DEA Models Overview

Abstract

We begin with the basic DEA Models and some useful extensions (although we expect that some will see it as too much while others as too little). While we promised to minimize the mathematics, some are, unfortunately, unavoidable. We have excluded any specific discussion of the underlying linear programming (LP) mathematics that drives DEA, and while some general understanding of this is helpful for understanding the academic literature, it is not needed to understand the benefits and ways to apply DEA.

Appendices

Appendix: Chapter 1 (Sherman and Zhu 2006)

How DEA Works and How to Interpret the Results

We now illustrate how DEA is used to evaluate efficiency by means of the simplified bank branch example noted in Table 1.2. This analysis assumes only one type of transaction and two types of resources used to process these transactions – bank teller hours (H) and supply dollars (S). This example was selected because it lends itself to graphic description, and because it is simple enough to be analyzed without DEA. Hence, the results can be compared to an independent analysis of efficiency. Note that DEA is most valuable in complex situations where there are multiple outputs and inputs which cannot be readily analyzed with other techniques such as ratios, and where the number of service organization units being evaluated are so numerous that management cannot afford to evaluate each unit in depth. For example, an actual bank application involved the evaluation of fourteen branches and seventeen types of transactions as output measures. DEA was used to help direct management’s efforts to improve the efficiency of units that were first identified as inefficient with this technique.

Table 1.2 Illustrative example of five bank branches

Assume that there are five bank branches (B1, B2, B3, B4, and B5) and that each processes 1,000 transactions (e.g. deposits) during one common time period (e.g. week, month, year) by jointly using two inputs: tellers measured in labor hours (H) and supplies measured in dollars (S). (See Table 1.2 for a summary of the outputs and inputs.)

The problem facing the manager is to identify which of these branches are inefficient and the magnitude of the inefficiency. This information can be used to locate the branches that require remedial management action, to reward the more efficient managers, and/or to determine the management techniques used in the more efficient branches that should be introduced into less efficient branches. While the manager can observe the number of transactions processed and the amount of resources (H and S) used, he or she does not know the efficient output-to-input relationship. That is, the efficient amount of labor and supplies needed for each transaction is not readily determinable. The problem is illustrated in Fig. 1.2.

Fig. 1.2

Graphic representation of the five bank branches

In this example, it can be observed that B1 and B2 are relatively inefficient. B1 produced the same output level as B4 but used 100 more supply dollars (S) than B4. B2 also produced the same output level as B4 but achieved this by using 10 more teller labor hours. With the information available in Table 1.2, it is not possible to determine whether B3, B4, or B5 is more or less efficient. While information about relative prices might allow one to rank B3, B4 and B5, the finding that B1 and B2 are inefficient would not change. That is, B1 and B2 should be able to reduce inputs without reducing outputs regardless of the price of the inputs.

DEA compares each service unit with all other service units, and identifies those units that are operating inefficiently compared with other units’ actual operating results. It accomplishes this by locating the best practice or relatively efficient units (units that are not less efficient than other units being evaluated). It also measures the magnitude of inefficiency of the inefficient units compared to the best practice units. The best practice units are relatively efficient and are identified by a DEA efficiency rating of θ = 1. The inefficient units are identified by an efficiency rating of less than 1 (θ < 1 or θ < 100%). Table 1.3 gives the input-oriented efficiency ratings for the branches in the example.

Table 1.3 DEA results for five bank branches

Table 1.3 indicates that DEA identified the same inefficient branches that were identifiable through observation of the data. B1 and B2 have efficiency ratings below 100%, which identifies them as inefficient. In addition, DEA focuses the manager’s attention on a subgroup of the bank branches referred to as the efficiency reference set in Table 1.3. This efficiency reference set (ERS) includes the group of service units (or DMUs in standard DEA terminology) against which each inefficient branch was found to be most directly inefficient in comparison. (If a service unit’s efficiency rating is 100%, then this unit itself is its ERS.) For example, B1 was found to have operating inefficiencies in direct comparison to B4 and B5. The value in parentheses in Table 1.3 represents the relative weight assigned to each efficiency reference set member to calculate the efficiency rating (θ). Figure 1.2 illustrates this using B2 as an example.

DEA has determined that, among the five bank branches, B5, B4, and B3 are relatively efficient . In this simple case, this can be represented by the solid line in Fig. 1.2, which locates the units that used the least amount of inputs to produce their output level. These three branches, B5, B4 and B3 comprise the best practice set or best practice frontier. No indication is provided as to which, if any, of these three is more or less efficient than the other two. As noted earlier, all three could be somewhat inefficient. The best practice units are those which are not clearly inefficient compared with other units being evaluated.

