Efficiency of two-stage systems in stochastic DEA

In the present world, there are many two-stage systems which provide information of inputs, outputs and intermediate measures which are imprecise, such as, (stochastic, fuzzy, interval etc). In these conditions, a two-stage data envelopment analysis or a (two-stage DEA method) cannot evaluate the efficiencies of these systems. In several two-stage systems, the simultaneous presence of stages is necessary for the final product. Hence, in this paper, we shall propose the stochas- tic multiplicative model and the deterministic equivalent, to measure the efficiencies of these systems, primarily, in the presence of stochastic data, under the constant returns to scale (CRS) assumption, by using the non-compensatory property of the multiplication operator.Then, we will use the reparative property of the additive operation to propose the additive models as well as the deterministic equivalents, to calculate the efficiencies of two-stage systems, in presence of stochastic data, under the constant returns to scale (CRS) and variable returns to scale (VRS) assumptions. This is to illustrate that a simultaneous presence of the stages is not necessary for the final product and one stage compen- sates the shortcomings of another stage. Likewise, we shall convert each of these deterministic equivalents to quadratic programming prob- lems. Based on the proposed stochastic models, the whole system is efficient if and only if, the first and the second stages are efficient. Ul- timately, in the proposed multiplicative model, we will illustrate the proposed multiplicative model, by employing the data of the Taiwanese non-life insurance companies, which has been extracted from the extant literature.