and FOSTER, M.J. (1987) Relative efficiency assessments using data enveloprnent analysis: an application to data on rates departments, J. Opl. Res. Soc. 38, 397-412.
5. BANKER R.D. and MOREY R.C. (1986) Efficiency analysis for exogenously fixed inputs and outputs, Ops. Res., 34, 513-521.
6. THANASSOULIS E. and DYSON R.G. (1988) Setting target input output levels for relative efficiency under different priorities over individual input output improvements, Warwick Papers in Management, No. 25, University of Warwick.
7. DYSON R.G. and THANASSOULIS E., (1988), Reducing weight flexibility in data envelopment analysis, J. Op'. Res. Soc., 39, 563-576.
8. SEIFORD L.M. (1989) A bibliography of data envelopment analysis, Working paper, Dept of Industrial Engineering and Operations Research, University of Amherst, MA 01003, USA.