**2. Material and methods**

### **2.1. Generally about Data Envelopment Analysis**

The story about DEA begins with the doctoral dissertation of Edward Rhodes, who has tried to evaluate the curriculum of public schools in Texas in the United States. At that time, it was a challenge to assess the relative efficiency of schools with multiple inputs and outputs, and without the usual information on prices and costs. As a result, the formulation of CCR model2 was developed and the first DEA paper was published in the European Journal of Operational Research in year 1978 [14].

DEA was originally developed as a tool to measure the effectiveness of organizations working on non-profit basis (public schools and hospitals, military establishments), where it is not possible to determine efficacy based on the value of their inputs and outputs. Later, the DEA has found application in profit organizations (companies, banks), and its development has resulted in over 3,000 papers published by the year 2001 [9].

Today, we find the application of DEA in many areas, such as education (public schools and universities), health care (hospitals, clinics, health centers), banking, sports, market research, agriculture, retail, transportation, hospitality, construction, etc. Bibliography of DEA which was published in 1994 [15] recorded 472 papers which were published in the period 1978. - 1992. References from 2002 [9] state the number of 3203 papers in the period of 1978. - 2001. Such a number of articles shows the great importance and interest for the DEA methodology and its applications.

The reasons for the rapid growth probably lie in the fact that DEA is an interdisciplinary applicable methodology, which is also suitable in cases where other approaches do not provide satisfactory results because of complex or unfamiliar nature of relationship between multiple inputs and outputs. So, in recent years, data envelopment analysis has become a central technique in the analysis of productivity and efficiency.

achievable efficiency frontier i.e. production possibilities, and sets the maximum output for

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**Figure 1.** Graphical description of efficiency frontier in DEA model (two-input example)

DEA is based on the extreme values and each DMU is compared only with the best units. The basic assumption is that if a particular unit with X inputs (resources) can produce Y outputs (products), the other units should be able to do the same if they are working efficiently. The center of the analysis lies in finding the 'best' virtual production unit for each actual/real unit. If the virtual unit is better than the original, whether it achieves more outputs with the same inputs or it achieves the same outputs with fewer inputs, then the real unit is inefficient.

In the next section a simple example will be given to explain the theoretical basis on which the Data envelopment analysis stands. First, numerical example of mathematical assumptions and procedures necessary in different DEA models will be presented. And then the graphical representation of the same example will describe the concept of Data envelopment analysis.

A simple numerical example might help show what is going on. Assume that there are three baseball 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.

each unit at a given level of its inputs.

*2.2.1. Simple numerical example*

**•** 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

### **2.2. Mathematical and statistical basics of DEA**

Data envelopment analysisis is a deterministic, non-parametric methodology for determining the relative efficiency of comparable decision making units considering their similar work technology and performance of similar tasks. Decision making units (DMUs) represent any production or non-production units that have the same types of inputs and outputs, and they differentiate one from another according to the level of available resources and the level of activity within the transformation process (inputs to outputs). DEA determines the relative efficiency of analysed units by constructing the empirical efficiency frontier i.e. frontier or margin of production possibilities (this term is used although the analysis may consider unproductive sectors) based on the information about used inputs and achieved outputs of all units included in the analysis. The most successful units (best practice units), the ones that determine the efficiency frontier, gain the grade '1', and the degree of technical inefficiency of all other units is calculated based on the distance of their input-output ratio in relation to the efficiency frontier (Figure 1).

For each unit included in the analysis a particular problem of linear programming is solved and its maximum efficiency, regarding other units in the reference set, is determined. The relative efficiency of the unit is calculated as the ratio of weighted sum of outputs and weighted sum of inputs. Weight of outputs and inputs for each unit is determined so to make its measure of efficiency maximum possible, with the limitation that the result of the relative efficiency can not be over one ('1'). A model defined in such a way maximizes the result of the relative efficiency of each unit provided that the gained set of weights must be feasible and attainable for any other unit in the observed group. This means that DEA defines the best possible

<sup>2</sup> Today, there are many DEA models that differ regarding returns to scale (models which assume constant and models aimed at increasing outputs - output oriented). CCR model (by Charnes, Cooper and Rhodes) is one of the basic and most frequently used models.

**Figure 1.** Graphical description of efficiency frontier in DEA model (two-input example)

achievable efficiency frontier i.e. production possibilities, and sets the maximum output for each unit at a given level of its inputs.

