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Data Envelopment Analysis (DEA) is a mathematical programming approach to evaluate the relative efficiency of decision-making units (DMUs) which use multiple inputs to produce multiple outputs. Identification of the DMUs forming the frontier before using DEA is of great importance to have an effective calculation. This article introduces the worst efficiency analysis approach in which an inefficient production frontier is used to determine the worst relative efficiency score that can be assigned to any DMU. Furthermore, mathematical properties determining the intrinsic relationships between the inefficient frontier DMUs and the output-input ratios are discussed. It was observed that a high ranked performance in the ratio analysis indicates a DEA frontier. This in turn allows the identification of membership of frontier DMUs without solving a DEA program. This finding is helpful in simplifying the DEA solution.

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In the present world, calculating the efficiency (or the performance) of systems with an internal structure, such as two-stage systems, is principally imperative during multi-time periods. In the present approach, the traditional two-stage Data Envelopment Analysis (DEA) model is developed to a multi-period two-stage DEA model, which evaluates the overall and periods of efficiencies synchronously. This approach which is used, is not only alone incapable of having functional capacities under the assumption of variable returns to scale (VRS), but is also inattentive to the importance or magnitude of data in different periods. In this study, in order to surmount these shortcomings, we expand the existing approaches and introduce a generalized model to measure the overall efficiency of a multi-period two-stage system under the VRS assumption, wherein, the importance of data in time- periods is considered in a diverse manner. According to this generalized model, theorems are also being presented to determine the type of returns to scale (RTS) of both stages, as well as the system of the entire time periods and also each period. Finally, the real data of Taiwanese non-life insurance companies, which has been extracted from the extant literature, is used to illustrate the proposed approach.