‎f‎inding the defining hyperplanes of production possibility set with ‎variable‎ returns to scale using the linear independent ‎vectors‎‎

The Production Possibility Set (PPS) is defined as the set of all inputs and outputs of a system in which inputs can produce outputs‎. ‎In Data Envelopment Analysis (DEA)‎, ‎it is highly important to identify the defining hyperplanes and especially the strong defining hyperplanes of the empirical PPS‎. ‎Although DEA models can determine the efficiency of a Decision Making Unit (DMU)‎, ‎but they cannot present efficient frontiers of PPS completely‎. ‎The notion of defining hyperplanes is crucial to marginal discussions‎, ‎marginal rates‎, ‎marginal rates of substitution‎, ‎sensitivity analysis‎, ‎returns to scale‎, ‎and in particular‎, ‎the efficiency analysis of DMUs‎. ‎In this paper‎, ‎we propose a new method to determine all strong efficient(Pareto-efficient) DMUs and strong defining hyperplanes of the PPS with variable returns to scale which strong efficient DMUs are located on them‎. ‎Furthermore‎, ‎we apply the proposed method to find the normal vectors or gradient of the strong defining hyperplanes of the PPS including strong efficient DMUs‎. ‎Consequently‎, ‎the equations of these hyperplanes are determined‎. ‎To illustrate the ability of the proposed method‎, ‎some numerical examples are finally provided‎. ‎Our method can be easily implemented using existing packages for operation research‎, ‎such as GAMS.