DEA is based on the extreme values and each DMU is compared only with the best units. The basic assumption is that if a particular unit with X inputs (resources) can produce Y outputs (products), the other units should be able to do the same if they are working efficiently. The center of the analysis lies in finding the 'best' virtual production unit for each actual/real unit. If the virtual unit is better than the original, whether it achieves more outputs with the same inputs or it achieves the same outputs with fewer inputs, then the real unit is inefficient.

In the next section a simple example will be given to explain the theoretical basis on which the Data envelopment analysis stands. First, numerical example of mathematical assumptions and procedures necessary in different DEA models will be presented. And then the graphical representation of the same example will describe the concept of Data envelopment analysis.

### *2.2.1. Simple numerical example*

possible to determine efficacy based on the value of their inputs and outputs. Later, the DEA has found application in profit organizations (companies, banks), and its development has

Today, we find the application of DEA in many areas, such as education (public schools and universities), health care (hospitals, clinics, health centers), banking, sports, market research, agriculture, retail, transportation, hospitality, construction, etc. Bibliography of DEA which was published in 1994 [15] recorded 472 papers which were published in the period 1978. - 1992. References from 2002 [9] state the number of 3203 papers in the period of 1978. - 2001. Such a number of articles shows the great importance and interest for the DEA methodology

The reasons for the rapid growth probably lie in the fact that DEA is an interdisciplinary applicable methodology, which is also suitable in cases where other approaches do not provide satisfactory results because of complex or unfamiliar nature of relationship between multiple inputs and outputs. So, in recent years, data envelopment analysis has become a central

Data envelopment analysisis is a deterministic, non-parametric methodology for determining the relative efficiency of comparable decision making units considering their similar work technology and performance of similar tasks. Decision making units (DMUs) represent any production or non-production units that have the same types of inputs and outputs, and they differentiate one from another according to the level of available resources and the level of activity within the transformation process (inputs to outputs). DEA determines the relative efficiency of analysed units by constructing the empirical efficiency frontier i.e. frontier or margin of production possibilities (this term is used although the analysis may consider unproductive sectors) based on the information about used inputs and achieved outputs of all units included in the analysis. The most successful units (best practice units), the ones that determine the efficiency frontier, gain the grade '1', and the degree of technical inefficiency of all other units is calculated based on the distance of their input-output ratio in relation to the

For each unit included in the analysis a particular problem of linear programming is solved and its maximum efficiency, regarding other units in the reference set, is determined. The relative efficiency of the unit is calculated as the ratio of weighted sum of outputs and weighted sum of inputs. Weight of outputs and inputs for each unit is determined so to make its measure of efficiency maximum possible, with the limitation that the result of the relative efficiency can not be over one ('1'). A model defined in such a way maximizes the result of the relative efficiency of each unit provided that the gained set of weights must be feasible and attainable for any other unit in the observed group. This means that DEA defines the best possible

2 Today, there are many DEA models that differ regarding returns to scale (models which assume constant and models aimed at increasing outputs - output oriented). CCR model (by Charnes, Cooper and Rhodes) is one of the basic and most

resulted in over 3,000 papers published by the year 2001 [9].

technique in the analysis of productivity and efficiency.

**2.2. Mathematical and statistical basics of DEA**

and its applications.

454 Computational and Numerical Simulations

efficiency frontier (Figure 1).

frequently used models.

A simple numerical example might help show what is going on. Assume that there are three baseball 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.


Now, as a DEA analyst, we combine 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.

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.)

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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

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

To conclude this section, DEA models are linear programming methods that calculate the efficiency frontier of a set of DMUs and evaluate the relative efficiency of each unit, therby allowing a distinction to be made between efficient and inefficient DMUs. Those identified as "best practice units" (i.e., those determining the frontier) are given a rating of one, whereas the degree of inefficiency of the rest is calculated on the basis of the Euclidian distance of their

Compared to regression or stochastic frontier analysis methods, DEA shows several advan‐ tages. First, DEA allows handling multiple inputs and outputs (with different units) in a noncomplex way. Second, DEA does not require any initial assumption about a specific functional form linking inputs and outputs. While a typical statistical approach (regression analysis) is based on average values, DEA is an extreme point method and compares each

OC/OV, players A and C will have efficiency scores equal to 1.0.

an LP formulation of DEA (*Source:* [16]).

**Figure 2.** Graphical example of DEA for player B

input-output ratio from the frontier [17].

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. (*Source:* [16]).
