The measurement of relative efficiency using data envelopment analysis with assurance regions that link inputs and outputs

assura tegoriz nput w t weigh he DEA ss the es the e effic velope The most popular weight restrictions are to be within certain ranges. ARs can be ca ARI specify bounds on ratios between i bounds on ratios that link input to outpu efficiency, but in the presence of ARII t become infeasible. In this paper we discu pose a new nonlinear model that overcom which enables the assessment of relativ specified. The application of the model de secondary schools. * Corresponding author. Address: Iran University o E-mail addresses: [email protected] (M. Khalili nce regions (ARs), which impose ratios between weights ed into two types: ARs type I (ARI) and ARs type II (ARII). eights or between output weights, whilst ARII specify ts. DEA models with ARI successfully maximize relative models may under-estimate relative efficiency or may problems that can occur in the presence of ARII and prolimitations discussed. Also, the dual model is described, iency when trade-offs between inputs and outputs are d is illustrated in the efficiency assessment of Portuguese The efficiency of decision making units (DMUs) in data envelopment analysis (DEA) is defined as the ratio of the weighted sum of outputs to the weighted sum of inputs. The weights are the variables of the DEA model, and DMUs have complete freedom to choose the weights associated with each input and/or output that maximise their relative efficiency. This complete flexibility in the selection of weights is especially important for identifying inefficient DMUs, as when the unit under assessment does not score 100% efficiency, this tells us that its peers are more productive even when the weights of all units are set to maximise the score of the unit assessed. Therefore, no inefficient unit can complain that its score would have been better if a different set of weights was used. However, this complete flexibility may result in some inputs and/or outputs being assigned a zero or negligible weight, meaning that these factors are in fact ignored in the efficiency assessment. One way to limit the range of values that the weights can take is to use weight restrictions. Literature reviews on the use of weight restrictions in DEA can be found in Allen et al. (1997) and Thanassoulis et al. (2004). Several types of weight restrictions (WRs) have been proposed in the DEA literature. In Allen et al. (1997) the direct weight restrictions are categorized into three types: assurance regions type I (first proposed by Thompson et al., 1986), assurance regions type II (first proposed by Thompson et al., 1990 and often called linked cone assurance regions), and absolute weight restrictions (first proposed by Dyson and Thanassoulis, 1988). Assurance regions (AR) are distinct from absolute weight restrictions because instead of imposing the weights to be within a certain range of values, they impose ratios between weights to be within certain ranges. ARI specify these ratios either between input or output weights separately, and ARII specify ratios that link input to output weights. When absolute weight restrictions are imposed directly on DEAmodels with constant returns to scale (CRS) technology, the models may be infeasible, or the efficiency scores may be under-estimated (see e.g. Allen et al., 1997; Podinovski and Athanassopoulos, 1998). In order to obtain a correct estimate of relative efficiency in the presence of absolute weight restrictions, Podinovski and Athanassopoulos (1998) proposed the use of a Maximin model, and developed an equivalent linear programming formulation to enable an easy computation of relative efficiency scores, whilst avoiding all the problems of absolute weight restrictions (see also Podinovski, 1999). The problems mentioned above do not occur in DEA models that use ARI (see Charnes et al., 1990; Thompson et al., 1990; Podinovski, 2001). This may explain the widespread use of this approach in DEA assessments (see e.g. Thompson et al., 1992 for oil/gas producers, Introduction f Science and Technology, Tehran, Iran. Tel.: +98 (0) 21 77454199. ), [email protected] (A.S. Camanho), [email protected] (M.C.A.S. Portela), [email protected] (M.R. Alirezaee). Schaffnit et al., 1997 for bank branches and Olesen and Petersen, 2002 for hospitals). However, similar problems occur in the presence of ARII, which may justify the small number of empirical applications that used DEA models with ARII (see e.g. Thanassoulis et al., 1995, that applied ARII for English Perinatal Care units). The use of ARII has been more frequent in ‘‘profit ratio models”, which differ from standard DEA because the constraints imposing the ratio of virtual outputs to virtual inputs to be lower or equal to one are removed for all DMUs (see e.g. Thompson et al., 1995, 1996, 1997). Thompson et al. (1990) and Thompson and Thrall (1994) pointed a limitation associated with the use of ARII in DEAmodels, which is the possibility of the model being infeasible for some or all DMUs. To overcome this limitation, Thompson and Thrall (1994) introduced a nonlinear DEA model that can retrieve the correct relative efficiency scores in the presence of ARII. As mentioned by the authors, the model was only solved for the special case of a single output and two inputs. Tracy and Chen (2005) also addressed this issue for a generalised form of weight restrictions, which encompass all forms of weight restrictions described in the literature. In fact, as explained in the next sections, our approach is similar to that of Tracy and Chen (2005) and solves some of its problems for the special case of ARII. In this paper we explore the use of DEA models with ARII. The main limitations of these models, in addition to the infeasibility problem mentioned above, include the under-estimation of relative efficiency and the definition of a frontier for the production possibility set that may not include any of the observed DMUs. To overcome these limitations, inspired by the ideas of Thompson and Thrall (1994) and Podinovski and Athanassopoulos (1998), we propose the use of a nonlinear model that is equivalent to a Maximin model. This paper is organized as follows. Section 2 explores the problems of using ARII associated to a standard formulation of the DEA model. The limitations of this procedure are illustrated using an example. Section 3 develops a new formulation of the DEA model that overcomes the problems described, and shows how the correct relative efficiency estimates can be retrieved in the presence of ARII. Section 4 applies the new model to Portuguese secondary schools. Section 5 concludes the paper. The original DEA model for the estimation of relative efficiency was proposed by Charnes et al. (1978). The rationale of a relative efficiency measure is to compare the ratio output/input of the DMU assessed with the best value of this ratio observed in other DMUs. In the case of multiple input and multiple output assessments, the relative efficiency notion can be generalised to a comparison of ratios of the weighted sum of outputs to the weighted sum of inputs. To illustrate this idea, consider the assessment of n DMUs ðj 1⁄4 1; . . . ;nÞ. Each DMU uses m inputs xijði 1⁄4 1; . . . ;mÞ, to produce s outputs yrjðr 1⁄4 1; . . . ; sÞ. The input and output weights used for the efficiency assessment of a DMU are v i and ur , respectively. The concept of relative efficiency can be defined using mathematical programming. The use of a Maximin model for this purpose was proposed by Thompson and Thrall (1994), and we reproduce in (1) the formulation described in Cooper et al. (1996). The interpretation of DEA effciency assessments with ARII