How Data Envelopment Analysis works
Frontier Analyst is designed to help you measure and improve the performance of your organisation. The quest for greater efficiency is never ending as managers are always under pressure to improve the performance of their organisations. In the public sector, governments are constantly seeking better value for tax payers’ money, while the emergence of a more global economy has intensified competitive pressures on commercial companies. The onus is therefore on managers to achieve better results from the resources available to them. Frontier Analyst uses a powerful technique called Data Envelopment Analysis (DEA) to assist you in doing this.
The analysis compares the relative efficiency of organisational “units” such as bank branches, hospitals, vehicles, shops and other instances where units perform similar tasks. These units utilise similar resources, referred to as inputs, to generate similar outputs. For example, a shop has inputs of staff and floor space, and has outputs of sales volume and total revenue. However, there can be considerable differences in the way in which individual units combine inputs to produce outputs. In addition there may also be differences in potential among units caused by the technology they have available, their geographical location or catchment population.
Frontier Analyst allows you to take account of all the important factors that affect a units performance to provide a complete and comprehensive assessment of efficiency. Frontier Analyst does this by converting the multiple inputs and outputs into a single measure of productive efficiency. By doing so it identifies those units which are operating relatively efficiently and those which are not. The efficient units, those making best use of resources, are rated as being 100% efficient whilst the inefficient ones obtain lower scores.
Frontier Analyst generates efficiency scores for all units being analysed. It shows how much inefficient units need to reduce their inputs or increase their outputs in order to become efficient. Frontier Analyst therefore not only helps managers answer the question “How well are the units doing?” but also “How much could they improve?”. It suggests performance targets, such as, unit A should be able to produce 15% more output with their staffing level or unit B should be able to reduce costs by 25% and still produce the same level of outputs. It also identifies the units which are performing best and their operating practices can then be examined to establish a guide to “best practice” for others to emulate.
Frontier Analyst presents the results of an efficiency study very effectively, using high powered graphics, so that you can see and understand the information that the analysis provides more clearly. It offers various ways of visualising the results and shows in detail which units are performing the best and why they are doing so. It graphically displays performance information relating to an inefficient unit and shows the difference between its performance and the “best practice” units to which it has been compared.
Many organisations such as banks, hospitals, airlines, government departments and local authorities are using this analysis. It is used by managers in these organisations to perform a number of tasks including:
- Resource allocation: reallocating from the inefficient to the efficient
- Identification of “best practice”
- Identification of “poor practice”
- Target Setting
- Monitoring efficiency changes over time
- Rewards for good performance
- Planning site locations
Frontier Analyst has been designed to make managing the data, visualising the results and understanding the process a whole lot simpler. As a result undertaking an efficiency study has never been easier.
The Manager’s Challenge
Today, managers have to cope with and make sense of an enormous amount of data relating to their organisation. Data on sales, costs, stocks, markets, demographics etc. The challenge is to somehow derive useful insights from all these numbers that will lead to improvements in the performance of the organisation.
Consider the case of a bank which has a number of branches. The goal of management is to ensure that each of these branches achieves the best possible performance; the problem though is deciding what that means and how best to go about measuring it. The outputs of the branches such as sales, sales growth, accounts, market share, etc., can be studied and compared. Similarly, a branch’s inputs, such as, staff, office space, materials costs, etc., can be measured. Managers can then develop performance ratios such as sales per member of staff or profit per unit of office space utilised.
All these attempts to measure performance may however produce no overall clear picture as branches may exhibit considerable variation depending on the performance indicator chosen. This is where Frontier Analyst helps as it provides a more comprehensive measure of efficiency by taking into account all the important factors that affect a branch’s performance.
A Performance Ratio
If the branches of our example bank had a single input and a single output, the efficiency in converting their inputs into outputs of each branch is defined simply as:
If we decide to measure the efficiency of the branches by using the number of account transactions that a branch processes each year per number of staff employed as an indicator of performance. The bank has 6 branches located in various cities in the UK. The data for number of staff employed, total transactions processed and the ratio of transactions to staff is contained in the following table:
Branches Input Output Output/Input
No. of Staff Transactions Transactions
Employed Processed /
('000s) Staff
Birmingham 16 64 4
Cardiff 10 35 3.5
Glasgow 20 175 8.75
Leeds 22 132 6
London 30 180 6
Manchester 12 100 8.333
The performance ratio of “number of transactions processed per number of staff” suggests that Glasgow is the most efficient branch at processing transactions. This is because it processes the highest number of transactions, 8750, for each member of staff employed. Cardiff, by contrast, is the least efficient (in our fictional bank) as it only processes 3500 transactions per staff employed.
