Data Envelopment Analysis Summer Ventures 2012
The Battle of AdjectivesOptimal Versus Efficient • Optimal – most favorable or desirable. • Efficient - performing or functioning in the best possible manner with theleastwasteoftime and effort; having and using requisite knowledge, skill,andindustry;competent;capable: a reliable, efficient secretary.
What is Dea? • DEA is a complex mathematical tool used to develop an efficiency scale for items that cannot be ranked in a traditional manner. • Example Applications • Sports Leisure Activities • Manufacturing • Higher Education • Marketing • Decision Sciences
Basic Terminology • Decision Making Unit – (DMU) The item(s) for consideration (the items to be rated for efficiency). • Input – Quantities describing the DMU such as cost, salary, etc. that we wish to minimize in the analysis process. • Output – Quantities describing what the DMU produces as a result of the input. InputsOutputs DMU
Process • DEA is a modified form of linear programming. • The assumptions of DEA are that we have a population of DMU’s, and of that population at least one must be most efficient (100%). • Once the most efficient DMU’s are selected each successive DMU’s efficiency is calculated using a vector analysis based upon what is considered the most efficient DMU’s of the population.
A simple example • Consider that a baseball club has three batters.
Who is most efficient? • By analysis of the graph we see that players A and C create an outline around the data, as a result players A and C are 100% efficient.
Scatterplot with efficiency frontier Home Runs Singles
Who is most efficient? • By analysis of the graph we see that players A and C create an outline around the data, as a result players A and C are 100% efficient. • Player B can be expressed as a linear combination of both A and C. • Player B: 43.75%A + 25%C = B for a 68.75% efficiency index
Activity 1 Marksmanship
Returns to Scale • Notice that in the previous problems, all three batters had an identical number of “at bats”. Also note that each of you had an equal number of “shots”. • In practice, many DEA applications will not have inputs of the same size. For example, it is not practical to believe that all baseball players will have an equal number of at bats thus changing the relationship between the number of inputs and the outputs. • Returns to scale refers to increasing or decreasing efficiency based on size.
Multiple Outputs • Even though DEA is a linear programming application, if we have a problem with more than two outputs a graphical solutions is illogical. • For larger problems we will employ a method called the simplex method to find our efficiency scales. • Since the elementary notions of the simplex methods are relatively complex, we will use some computer software to simplify the process.
Activity 2 GOAT debates
Strengths of DEA • 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.
Limitations of DEA • 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.