Continuous Optimization
A generalized model for data envelopment analysis

https://doi.org/10.1016/S0377-2217(03)00140-1Get rights and content

Abstract

Data envelopment analysis (DEA) is a method to estimate a relative efficiency of decision making units (DMUs) performing similar tasks in a production system that consumes multiple inputs to produce multiple outputs. So far, a number of DEA models have been developed: The CCR model, the BCC model and the FDH model are well known as basic DEA models. These models based on the domination structure in primal form are characterized by how to determine the production possibility set from a viewpoint of dual form; the convex cone, the convex hull and the free disposable hull for the observed data, respectively.

In this study, we suggest a model called generalized DEA (GDEA) model, which can treat the above stated basic DEA models in a unified way. In addition, by establishing the theoretical properties on relationships among the GDEA model and those DEA models, we prove that the GDEA model makes it possible to calculate the efficiency of DMU incorporating various preference structures of decision makers. Furthermore, we propose a dual approach to GDEA, GDEAD and also show that GDEAD can reveal domination relations among all DMUs.

Introduction

Data envelopment analysis (DEA) was suggested by Charnes, Cooper and Rhodes (CCR), and built on the idea of Farrell [10] which is concerned with the estimation of technical efficiency and efficient frontiers. The CCR model [5], [6] generalized the single output/single input ratio efficiency measure for each decision making unit (DMU) to multiple outputs/multiple inputs situations by forming the ratio of a weighted sum of outputs to a weighted sum of inputs. DEA is a method for measuring the relative efficiency of DMUs performing similar tasks in a production system that consumes multiple inputs to produce multiple outputs. The main characteristics of DEA are that (i) it can be applied to analyze multiple outputs and multiple inputs without preassigned weights, (ii) it can be used for measuring a relative efficiency based on the observed data without knowing information on the production function and (iii) decision makers' preferences can be incorporated in DEA models. Later, Banker, Charnes and Cooper (BCC) suggested a model for estimating technical efficiency and scale inefficiency in DEA. The BCC model [2] relaxed the constant returns to scale assumption of the CCR model and made it possible to investigate whether the performance of each DMU was conducted in region of increasing, constant or decreasing returns to scale in multiple outputs and multiple inputs situations. In addition, Tulkens [20] introduced a relative efficiency to non-convex free disposable hull (FDH) of the observed data defined by Deprins et al. [9], and formulated a mixed integer programming to calculate the relative efficiency for each DMU. Besides basic models as mentioned in the above, a number of extended models have been studied, for example, cone ratio model [8], polyhedral cone ratio model [7], Seiford and Thrall's model [16], Wei and Yu's model [21], and so on.

On the other hand, relationships between DEA and multiple criteria decision analysis (MCDA) have been studied from several viewpoints by many authors. Belton [3], and Belton and Vickers [4] measured an efficiency as a weighted sum of input and output. Stewart [17] showed the equivalence between the CCR model and some linear value function model for multiple outputs and multiple inputs. Joro et al. [13] proved structural correspondences between DEA models and multiple objective linear programming using an achievement scalarizing function proposed by Wierzbicki [22]. Especially, various ways of introducing preference information into DEA formulations have been developed. Golany [11] suggested a so-called target setting model, which allows decision makers to select the preferred set of output levels given the input levels of a DMU. Thanassoulis and Dyson [19] introduced models that can be used to estimate alternative output and input levels, in order to render relatively inefficient DMUs efficient. Zhu [23] proposed a model that calculates efficiency scores incorporating the decision makers' preference informations, whereas Korhonen [14] applied an interactive technique to progressively reveal preferences. Halme et al. [12] evaluated an efficiency of DMU in terms of pseudo-concave value function, by considering a tangent cone of the feasible set at the most preferred solution of decision maker. Agrell and Tind [1] showed correspondences among the CCR model [5], the BCC model [2] and the FDH model [20] and MCDA model according to the property of a partial Lagrangean relaxation. Yun et al. [24] suggested a concept of “value free efficiency” in the observed data.

In this study, we propose a generalized model for DEA, so-called GDEA model, which can treat basic DEA models, specifically, the CCR model, the BCC model and the FDH model in a unified way. In addition, we show theoretical properties on relationships among the GDEA model and those DEA models, and the GDEA model makes it possible to calculate the efficiency of DMUs incorporating various preference structures of decision makers. Finally, we suggest a dual approach GDEAD to GDEA and show also that GDEAD can reveal domination relations among all DMUs.

Section snippets

Basic DEA models

In the following discussion, we assume that there exist n DMUs to be evaluated. Each DMU consumes varying amounts of m different inputs to produce p different outputs. Specifically, DMUj consumes amounts xj:=(xij) of inputs (i=1,…,m) and produces amounts yj:=(ykj) of outputs (k=1,…,p). For these constants, which generally take the form of observed data, we assume xij>0 for each i=1,…,m and ykj>0 for each k=1,…,p. Further, we assume that there are no duplicated units in the observed data. The p×n

GDEA based on parametric domination structure

In this section, we formulate the GDEA model based on a domination structure and define a new `efficiency' in the GDEA model. Next, we establish relationships between the GDEA model and basic DEA models mentioned in Section 2.