DEA indicates that B2 is inefficient compared to e on the line connecting B4 and B3. One way for B2 to become efficient is for it to reduce its inputs to 85.7% of their current levels. This would move B2 onto the relatively efficient production segment at point e in Fig. 1.2, which reflects the use of 25.7 teller hours (0.857 × 30) and use of 171 supply dollars (0.857 × 200). DEA provides information to complete the calculation suggested in Fig. 1.2. This is illustrated in Table 1.4

Table 1.4 Inefficiency in branch B2 calculated by DEA

Table 1.4 indicates that a mixture of the operating techniques utilized by B3 and B4 would result in a composite hypothetical branch that processes the same number of transactions (1,000) as B2, but that requires fewer inputs than B2. Hence, by adopting a mixture of the actual techniques used by B3 and B4, B2 should be able to reduce teller hours by 4.3 units and supply dollars by 29 units without reducing its output level. A similar calculation can be completed for each inefficient unit located by the DEA analysis.

At this point it must be re-emphasized that DEA results are most useful when there are multiple outputs and inputs, and where the type of intuitive analysis that could be applied to verify the DEA results in the above example would not be possible. Nevertheless, the efficiency rating, the efficiency reference set, the analysis performed in Table 1.4, and the ability to determine alternative paths that would make an inefficient unit efficient would all be readily available to management. Applications to numerous organizations suggest that the representation in Table 1.4 is one of the more direct ways to summarize and explain what DEA has achieved and its implications for management.

In summary, the interpretation of DEA results tends to proceed in the following order:

• The efficiency ratings are generated as in Table 1.3. Units that are efficient (θ = 100%) are relatively, and not strictly, efficient. That is, no other unit is clearly operating more efficiently than these units, but it is possible that all units, including these relatively efficient units, can be operated more efficiently. Therefore, the efficient branches (B3, B4, and B5) represent the best existing (but not necessarily the best possible) management practice with respect to efficiency.

• Inefficient units are identified by efficiency ratings of θ < 1 or θ < 100%. These units (B1 and B2) are strictly inefficient compared to all other units and are candidates for remedial action by management. In fact, the inefficiency identified with DEA will tend to understate, rather than overstate, the inefficiency present.

• The efficiency reference set indicates the relatively efficient units against which the inefficient units were most clearly determined to be inefficient. The presentation in Table 1.4 summarizes the magnitude of the identified inefficiencies by comparing the inefficient unit with its efficiency reference set .

• The results in Table 1.4 indicate the following: B2 has been found to be relatively less efficient than a composite of the actual output and input levels of B3 and B4. If a combination of the operating techniques used in B3 and B4 were utilized by inefficient B2, B2 should be able to reduce the number of hours used by 4.3 units and the amount of supplies used by 29 units while providing the same level of services. Of course, management can also use DEA to identify other methods or combinations of methods to improve the efficiency of inefficient units.

The Mathematical Formulation of DEA

The linear programming technique is used to find the set of coefficients (u’s and v’s) that will give the highest possible efficiency ratio of outputs to inputs for the service unit being evaluated. Table 1.5 provides a fractional programming DEA mathematical model.

Table 1.5 Multiplier form of DEA mathematical model

In the model,

n = number of decision making units (DMU) being compared in the DEA analysis

DMU_j = decision making unit number j

θ = efficiency rating of the DMU being evaluated by DEA

y _rj = amount of output r used by DMU _j

x _ij = amount of input i used by DMU _j

m = number of inputs used by the DMUs

s = number of outputs generated by the DMUs

u _r = coefficient or weight assigned by DEA to output r, and

v _i = coefficient or weight assigned by DEA to input i.

The data required to apply DEA are the actual observed outputs produced y _rj and the actual inputs used x _ij , during one-time period for each DMU in the set of units being evaluated. Hence, x _ij is the observed amount of the i ^th input used by the j ^th service unit, and y _rj is the amount of r ^th output produced by the j ^th service unit.

If the value of θ for the DMU being evaluated is less than 100%, then that unit is inefficient, and there is the potential for that unit to produce the same level of outputs with fewer inputs. The theoretical development of this approach is discussed in detail in Cooper et al. (2007). Rather than reproduce this discussion, DEA will be explained with several simple applications and with emphasis on how to apply it, how to interpret the results and the implications for managing productivity.

