A new framework for the solution of DEA models

We provide an alternative framework for solving Data Envelopment Analysis (DEA) models which, in comparison with the standard Linear Programming (LP) based approach that solves one LP for each Decision Making Unit (DMU), delivers much more information. By projecting out all the variables which are common to all LP runs, we obtain a formula into which we can substitute the inputs and outputs of each DMU in turn in order to obtain its efficiency number and all possible primal and dual optimal solutions. The method of projection, which we use, is Fourier-Motzkin (F-M) Elimination. This provides us with the finite number of extreme rays of the elimination cone. These rays give the dual multipliers which can be interpreted as weights which will apply to the inputs and outputs for particular DMUs. As the approach provides all the extreme rays of the cone, multiple sets of weights, when they exist, are explicitly provided. Several applications are presented. It is shown that the output from the F-M method improves on existing methods of i) establishing the returns to scale status of each DMU, ii) calculating cross-efficiencies and iii) dealing with weight flexibility. The method also demonstrates that the same weightings will apply to all DMUs having the same comparators. In addition it is possible to construct the skeleton of the efficient frontier of efficient DMUs. Finally, our experiments clearly indicate that the extra computational burden is not excessive for most practical problems. Subject Classifications: Data Envelopment Analysis, Returns to Scale in DEA, Cross-efficiency in DEA, Weight Restrictions in DEA, Fourier-Motzkin elimination.