# OR-Notes

## J E Beasley

OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research (OR). They were originally used by me in an introductory OR course I give at Imperial College. They are now available for use by any students and teachers interested in OR subject to the following conditions.

A full list of the topics available in OR-Notes can be found here.

#### Extending data envelopment analysis

In order to illustrate how data envelopment analysis (DEA) can be extended through the use of value judgements we will construct a DEA model for comparing university departments concerned with the same discipline.

Consider business and management studies at Imperial College and Southampton. We have the following data:

```                        Imperial    Southampton
College     (S)
(IC)
Student numbers
Total number            304         119

Expenditure (£'000)
General expenditure     970         177
Equipment expenditure   64          10
Total expenditure       1034        187

Other data
Research income         220         22
Research rating         3           3
```

How then can we compare these two departments using this data? Note here that this is actually real data from a particular year.

A commonly used method is ratios - for example:

```                        Imperial    Southampton
College     (S)
(IC)
Expenditure per:
student  3.4         1.6
staff member  29.5        26.7

Research income per:
staff member  6.3         3.1
£ of expenditure  0.21        0.12

Students per:
staff member  8.7         17
(the staff/student ratio)

Equipment expenditure per:
student  0.21        0.08
staff member  1.83        1.43```

A problem with comparison via ratios is that different ratios give a different picture and it is difficult to combine the entire set of ratios into a single judgement.

#### Data envelopment analysis

We shall indicate how to develop a model for comparing our two example departments using data envelopment analysis (DEA). In words DEA:

• requires the inputs and outputs for each DMU to be specified;
• defines efficiency for each DMU as a weighted sum of outputs [total output] divided by a weighted sum of inputs [total input]; where
• all efficiencies are restricted to lie between zero and one (i.e. between 0% and 100%).
• in calculating the numerical value for the efficiency of a particular DMU weights are chosen so as maximise its efficiency, thereby presenting the DMU in the best possible light.

We illustrate this below.

#### Inputs and outputs

What shall we choose as our inputs and outputs?

The answer is not as obvious as it might seem and for the purposes of illustration I have chosen the following.

#### Inputs

• General expenditure
• Equipment expenditure

#### Outputs

• Research income

The basic DEA model is therefore:

• EIC = (161wUG+111wPGT+32wPGR+220wRES)/(970wgen+64w equip)
• 0 <= EIC <= 1
• ES = (105wUG+12wPGT+2wPGR+22wRES)/ (177wgen+10wequip)
• 0 <= ES <= 1
• wUG,wPGT,wPGR,wRES, wgen,wequip >= 0

where

• wUG is the weight attached to undergraduates
• wPGT is the weight attached to taught postgraduates
• wPGR is the weight attached to research postgraduates
• wRES is the weight attached to research income
• wgen is the weight attached to general expenditure
• wequip is the weight attached to equipment expenditure
• EIC is the efficiency of Imperial College (expressed as a fraction)
• ES is the efficiency of Southampton (expressed as a fraction)

To decide the value of EIC we maximise EIC subject to the constraints above.

To decide the value of ES we maximise ES subject to the constraints above.

#### Results

Suppose we solve to decide the value for EIC - what do we get?

In fact we find that EIC=1 (the maximum possible) when wUG=0.8236, wPGT=7.5445, wgen=1 and all other weights are zero.

These weights seem very unrealistic. They mean:

• all other input/output factors (e.g. research postgraduates) are ignored.

How then can we improve our basic model?

#### Model improvement

In order to improve our model we introduce more constraints.

This addition of constraints involves value judgements. Just as we exercised our judgement in choosing the inputs and outputs we use our judgement as to what are appropriate constraints to add to the basic DEA model above.

For example we might think that appropriate constraints are:

• wPGR >= 1.25wPGT
• this equation ensures that the weight associated with a research postgraduate is at least 25% greater than the weight associated with a taught postgraduate.
• wPGT >= 1.25wUG
• this equation ensures that the weight associated with a taught postgraduate is at least 25% greater than the weight associated with an undergraduate.
• wPGR <= 2wUG
• this equation ensures that the weight associated with a research postgraduate is at most twice that associated with an undergraduate.

We can add these constraints to our basic model and resolve to get a value for EIC.

If we do this we find that EIC=1 when wRES=4.4091, wgen=1 and all other weights are zero.

It is clear that we need to add yet more constraints to generate realistic results.

Currently all the weights associated with student numbers are zero. We can argue that for IC the weighted output associated with student numbers, namely (161wUG+111wPGT+32wPGR), as a proportion of the total weighted output, namely (161wUG+111wPGT+32wPGR+220wRES), should be at least 50% (for example)

i.e. we have the constraint

(161wUG+111wPGT+32wPGR)/(161wUG+111wPGT+32 wPGR+220wRES) >= 0.5

We can add this constraint to our basic model and resolve to get a value for EIC.

If we do this we find that EIC=0.872 when wUG=1.0507, wPGT=1.6812, wPGR=2.1014, wRES=1.9228, wgen=1

These weights seem more reasonable.

We could, of course, continue adding constraints. I hope though that the point is clear - to the basic DEA model we add whatever constraints we feel are appropriate in order to compare university departments.

Note here that it is a simple matter to extend the approach given above, used to compare just two university departments, to comparing any number of university departments in the same discipline.

#### Comment

We have illustrated here how DEA can be used to construct a model for the quantitative comparison of university departments. In a similar manner DEA can be used to build models for the quantitative comparison of decision-making units in any organisation. A key point to note here is the addition of extra constraints (value judgements) in order to make the DEA model more representative of the underlying situation being modelled.