- Simple Numerical Example
- Graphical Example
- Returns to Scale
- Applications
- Strengths of DEA
- Limitations of DEA
- Other Topics

DEA Software

Data Envelopment Analysis (DEA) is becoming an increasingly popular management tool. Questions about it have cropped up recently in the sci.op-research newsgroup but nothing underlines this fact more than the fact that it was featured in a recent issue of Fortune magazine (10/31/94, p.38). Groups such as the Productivity Analysis Research Network (PARN ) and others have supported DEA and other techniques but there hasn't been a central WWW link.

This WWW page is meant to be a basic introduction to DEA and link to other WWW resources. This introduction focuses more on gaining an intuitive understanding of DEA than mathematical rigor because of the varying backgrounds of the readers. Later versions may include more in the way of mathematics and advanced topics in DEA. People already proficient with DEA may find the links to other DEA related Internet sites useful. Eventually I may also add a calendar of events related to DEA.

Information on the DEA WWW page author: Tim Anderson

This section is an introduction to Data Envelopment Analysis (DEA) for people unfamiliar with the technique. For a more in-depth discussion of DEA, the interested reader is referred to Seiford and Thrall [1990] or the seminal work by Charnes, Cooper, and Rhodes [1978].

DEA is commonly used to evaluate the efficiency of a number of producers. A typical statistical approach is characterized as a central tendency approach and it evaluates producers relative to an average producer. In contrast, DEA is an extreme point method and compares each producer with only the "best" producers. By the way, in the DEA literature, a producer is usually referred to as a decision making unit or DMU. Extreme point methods are not always the right tool for a problem but are appropriate in certain cases. (See Strengths and Limitations of DEA.)

A fundamental assumption behind an extreme point method is that if a given producer, A, is capable of producing Y(A) units of output with X(A) inputs, then other producers should also be able to do the same if they were to operate efficiently. Similarly, if producer B is capable of producing Y(B) units of output with X(B) inputs, then other producers should also be capable of the same production schedule. Producers A, B, and others can then be combined to form a composite producer with composite inputs and composite outputs. Since this composite producer does not necessarily exist, it is sometimes called a virtual producer.

The heart of the analysis lies in finding the "best" virtual producer
for each real producer. If the virtual producer is better than the original
producer by either making more output with the same input or making the same
output with less input then the original producer is *inefficient*. Some
of the subtleties of DEA are introduced in the various ways that producers A and
B can be scaled up or down and combined.

The procedure of finding the best virtual producer can be formulated as a
linear program. Analyzing the efficiency of
*n* producers is then a set of *n *linear programming problems. The
following formulation is one of the standard forms for DEA. *lambda* is a
vector describing the percentages of other producers used to construct the
virtual producer.
*lambda X* and *lambda Y* and are the input and output vectors for
the analyzed producer. Therefore *X *and *Y
*describe the virtual inputs and outputs respectively. The value of theta is
the producer's efficiency.

**DEA Input-Oriented Primal Formulation **

It should be emphasized that an LP of this form must be solved for each of the DMUs. There are other ways to formulate this problem such as the ratio approach or the dual problem but I find this formulation to be the most straightforward. The first constraint forces the virtual DMU to produce at least many outputs as the studied DMU. The second constraint finds out how much less input the virtual DMU would need. Hence, it is called input-oriented. The factor used to scale back the inputs is theta and this value is the efficiency of the DMU.

A simple numerical example might help show what is going on. Assume that there are three players (DMUs), A, B, and C, with the following batting statistics. Player A is a good contact hitter, player C is a long ball hitter and player B is somewhere in between.

- Player A: 100 at-bats, 40 singles, 0 home runs
- Player B: 100 at-bats, 20 singles, 5 home runs
- Player C: 100 at-bats, 10 singles, 20 home runs

Now, as a DEA analyst, we play the role of Dr. Frankenstein by combining parts of different players. First let us analyze player A. Clearly no combination of players B and C can produce 40 singles with the constraint of only 100 at-bats. Therefore player A is efficient at hitting singles and receives an efficiency of 1.0.

