The multiplier and envelopment models in data envelopment analysis (DEA) have a primal-dual relationship, and produce the same efficiency measure for a decision making unit (DMU). In addition to measuring efficiency, the multiplier model is able to identify the production frontier facets defined by the DMUs being evaluated, and the envelopment model is able to identify the projection point, based on which the efficiencies are measured. For general two-stage systems where the whole operation of the system is divided into two smaller operations carried out by two divisions connected in series, the multiplier model is generally used to measure division efficiencies, and the envelopment model is used to identify the projection point. This paper shows that the projection point identified by the envelopment model is not the one used by the multiplier model to measure the division efficiencies for general two-stage systems. Based on the primal-dual relationship of the two models, the envelopment model is reformulated to be able to obtain the projection point and measure the division efficiencies at the same time. The input- and output-orientation of the two divisions lead to four forms of the model. A case of measuring the innovation efficiency of thirty-five countries is used to illustrate the characteristics of this model and its differences from the multiplier model.
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