The efficiency of all the branches relative to Glasgow can be determined by dividing their ratio of “transactions/staff” by Glasgow’s ratio of “transactions/staff”. The result of these calculations is shown in this table:
Branches Relative Efficiency Birmingham 4 / 8.75 = 0.46 Cardiff 3.5 / 8.75 = 0.40 Glasgow 8.75 / 8.75 = 1.00 Leeds 6 / 8.75 = 0.69 London 6 / 8.75 = 0.69 Manchester 8.333 / 8.75 = 0.95
If each of the branches relative efficiency scores is multiplied by 100 then Glasgow can be said to be 100% efficient whereas Birmingham, for example, is only 46% efficient.
Glasgow in terms of the performance ratio “transactions/staff” is an example of “best achieved performance” and is the branch which all the other branches should seek to emulate. The Glasgow branch could therefore be used as a reference branch in order to set targets for the improved performance of the other branches. For example, in order to equal Glasgow’s performance Cardiff could be given the output target of increasing its transactions processed from 35,000 to 87,500. Its output/input ratio would then be 87,500 / 10 = 8750, the same as Glasgow’s, thus making Cardiff 100% efficient as well.
An Illustration of the Analysis
It is seldom the case though that a unit has only a single input and a single output. There are usually a number of factors which determine the operational efficiency of a unit. In our bank example the manager of the bank may well have considered that the branches actually produced two distinct outputs from the single input of staff. These two outputs could be:
- the number of transactions a branch processes on personal customer accounts
- the number of transactions a branch processes on business customer accounts
The performance of the branches therefore needs to be assessed on how efficiently they use their single input of staff to produce the two distinct categories of transactions outputs. The data for the six branches showing the staff numbers and the transactions now split into two categories is shown in the following table:
Branches Input Output 1 Output 2
Staff Personal Business
Numbers Transactions Transactions
('000s) ('000s)
Birmingham 16 44 20
Cardiff 10 23 12
Glasgow 20 125 50
Leeds 22 80 52
London 30 140 40
Manchester 12 55 45
The efficiency of each branch in producing the two outputs can be found by dividing each of their outputs by their input and seeing which branches have the highest ratios. The next table shows the results of doing this:
Branches Personal Transactions Business Transactions
/ /
Staff Staff
Birmingham 2.75 1.25
Cardiff 2.3 1.2
Glasgow 6.25 2.5
Leeds 3.636 2.363
London 4.666 1.333
Manchester 4.583 3.750
The higher the ratio of an output to input the more efficient a branch is in producing that output. The branch with the highest ratio of Personal Transactions/Staff is Glasgow and consequently this branch is the most efficient in processing transactions on personal accounts. However, the branch with the highest ratio of Business Transactions/Staff is Manchester and so this branch is the most efficient in processing transactions on business accounts. In this situation, the picture of efficiency is less clear. Which measure should be used? The Frontier Graph is a method of resolving this graphically by plotting Personal Transactions/Staff against Business Transactions/Staff for all the branches. This is shown in the following ‘frontier graph’:
This diagram shows the which is the fundamental concept of DEA. The “Cardiff” and “intersection” labels indicate the items referenced in the text. In the Frontier Analyst software, simply move the cursor over the data points to see this detail in the status line of the main window.
The positions on the graph represented by Glasgow and Manchester demonstrate a level of performance which is superior to all the other branches. A line can be drawn on the graph between Glasgow and Manchester which, together with the straight lines from Glasgow to the Y axis and Manchester to the X axis, represent what is called the efficiency frontier. The efficiency frontier, derived from the most efficient branches in the data-set, represents a standard of best achieved performance. As a result it can be used as a threshold against which to measure the performance of all the other branches. (This does not imply that the branches on the frontier cannot improve their performance but that there is no demonstrable basis to measure the extent to which they can do so.)