Now we move on to analyze player B. Suppose we try a 50-50 mixture of players A and C. This means that lambda=[0.5, 0.5]. The virtual output vector is now,

*lambda Y *= [0.5 * 40 + 0.5 * 10, 0.5 * 0 + 0.5 * 20] = [25, 10]

Note that *X *= 100 = X(0) where X(0) is the input(s) for the DMU being
analyzed. Since *lambdaY *> Y(0) = [20, 5], then there is room to scale
down the inputs, *X *and produce a virtual output vector at least equal to
or greater than the original output. This scaling down factor would allow us to
put an upper bound on the efficiency of that player's efficiency. The 50-50
ratio of A and C may not necessarily be the optimal virtual producer. The
efficiency, theta, can then be found by solving the corresponding
linear program .

It can be seen by inspection that player C is efficient because no combination of players A and B can produce his total of 20 home runs in only 100 at bats. Player C is fulfilling the role of hitting home runs more efficiently than any other player just as player A is hitting singles more efficiently than anyone else. Player C is probably taking a big swing while player A is slapping out singles. Player B would have been more productive if he had spent half his time swinging for the fences like player C and half his time slapping out singles like player A. Since player B was not that productive, he must not be as skilled as either player A or player C and his efficiency score would be below 1.0 to reflect this.

This example can be made more complicated by looking at unequal values of inputs instead of the constant 100 at-bats, by making it a multiple input problem, or by adding more data points but the basic principles still hold.

The single input two-output or two input-one output problems are easy to analyze graphically. The previous numerical example is now solved graphically. (An assumption of constant returns to scale is made and explained in detail later.) The analysis of the efficiency for player B looks like the following:

**Graphical Example of DEA for Player B**

If it is assumed that convex combinations of players are allowed, then the line segment connecting players A and C shows the possibilities of virtual outputs that can be formed from these two players. Similar segments can be drawn between A and B along with B and C. Since the segment AC lies beyond the segments AB and BC, this means that a convex combination of A and C will create the most outputs for a given set of inputs.

This line is called the called the efficiency frontier. The efficiency frontier defines the maximum combinations of outputs that can be produced for a given set of inputs. The segment connecting point C to the HR axis is drawn because of disposability of output. It is assumed that if player C can hit 20 home runs and 10 singles, he could also hit 20 home runs without any singles. We have no knowledge though of whether avoiding singles altogether would allow him to raise his home run total so we must assume that it remains constant.

Since player B lies below the efficiency frontier, he is inefficient. His efficiency can be determined by comparing him to a virtual player formed from player A and player C. The virtual player, called V, is approximately 64% of player C and 36% of player A. (This can be determined by an application of the lever law. Pull out a ruler and measure the lengths of AV, CV, and AC. The percentage of player C is then AV/AC and the percentage of player A is CV/AC.)

The efficiency of player B is then calculated by finding the fraction of inputs that player V would need to produce as many outputs as player B. This is easily calculated by looking at the line from the origin, O, to V. The efficiency of player B is OB/OV which is approximately 68%. This figure also shows that players A and C are efficient since they lie on the efficiency frontier. In other words, any virtual player formed for analyzing players A and C will lie on players A and C respectively. Therefore since the efficiency is calculated as the ratio of OA/OV or OC/OV, players A and C will have efficiency scores equal to 1.0.

The graphical method is useful in this simple two dimensional example but gets much harder in higher dimensions. The normal method of evaluating the efficiency of player B is by using an LP formulation of DEA.

Since this problem uses a constant input value of 100 for all of the players, it avoids the complications caused by allowing different returns to scale. Returns to scale refers to increasing or decreasing efficiency based on size. For example, a manufacturer can achieve certain economies of scale by producing a thousand circuit boards at a time rather than one at a time - it might be only 100 times as hard as producing one at a time. This is an example of increasing returns to scale (IRS.)

On the other hand, the manufacturer might find it more than a trillion times as difficult to produce a trillion circuit boards at a time though because of storage problems and limits on the worldwide copper supply. This range of production illustrates decreasing returns to scale (DRS.) Combining the two extreme ranges would necessitate variable returns to scale (VRS.)

Constant Returns to Scale (CRS) means that the producers are able to linearly scale the inputs and outputs without increasing or decreasing efficiency. This is a significant assumption. The assumption of CRS may be valid over limited ranges but its use must be justified. As an aside, CRS tends to lower the efficiency scores while VRS tends to raise efficiency scores.

