This paper clarifies the role of non-Archimedean infinitesimal ε in DEA models so that the associated linear programs may be infeasible (for the multiplier side) and unbounded (for the envelopment side) for certain values of ε. It is shown that the bound of ε proposed by Ali and Seiford (1993) is invalid for feasibility and boundedness of the linear programs. A procedure is presented for determining an assurance interval of ε. It is also shown that an assurance value for ε can be found using a single linear program.

Abstract. In Data Envelopment Analysis, when the number of decision making units is small, the numb e r o f u n i t s o f t h e d o m i n a n t or e cient set is relatively large and the average e ciency is generally high. The high average e ciency is the result of assuming that the units in the e cient set are 100% e cient. If this assumption is not valid, this results in an overestimation of the e ciencies, which will be larger for a smaller number of units. Samples of various sizes are used to nd the related bias in the e ciency estimation. The samples are drawn from a large scale application of DEA to bank branch e ciency. The e ects of di erent assumptions as to returns to scale and the number of inputs and outputs are investigated.

The most popular weight restrictions are assurance regions (ARs), which impose ratios between weights to be within certain ranges. ARs can be categorized into two types: ARs type I (ARI) and ARs type II (ARII). ARI specify bounds on ratios between input weights or between output weights, whilst ARII specify bounds on ratios that link input to output weights. DEA models with ARI successfully maximize relative efficiency, but in the presence of ARII the DEA models may underestimate relative efficiency or may become infeasible. In this paper we discuss the problems that can occur in the presence of ARII and propose a new nonlinear model that overcomes the limitations discussed. Also, the dual model is described, which enables the assessment of relative efficiency when trade-offs between inputs and outputs are specified. The application of the model developed is illustrated in the efficiency assessment of Portuguese secondary schools. 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 DEA models with constant returns to scale (CRS) technology, the models...

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The analytical hierarchical process/data envelopment analysis (AHP/DEA) methodology for ranking decision‐making units (DMUs) has some problems: it illogically compares two DMUs in a DEA model; it is not compatible with DEA ranking in the case of multiple inputs/multiple outputs; and it leads to weak discrimination in cases where the number of inputs and outputs is large. In this paper, we propose a new two‐stage AHP/DEA methodology for ranking DMUs that removes these problems. In the first stage, we create a pairwise comparison matrix different from AHP/DEA methodology; the second stage is the same as AHP/DEA methodology. Numerical examples are presented in the paper to illustrate the advantages of the new AHP/DEA methodology.