The efficiency frontier ‘envelops’ the inefficient units within it and clearly shows the relative efficiency of each branch. Branches which are located on the frontier are performing better than any branches below the frontier. Any branch on the frontier is considered 100% efficient and any branch below it is relatively less efficient and has an efficiency rating of less than 100%. In this example Glasgow and Manchester are therefore considered 100% efficient: Glasgow because it is the most efficient at processing personal transactions and Manchester because it is the most efficient at processing business transactions. The other branches are not 100% efficient as they are not on the frontier.
Cardiff, for example, could become efficient if it increased its outputs, in the same proportions, whilst keeping its input the same. If it did this it would eventually reach the efficiency frontier at the point marked (“intersection”). Alternatively, it could reduce its input while keeping its outputs constant which would have the same effect. Its actual efficiency is calculated simply by the ratio of its distance from the origin over the distance from the origin to the point marked. This gives Cardiff an efficiency of 41%. Similarly London is 75% efficient, Leeds 71% and Birmingham 46%. Frontier Analyst automatically generates these efficiency scores for you. If there are only two active inputs and one output, or one active input and two outputs, then Frontier Analyst is able to display a “frontier graph” a 2-dimensional representation of this.
When more than two outputs and one input or one output and two inputs are active the problem becomes ‘multidimensional’ and is no longer suitable to be represented graphically. In this situation the analysis is much more complicated and Frontier Analyst becomes a requirement. In addition, Frontier Analyst is able to determine the potential improvements and other information from the mix of data, to provide a good insight into efficiency.
Efficiency Scores
The Efficiency Scores window is now displayed which shows the efficiency of the bank branches in descending order of efficiency. It can be seen that Glasgow and Manchester are found to be 100% efficient with the other branches having efficiency scores of less than 100%.
Frontier Analyst enables you to sort the order in which the units and their efficiency scores are displayed. If you click the button on the Efficiency Scores window toolbar called “Sort 1-9” the bank branches will be sorted in ascending order of efficiency. “Sort A-Z” orders the branches alphabetically and “Unsort” orders the branches in the order they were entered into the data set.
The buttons on the Efficiency Scores window toolbar called “Show All”, “100%” and “<100%” enable you to choose whether to display either:
* all the branches
* only those which are 100% efficient
* or only those which are inefficient and hence have an efficiency rating of less than 100%.
The Frontier Plot
In an efficiency analysis where only one input and two outputs have been used the results can be displayed graphically, in a “frontier plot”.
Each point on the plot represents a branch in the analysis. One of the inefficient branches is currently highlighted in red. If you move the cursor over this red diamond the name of this branch and its efficiency will be displayed in the panel at the bottom of the window. A line has been drawn on the plot from the origin through Birmingham to a red star located on the efficiency frontier. This red star indicates the position on the efficiency frontier that the Birmingham branch would reach if it either:
* reduced the amount of the input staff it is currently using, or
* increased both of its outputs in the same proportions by a sufficient amount until it arrived at this point on the frontier. It would then be 100% efficient.
If you place the cursor over any diamond on the plot that branches name and efficiency will be shown in the panel at the bottom of the window. If you click any diamond then a line will be drawn from the origin, through that branch, and onto the frontier.
Looking at an Inefficient Unit
Frontier Analyst graphically displays performance information relating to inefficient units and shows the difference between their performance and the “best practice” (100% efficient) units to which they have been compared.
Performance information on any inefficient unit can be obtained from the Efficiency Scores window. This window shows that Cardiff has an efficiency score of 40.7%. It also contains three ‘tabs’ labelled “Potential Improvements”, “Reference Comparison” and “Reference Contributions”. The tab which is initially visible is Potential Improvements.
Potential Improvements
An efficiency study not only provides an efficiency score for each unit but also indicates by how much and in what areas an inefficient unit needs to improve in order to be efficient. This information can enable targets to be set which could help guide inefficient units to improved performance.
If a unit is found to be inefficient then it should be able to produce its current level of outputs with fewer inputs (input minimisation) or generate a higher level of outputs given the same inputs (output maximisation). The Potential Improvements graph shows what percentage a unit needs to either decrease its inputs or increase its outputs in order to become 100% efficient. For the Cardiff branch the graph shows it needs to reduce its number of staff by nearly 60% while maintaining the same level of outputs in order to become efficient.
The target data for the Cardiff branch is also displayed in a table which can be viewed by clicking on the “Table” button.
