Slack Variables Measure The Super-effective Model

Data envelopment analysis (DEA for short), originally formulated by A.Charnes, Cooper and Rhodes (CCR 1978), measure the relative efficiencies among the decision making units(DMUs) with multiple input and multiple output as a linear programming formulation. It is a new cross-subject that covers mathematics, operation research, economics as wel as management science. DMU may be efficiency simultaneously when evaluated by DEA model, where the comparison of DMU is not sufficient. In the recent decade, ranking DEA efficient units has become the interests of many DEA researchs, and a method in which the DMU efficient units is classified has been proposed, called super-efficiency models. The slacks-based measure (SBM) of super-efficiency is developed by Tone based on input/output slacks. The rationality for this measure is to minimize a sort of weighted l1-distance from an efficient decision making units to the production possibility set excluding the decision making units. The SBM super-efficiency model is a non-radial super-efficiency model compared to the traditional radial super-efficiency DEA models:In which we will deal with n DMUs with the input and output matrices X= (xij)∈Rm×n and Y= (yrj)∈Rs×n. It is assume that the data set is positive, i.e X> 0, and Y> 0.Super-efficiency models not only comparison of DMU efficient but also it can classifies all the decision making units. However, the SBM super-efficiency model evaluated only for decision making units of efficiency, but cannot be classify all the decision making units. This paper extends the SBM super-efficiency model based on the work of Andersen. Our new model express as: called SCI model and called SCI super-efficiency modelUnlike the traditional SBM super-efficiency model, our new model are always feasible under constant or variable return to scale assumption, furthermore it can be classify all the decision making units, we have the following result:Result 1:SCI super-efficiency model are always feasible under constant or variable return to scale assumption.Result 2:For SCI model and SCI super-efficiency model considering any DMUs. Then there are three possible cases:Case 1:the SCI super-efficiency optimal value of DMU(xo,yo)θ*<1, then DMU(xo,yo) is SCI model of inefficient decision making unit.Case 2:the SCI super-efficiency optimal value of DMU[xo,yo)θ*=1, then DMU(xo, yo) is SCI model of inefficient decision making unit or DMU0∈E(?)F.Case 3:the SCI super-efficiency optimal value of DMU(xo,yo)θ*> 1, then DMUo∈E.In which we partition DMUs of frontier into three set:E is the set of extremely efficient decision making units; E is the set of efficient but not extremely efficient decision making units; F is the set of weak-efficient with non-zero slacks decision making units.At the same time, the returns to scale is also be considered. Finally, we have compared the application research, and the results indicate that our new model performs well.