Now, we formulate a generalized DEA model by employing the augmented Tchebyshev scalarizing function [15]. The GDEA model, which can evaluate the efficiency in several basic models as special cases, is the following:maximizeΔ,μk,νiΔsubjecttoΔ≦d̃j+α∑k=1pμk(yko−ykj)+∑i=1mνi

GDEA based on production possibility

In this section, we consider a dual approach to GDEA introduced in Section 3. We formulate the GDEAD model based on the production possibility set and define `efficiency' in the GDEAD model. Next, we establish relationships between the GDEAD model and dual models to basic DEA models mentioned in Section 2.

To begin with, an output–input vector zj of a DMUj,j=1,…,n, and output–input matrix Z of all DMUs respectively, denoted byzj:=yj−xj,j=1,…,n,andZ:=Y−X.In addition, we denote a (p+m)×n matrix Zo

Comparison between GDEA and DEA models

Now, we compare the efficiency in basic DEA models and the GDEA model for the data in Taylor et al. [18]. The data for thirteen Mexican commercial banks in two years (1990–1991) is from Taylor et al. [18]. As shown in Table 8, each bank has the total income as the single output. Total income is the sum of a bank's interest and non-interest income. Total deposits and total non-interest expense are the two inputs used to generate the output. Interest income includes interest earned from loan

Conclusions

In this paper, we suggested the GDEA model based on parametric domination structure, and defined α-efficiency in the GDEA model. In addition, we investigated theoretical properties on relationships between the GDEA model and existing DEA models, specifically, the CCR model, the BCC model and the FDH model. And then, it was proved that the GDEA model makes it possible to evaluate efficiencies of several DEA models in a unified way, and to incorporate various preference structures of decision

References (24)

  • A. Charnes et al.

    Cone ratio data envelopment analysis and multi-objective programming

    International of Systems Science

    (1989)
  • D. Deprins et al.

    Measuring labor-efficiency in post offices

    • Cited by (47)

    • The grain Food-Energy-Water nexus in China: Benchmarking sustainability with generalized data envelopment analysis

      2023, Science of the Total Environment

    • Sustainability based perspective on the utilization efficiency of urban infrastructure — A China study

      2019, Habitat International
      Citation Excerpt :

      In recent years, DEA model has been widely applied in different areas to assess efficiencies such as low-carbon economy efficiency (Wang et al., 2017; Zhang, Zeng, Wang, Yang, & Jiang, 2017), environmental efficiency (Li, Fang, Yang, Wang, & Hong, 2013), water efficiency (Wu et al., 2018), and energy efficiency (Li & Shi, 2014). The advantages of DEA have been well recognized (Cook, Tone, & Zhu, 2014; Cook & Seiford, 2009; Yun, Nakayama, & Tanino, 2004). DEA model can be applied to assess the relative efficiencies of DMUs without demand for specifying functional relation between inputs and outputs.

    • Investment efficiency of urban infrastructure systems: Empirical measurement and implications for China

      2017, Habitat International
      Citation Excerpt :

      The DMUs with minimum input levels given output levels or with maximum output levels given input levels form the DEA frontier (Chen & Ali, 2004). DEA has been commonly used for many areas of efficiency measurement for the following benefits (Chen, 2003; Lee & Pai, 2011; Yun, Nakayama, & Tannino, 2004; Zhu, 2002). Firstly, it can be applied to measure the relative efficiency with no need for previously knowing the functional form between inputs and outputs.

    • A fuzzy expected value approach under generalized data envelopment analysis

      2015, Knowledge-Based Systems
      Citation Excerpt :

      Concurrently, the Free Disposal Hull (FDH) model [11] was developed as an alternative DEA model which benefits from a mixed integer programming to calculate the relative efficiencies of decision making units (DMUs). In order to treat basic CCR, BCC and FDH models in a unified way, a generalized DEA model (GDEA) was proposed by Yun et al. [36]. Since traditional DEA models do not account for subjective input and output values, another class of DEA models emerged; that is, fuzzy DEA models [12,17].

    • On efficiency in convex hull of DMUs

      2013, Applied Mathematical Modelling
      Citation Excerpt :

      Then the obtained result will be compared with efficiency amount obtained from other classic DEA models. Assume that there are six DMUs which consume one input to produce one output, as seen in Table 1(taken from [17]). Also Table 2 shows the results of efficiency in the basic DEA models.

    • Environmental efficiency under weak disposability: an improved super efficiency data envelopment analysis model with application for assessment of port operations considering NetZero

      2023, Environment, Development and Sustainability
    View all citing articles on Scopus
    View full text
    Copyright © 2003 Elsevier B.V. All rights reserved.