This table shows the inputs and outputs used in the analysis. The “Actual” column shows the values of the inputs used and the values of the outputs actually produced by Cardiff. The “Target” column shows the amount of inputs and outputs that Cardiff should be using or producing in order to be efficient. The “Potential Improvement” column shows how much, in percentage terms, Cardiff’s use of inputs or production of outputs needs to change by in order for it to be efficient. The table shows that Cardiff should reduce its use of staff from 10 to 4, a reduction of almost 60%, in order to become as efficient as either Glasgow or Manchester. It is of course important to realise that this is just a guide to improvement potential. There may be reasons why a unit will find it hard to ever match the best performers, but such indicators can help make the best of your resources.
If the assessment of a unit as inefficient is felt to be justified then the information provided can be used as a basis for setting targets for the unit. As a first step in setting targets, the inefficient unit should be compared with the units in its reference set.
The Reference Set is the set of efficient branches to which the unit has been most directly compared when calculating its efficiency rating. Cardiff has efficient branches Glasgow and Manchester in its reference set and is only operating 40% as efficiently as they are. The reference set of a unit can help in providing an insight into why it is under-performing and show clearly the areas in which it is weak.
The efficient branches forming Cardiff’s reference set are listed on the left. Glasgow is currently highlighted and the graph shows a comparison between Glasgow and Cardiff. Click on Manchester in the list in order to view the comparison between Manchester and Cardiff.
The input and output values for Cardiff have all been scaled to equal 100%. Manchester’s values for its input and outputs are then expressed as a percentage of Cardiff’s values thus making for an easy comparison between the branches. The bars representing Cardiff’s values are all coloured blue and the bars representing the values of the efficient branch it is being compared with are red.
If you now use the mouse to place the cursor over the red bar representing Manchester’s input of staff the status panel of the main window will show “Manchester uses 120% of Staff compared with Cardiff”. It can be seen that Manchester is using 20% more staff than Cardiff is. Manchester has 12 members of staff employed and Cardiff 10.
Although Manchester is only using 20% more of the input staff than Cardiff is the graph shows that the output it achieves is substantially more then 20% higher than Cardiff’s. Use the mouse to move the arrow over the bars representing Manchester’s personal and business transactions. You will find that Manchester is achieving 239% of Cardiff’s output of personal transactions, more than double, and is achieving 375% of Cardiff’s output of business transactions.
The result of such a comparison should prompt an investigation into why Manchester is able to achieve much greater outputs from nearly the same level of inputs that Cardiff uses.
Cardiff can be compared in this manner with any of the branches in its reference set. If you click any of the branches in the reference set list then the graph will change to show a comparison between that branch and Cardiff.
An inefficient unit’s reference set contains the efficient units which have the most similar input/output orientation to itself and they should therefore provide examples of good operating practice for it to emulate. However, the reference set units do not all contribute equally to the target values for an inefficient unit. Some reference set units are more important than others. Frontier Analyst enables you to see which of the reference set units have been the biggest contributors.
The Reference Contributions graph shows all the branches in Cardiff’s reference set and how much in percentage terms they have each contributed to forming the target values for each of Cardiff’s inputs and outputs.
The vertical axis of the graph shows the amount in percentage terms that a reference set branch has contributed. All the values for each of the inputs and outputs add up to 100%. The horizontal axis lists all the inputs and outputs used in the analysis. Each reference set branch is displayed in a different colour.
The reference set branches which have the highest percentages have contributed the most. If a particular reference set branch predominates then it would be the main comparison for the inefficient branch. The identification of the reference set and within it those branches which are the biggest contributors enables the manager to compare the inefficient branch with the branches which are most similar to it. This avoids the need to study all the branches in order to understand the nature of the inefficiencies present.
It can be seen that Glasgow contributes more to the targets for Cardiff than Manchester does which would suggest that Glasgow might be the best branch to compare Cardiff’s performance too. Even so, both could certainly be used and Manchester may be more appropriate if it is of a similar size to Cardiff.
There are 3 main stages involved when conducting an efficiency study.
- defining and selecting the units to use in the analysis;
- deciding which factors to use for inputs and outputs; and
- using Frontier Analyst and interpreting the results.
The Frontier Analyst manual describes these stages in detail. This ranges from general considerations through use of the correlation facility to help determine appropriate inputs and outputs. The various optimisation models of Data Envelopment Analysis, such as input minimisation or output maximisation, are described in detail.
If measuring efficiency for performance indicators is important in your organisation, you need Frontier Analyst now.
