DEA-like models for efficiency evaluations of specialized and interdependent units

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Abstract

The problem of evaluating the efficiency of a set of specialized and interdependent decision making subunits (DMSUs) that make up a larger decision making unit (DMU) is considered. The DMSUs are interdependent, in the sense that part of the output produced by each of them may be used as an input by the other ones. They are also specialized, hence non-homogeneous, as they may have not the same inputs and outputs. For this problem, some efficiency indexes are introduced, and they are shown to satisfy some basic properties.

Introduction

Within classical data envelopment analysis (DEA) models (see, e.g., [3]), decision making units (DMUs) are assumed to be homogeneous, in the sense that they transform the same type of resources, or inputs, into the same type of products, or outputs. For this reason, each DMU naturally considers all the other DMUs as possible terms of comparison to assess its relative efficiency.

It is less trivial for a DMU to decide which DMUs should be chosen for comparison when some differences are present. In this context, as an example, DEA models have been developed to deal with DMUs that are still homogeneous but subject to different environmental factors (see, e.g., [12]).

In this paper, a new approach to assess the efficiency of specific units with possibly unique duties inside larger organizations is proposed. In particular, a set of decision making subunits (DMSUs) that make up a larger DMU is considered. The DMSUs are interdependent, in the sense that part of the output produced by each of them may be used as an input by the other ones. Furthermore, the DMSUs are specialized, hence non-homogeneous, since both their outputs and inputs may be different from one to the other.

Practical reasons lie behind this work. For instance, they are relevant to economic incentives that are often offered to the members of efficient DMUs to spur production. In the first case, the incentives may be given to the different DMSUs on the basis of the difference between the current and the previous year values of the output production. For each DMSU, the values of its products are expressed in arbitrary units, possibly different for each DMSU. An internal commission may assess the value of each new product comparing it with the others produced by the DMSU under concern. In the second case, in some way complementary to the first one, incentives may be equally distributed within the DMU. However, some DMSU of the DMU may feel itself unjustly penalized, since it considers itself more efficient than other colleagues. Then, complaints arise and the effect of incentives is reduced. For both the above situations, the possibility of introducing a new reward policy, which promotes efficiency also by means of spurring competition between the different DMSUs is currently being considered. Such a policy should take into account the interactions between the different DMSUs, such that the competition does not induce an overall decrease of the efficiency of the whole DMU. The aim of such a policy is to promote efficiency and to penalize opportunistic behavior. To this end, a formal model would turn out to be of great help.

The aim of this work is to present properties and limits for a generalization of DEA models to the case in which the relative efficiency of interdependent and specialized, hence non-homogeneous, units must be assessed.

The basic notations and definitions required to formalize the above issues are introduced in Section 2. In addition, an example referring to the structure of the administrative branch of a university is described in Section 3.

Since the DMSUs considered are not necessarily all homogeneous, as it is normally assumed in ordinary DEA models, some “comparability” questions arise. For instance, it might be non trivial to decide whether it is fair to compare DMSUs having different inputs and outputs, and, if this is the case, in which way the comparison could be achieved.

In fact, DMSUs with different inputs and outputs appear as different entities, so comparing them seems like comparing apples with oranges. However, our attitude towards this problem is different: considering that both apples and oranges are fruits, they can still be compared. Moreover, we may produce different comparisons (hence different efficiency evaluations) for the same DMSU: first each apple is compared with all other apples, and the best apple can be determined in this way. Then, apples can be compared also with all other fruits, thus determining the best fruit.

The approach in this paper is to identify groups of DMSUs that can be sensibly compared. This is accomplished in Section 4 where the concepts of H, N and A efficiency are introduced.

The second basic issue we consider deals with independence of the DMSUs: they are now in general not independent, as they all make up a larger DMU, like different departments of a large firm. To this respect, in Section 5 we are interested in assessing not only the relative efficiency of each department, but also its contribution to the performance of the whole firm. We show in Section 5.1 that this can still be done using DEA concepts, by the W efficiency.

In Section 6, some basic properties of the proposed efficiency indexes are discussed. Finally, some numerical results are reported in Section 7 and conclusions are drawn in Section 8.

Section snippets

Definitions

Assume a DMU consists of n interdependent DMSUs. Each DMSU j transforms resources, or inputs, into products, or outputs. In particular, it produces Kj different types of outputs and consumes Ij different types of external inputs (i.e., inputs coming from outside the whole DMU) and, possibly, a fraction of the outputs produced by the other DMSUs.

All the DMSUs considered have the same types of external inputs. Furthermore, both the external input levels and the output levels may take different

An example

An example referring to the structure of the administrative branch of a university is introduced to be used as a reference throughout the paper (see Fig. 1).

There are some department administrative offices which do not directly interact with each other and can be considered homogeneous, or at least belonging to the same necessary set. In addition, there is a set of central administrative offices, each of which exchanges material both with each other and with the department offices. Then, the

Problem statement

When a set Γ of DMSUs is considered, a DMSU j0 in Γ may be interested in determining the weights which assess its maximum relative efficiency, as defined in (1), with respect to the best (i.e. most efficient) DMSU in Γ.

For the sake of clarity, the problem is formally stated as follows.

Problem

Let a set Γ of DMSUs be given, and consider a DMSU j0Γ. Determine the value of the weights such that the relative efficiency of DMSU j0 is maximized in Γ.

Then, we say that a DMSU is efficient, with respect to the

Mathematical formulation

The problem under concern can be formally stated ash(j0)=supv,w>0minj∈ΓE(j0,v,w)E(j,v,w),where j0Γ and Γ is equal to either ΓH(j0), or ΓN(j0), or ΓA(j0), depending on which efficiency has to be assessed.

Note that the feasible region of (2) is connected and non-empty, and its objective function is continuous and bounded above by 1. Then, h(j0) exists and is obtained as w and w tend to an accumulation point (v,w) of the feasible region. The point (v,w), in the following referred to as

Properties

In this section, some properties of the above efficiencies are addressed and their relations with the input distance function (see, e.g., [10]) are investigated.

Single period

Consider the case in which models , , , , and , , , , , , are applied only to period 1 of the example of Section 3. Results are shown in Table 3. Note that, in this section and in the following one, the A efficiency is not considered since it coincides with the N efficiency. On the other hand, the value of the estimators hW(.,1) and hA(.,1) are reported. In particular, DMSU 2 is N and H efficient and has hA(2,1)=100%, i.e., it also lies on the frontier of the set S defined by the values of

Conclusions

In this paper, the problem of generalizing DEA in order to evaluate the relative efficiency of a set of specialized and interdependent DMSUs has been faced. Some properties have been presented and the relationship between efficiency measures and the inverse of the distance function introduced in [7], [10] has been investigated. In particular, it is shown that the proposed models, except the one concerning the W efficiency, straightforwardly generalize the DEA–CCR model.

It also turns out that a

Acknowledgements

We are grateful to an anonymous referee for his/her comments. According to them, the paper has been subject to major modifications which in our opinion significantly improved its value. Certainly, part of the results we propose could not have been obtained without his/her precious suggestions.

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