DEA differs from a simple efficiency ratio in that it accommodates multiple inputs and outputs and provides significant additional information about where efficiency improvements can be achieved and the magnitude of these potential improvements. Moreover, it accomplishes this without the need to know the relative value of the outputs and inputs that were needed for ratio analysis.

Assume that the DEA evaluation would begin by evaluating the efficiency of bank branch B2 in Table 1.2. Based on the DEA model (Table 1.5), the problem would be structured as described below using the data in Table 1.2.

Calculate the set of values for u ₁ , v ₁ , and v ₂ that will give branch B2 the highest possible efficiency rating:

$$ \operatorname{Maximize}\kern0.5em \theta =\frac{u_1(1000)}{v_1(30)+{v}_2(200)}. $$

This is subject to the constraint that no DMU (in this case bank branch) can be more than 100% efficient when the same values for u ₁ , v ₁ , and v ₂ are applied to each unit:

$$ \mathrm{B}1\kern3.75em \frac{u_1(1000)}{v_1(20)+{v}_2(300)}\le 1, $$

$$ \mathrm{B}2\kern3.75em \frac{u_1(1000)}{v_1(30)+{v}_2(200)}\le 1, $$

$$ \mathrm{B}3\kern3.75em \frac{u_1(1000)}{v_1(40)+{v}_2(100)}\le 1, $$

$$ \mathrm{B}4\kern3.75em \frac{u_1(1000)}{v_1(20)+{v}_2(200)}\le 1, $$

$$ \mathrm{B}5\kern3.75em \frac{u_1(1000)}{v_1(10)+{v}_2(400)}\le 1. $$

DEA calculates the efficiency rating for B2 to be 85.7% and the values for v ₁  = 1.429, v ₂  = 0.286, and u ₁  = 0.0857. DEA would be rerun for each branch in the objective function as was done above for branch B2.

To run DEA on a standard linear programming package, the fractional forms in Table 1.5 are algebraically reformulated as follows:

$$ \operatorname{Maximize}\ \theta ={u}_1{y}_{1o}+{u}_2{y}_{2o}+\dots +{u}_r{y}_{ro}\left(=\sum \limits_{r=1}^s{u}_r{y}_{ro}\right) $$

Subject to the constraints that

$$ {v}_1{x}_{1o}+{v}_2{x}_{2o}+\dots +{v}_m{x}_{mo}=\sum \limits_{i=1}^m{v}_i{x}_{io}=1 $$

$$ {u}_1{y}_{1j}+{u}_2{y}_{2j}+\dots +{u}_m{y}_{mj}\le {v}_1{x}_{1j}+{v}_2{x}_{2j}+\dots +{v}_m{x}_{mj},\mathrm{for}\ \mathrm{all}\ j. $$

That is, the DEA model presented in Table 1.5 is actually calculated as:

$$ {\displaystyle \begin{array}{l}\operatorname{Maximize}\ \sum \limits_{r=1}^s{u}_r{y}_{ro}\\ {}\mathrm{subject}\ \mathrm{to}\ \sum \limits_{r=1}^s{u}_r{y}_{rj}-\sum \limits_{i=1}^m{v}_i{x}_{ij}\le 0,j=1,\dots, n\\ {}\kern5em \sum \limits_{i=1}^m{v}_i{x}_{io}=1\\ {}\kern5.25em {u}_r,{v}_i\ge 0\end{array}} $$

(1.16)

where we assume that we have n DMUs.

To obtain the information provided in Table 1.3, one needs to employ the dual linear program to model (1.16). That is,

$$ {\displaystyle \begin{array}{l}\min \theta \\ {}\mathrm{subject}\ \mathrm{to}\ \sum \limits_{j=1}^n{\lambda}_j{x}_{ij}\le \theta {x}_{io}\kern1.5em i=1,2,\dots, m;\\ {}\kern5.25em \sum \limits_{j=1}^n{\lambda}_j{y}_{rj}\ge {y}_{ro}\kern2em r=1,2,\dots, s;\\ {}\kern5.25em {\lambda}_j\ge 0\kern4.5em j=1,2,\dots, n.\end{array}} $$

(1.17)

In DEA, model (1.16) is referred to as the “ multiplier model,” where u _r and v _i represent output and input multipliers (weights), respectively. Model (1.17) is referred to as the “envelopment model”.