DEA-like models for the efficiency evaluation of hierarchically structured units

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Abstract

The knowledge of the internal structure of decision making units (DMUs) gives further insights with respect to the “black box” perspective when considering data envelopment analysis models. We present one-level and two-level hierarchical structures of the DMUs under evaluation. Each unit is composed of consecutive stages of parallel subunits all with constant returns to scale. In particular, the maximization of the relative efficiency of a DMU is studied. For the two-stage situation, different degrees of coordination among the subunits of the hierarchical levels are discussed. When some form of coordination has to be guaranteed, we introduce balancing constraints and we compare two different models. In both cases, we prove that the maximum relative efficiency of a DMU is assessed by comparing it with all the existing subunits.

Introduction

The performance assessment of production or administrative units may rely on indexes which are function of the input and output flow levels. In particular, an evaluation measure may be defined as the weighted ratio of the products or outcomes yielded by the unit over the used or consumed resources. This index is generally referred to as efficiency (see, e.g., [4]). Although different features may be detected by the different performance measures (i.e., productivity, technical and allocative efficiency, etc.), the common underlying assumption is that each unit is always considered as a “black box”, i.e., its internal structure is not considered. As a consequence, generally there is no clear evidence of the transformations the resources are subject to within the considered units.

The knowledge of the internal structure of the decision making units (DMUs) might give further insights for their performance evaluation. For instance, it might be possible to determine whether better performances could be theoretically obtained by merging the technologies of some substructures of the observed DMUs. In addition, assessing the efficiency of each of the processes, or subunits, that yield DMU outputs might prevent that the inefficiency of some of them may be compensated by the efficiency of other ones.

In recent years, data envelopment analysis (DEA) has been proved to be a suitable tool to model the interdependencies among distributed decision makers operating in a structured framework (see, e.g., [2], where efficiency and incentives in regulated markets are considered). On one side, the relative efficiency evaluation process may be embedded in more aggregate models (see, e.g., [1], where classical DEA models, i.e., dealing with homogeneous decision makers and disregarding their internal structure, are considered). On the other side, the input to output transformation process performed by a DMU may also be disaggregated taking into account the intermediate steps the resources are subject to before being delivered as final products [5]. A network model appropriately represents the connections among all the subunits making up the whole DMU [6]. As a consequence, these subunits are interdependent, in the sense that part of the output flow produced by each of them is used as input by another one. This situation implies the non-homogeneity of the elements in the comparison set. However, the relative efficiency of a DMU with respect to other non-homogeneous units may be still evaluated using DEA-like models [3].

In this paper, we show that the knowledge of the internal structure of DMUs leads to different results with respect to the “black box” perspective when considering DEA models. In particular, the maximization of the relative efficiency of a DMU is studied (see, e.g., [7]). We always refer to the CCR model (see, e.g., [4]). Each unit is composed of consecutive stages of parallel subunits each of them exhibiting constant returns to scale. We prove that under some specific assumptions the maximum relative efficiency of a DMU is assessed by comparing it with all the existing subunits.

The following section presents a simple example which might help to sketch some relevant issues and to motivate the interest for studying them. Then, in Section 3 the notation used, the basic assumptions and the relevant definitions are introduced. In addition, the problem faced in this paper is stated. In 4 Single stage, 5 Two stages DEA models evaluating the maximum relative efficiency of DMUs with one-layer and two-layer hierarchical structures are presented. Their properties are discussed and compared. Finally, in Section 6 a short numerical example is provided.

Section snippets

Considering internal structure

A set of three DMUs U={u(a0,b0),u(a1,b1),u(a2,b2)}, where each DMU u(ai,bi) is made of two parallel subunits (DMSUs) ai, bi, i=0,1,2, is considered. DMUs are homogeneous, in the sense that they transform the same type of resources, or inputs, into the same type of products, or outputs. DMSUs ai, i=0,1,2, are homogeneous. Similarly, DMSUs bi, i=0,1,2, are also homogeneous. Each DMSU has one input, xk, and one output, yk, k=a,b. Fig. 1 shows the three DMUs and their internal structures.

When the

Definitions

In this section notation, basic assumptions and relevant definitions are introduced. Finally, the problem faced in this paper is formally stated.

Single stage

In this section, a model to evaluate the maximum relative efficiency of DMUs composed of a single layer of parallel subunits (see Fig. 1) is introduced. This is the simplest internal structure to be considered and the obtained results are later used in more complex environments. In this trivial case, obviously, no balancing constraints are necessary.

The maximum relative efficiency of a DMU u0 composed of a single stage of parallel subunits is evaluated by solving the following fractional

Two stages

In this section the maximum relative efficiency of DMUs whose internal structure is composed of two layers of parallel DMSUs is evaluated.

Denote as F and S the set of the subunits belonging to the first and the second stage, respectively. Thus D=FS. Again, C is the set of all the feasible combinations of subunits. F and S are the sets of the subsets of F and S of homogeneous subunits, respectively.

Since all the subunits within the same DMU are not homogeneous and they all have only one single

Numerical example

The different models presented in the previous section are compared by considering a set of 36 Italian Universities as two-level hierarchical DMUs. The internal structure of each unit is composed of one subunit (i.e., the central administration office) in the first layer and 14 subunits (i.e., departments and faculties) in the second stage. The central administration office receives funds from the Italian Ministry for University Research and Education (MIUR) and redistributes teaching personnel

Conclusions

When evaluating the maximum relative efficiency, this paper shows that the knowledge of the internal structure of DMUs gives further insights with respect to the “black box” perspective. We consider one-level and two-level hierarchical structures of the DMUs under evaluation assuming that each subunit exhibits constant returns to scale. For the two-stage situation, different degrees of coordination among the subunits of the hierarchical levels are discussed. When some form of coordination has

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