The CRS assumption can be made in the case of baseball batting since each at-bat is relatively independent and the cumulative batting statistics are then the sum of individual events. For example, it is expected that doubling the number of at-bats or plate appearances will double the number of home runs that a player hits which implies that the CRS assumption can be used. Therefore other situations such as VRS, IRS, and DRS are not covered here. This also explains why most of the examples concentrate on cases with equal inputs. In the one input model, the CRS assumption allows players to be scaled up or down and so the multiplication of player inputs to achieve some constant value is implied in some cases.

The simple baseball example described earlier may not convey the full view on the usefulness of DEA. It is most useful when a comparison is sought against "best practices" where the analyst doesn't want the frequency of poorly run operations to affect the analysis. DEA has been applied in many situations such as:

- health care (hospitals, doctors)
- education (schools, universities)
- banks
- manufacturing
- benchmarking
- management evaluation
- fast food restaurants
- retail stores

If there are requests for it, I may choose to add a pointer reference into the literature for each of the above applications but I think that would be overkill. Most of the papers in the References section contain applications.

By the way, the analyzed data sets vary in size. Some analysts work on problems with as few as 15 or 20 DMUs while others are tackling problems with over 10,000 DMUs. (The largest that I know of is Barr and Durchholz with 25,000 DMUs analyzed on a Sequent parallel computer with custom software.)

As the earlier list of applications suggests, DEA can be a powerful tool when used wisely. A few of the characteristics that make it powerful are:

- DEA can handle multiple input and multiple output models.
- It doesn't require an assumption of a functional form relating inputs to outputs.
- DMUs are directly compared against a peer or combination of peers.
- Inputs and outputs can have very different units. For example, X1 could be in units of lives saved and X2 could be in units of dollars without requiring an a priori tradeoff between the two.

The same characteristics that make DEA a powerful tool can also create problems. An analyst should keep these limitations in mind when choosing whether or not to use DEA.

- Since DEA is an extreme point technique, noise (even symmetrical noise with zero mean) such as measurement error can cause significant problems.
- DEA is good at estimating "relative" efficiency of a DMU but it converges very slowly to "absolute" efficiency. In other words, it can tell you how well you are doing compared to your peers but not compared to a "theoretical maximum."
- Since DEA is a nonparametric technique, statistical hypothesis tests are difficult and are the focus of ongoing research.
- Since a standard formulation of DEA creates a separate linear program for each DMU, large problems can be computationally intensive.

There are a lot of other items that I haven't covered in this brief introduction to DEA. I would like to expand on this work but covering all of these topics could entail as much work as writing a book and given the limitations of HTML publishing, be even more difficult. If there is sufficient interest, I may continue to revise and expand this coverage to include some of these topics but in the meantime, you will need to refer to the references listed later for more discussion.

- Input vs. output orientation
- Dual formulations and interpretations
- Bounds on relative factor weights
- Related models:
- Free Disposal Hull, FDH
- Additive model
- Multiplicative model

- Discretionary vs. nondiscretionary variables
- Possibility of being inefficient despite theta = 1 (hint, look at the slacks)
- Non-Archimedean vs. 2-phase solution
- Window analysis for examining panel data
- Techniques for handling stochastic variation (ex. chance constraints)
- Parametric alternatives to DEA
- Post-processing of efficiency and hypothesis testing (ex. "are private hospitals more efficient than public hospitals?")

PARN Sites

- Information about joining PARN .
- US - Brigham Young - FTP
- Maintained by Larry Walters - Please notify him of uploads.

- Europe - Odense University -
FTP
- Includes PARN news and a DEA system for GAMS

CMTE Centre for Management of Technology and Entrepreneurship (CMTE) at the University of Toronto

CMTE is doing some very interesting work including some significant applications of DEA in the banking industry.

Imperial College (Their OR Library has a few DEA test problems)

WORMS World-Wide-Web for Operations Research and Management Science

This is a good site containing a grass roots movement to support OR and MS techniques via the WWW. This DEA page is actually a part of the WORMS network.

Michael Trick's OR Home Page (Not really a DEA site but lots of connections to other OR sites)

The following site is probably the most comprehensive DEA WWW page. Ali Emrouznejad's DEA Home page

Send Me a DEA Site URL Address to be Added to the List

*This introduction to DEA is a constantly evolving document and feedback is
appreciated.*

*Last updated: 6/5/96*