In primal form of DEA CRS model, we are going to maximize the efficiency of the reference firm. But how is it possible that in the dual form also, we are minimizing the efficiency of the reference firm? What is the physical interpretation of the dual version? Please see the attached file.
You won’t find a physical interpretation, because this is not a physical problem.
One way to think about the dual version is that you are finding the point on the frontier that minimises the maximum horizontal distance from the frontier to each input of the observed unit.
The interpretation changes depending on things like whether the model is input or output oriented. I don’t know of a case where the returns to scale changes the interpretation. It will, of course, change the frontier.
Dear Parag
As John states, that is the explanation of the 'apparent' contradiction between the primal and the dual
I do not know what is DEA (CRS) Model so my response will be in general terms for a Linear Programming (LP) Problem called Primal that requires maximization of some objective function and its dual that requires minimization.
There is an excellent explanation as 'Economic Interpretation of Dual' in Linear Programming by Vasek Chvatal (Pages 63-65). Soft copy of this book is available on net.
Dear Parag, John is right. Solving dual gives you an idea about how slacks are behaving. This tells you about the potential improvements needed in input and output variables. Added convexity constraint in the dual tells you about returns to scale concept.
It's not this way in DEA I think. We find the solution of the primal by solving it's dual because solving the dual gives more information and is less time consuming.
You can interpret theta as the fraction of input resourced of observed DMU required to produce its current outputs. Or in other words, frontier DMU can always produce the outputs of observed DMU just by taking a fraction (which is equal to theta) of the observed DMU inputs.
In the Dual LP (Input-Oriented), Inputs are multiplied by Theta, and Minimizing Theta signifies "proportional reduction of inputs". Similarly, in the Output-Oriented model, Outputs are multiplied by Theta, and Maximizing Theta signifies "proportional augmentation of outputs." In that context, Theta in either DEA model can be interpreted as INTENSITY at which Inputs/Outputs are Incremented/Decremented.
There are two things to keep in mind when dealing with DEA models. Input oriented and output oriented objective function. In the input oriented you minimize the ratio of inputs to outputs. In otherwords, you are seeking here the minimum input combination that produce specific level of output, Where as in the output oriented objective function you maximie the ratio of outputs combination at specific level of inputs. Your case of theta minimiation refers to the first case mentioned above.
There is an LP solved for each DMU. In the primal problem, you are finding the efficiency of a particular DMU with respect to all other DMUs. In the problem for DMU x, you fix the amount of inputs for one unit of x's output to 1, and try to find out if this DMU is the most efficient or not. If it is, the efficiency of x will be found as 1. If it is say, 0.8, then that means the other DMUs are more efficient, i.e. it is possible to produce the same amount of output as x by using only 80% of the x's inputs. In the dual version of the same problem, you are trying to find the equivalent of the DMU x as a linear combination of all the DMUs. That linear combination of the DMUs can produce the same amount of the output ox x, given the same amount of inputs. In either formulation, the objective value gives you the efficiency of the DMU x. If it is 1, then it is efficient. If it is less than 1, then it is not as efficient as the others. You can reverse this and think of the dual as the primal and the primal as the dual, and formulate the problem either way.