Average lexicographic efficiency for data envelopment analysis

Cross evaluation measures the efficiency of each DMU based on both its self-evaluation and the peer evaluations of all other DMUs. One problem of cross evaluation is that the cross efficacy scores may be not unique because of the non-uniqueness of the optimal weights of the conventional DEA model used. We propose a new approach for measuring the efficiencies of DMUs based on their lexicographical evaluations. It does not require a common set of weights for all DMUs but can provide a unique efficiency score for each DMU. In the approach, the efficiencies of the DMUs are lexicographically evaluated for each of their possible orders, subject to the constraints that the efficiencies of all DMUs already evaluated cannot be changed. The average lexicographic efficiency of each DMU is then obtained by averaging its lexicographic efficiencies. This efficiency has some desirable properties. Since all DMUs have an equal right to claim a position in the lexicographic evaluation, it is fair in some sense, as the Shapley value for payoff allocation in cooperative games. A procedure for efficiently computing the lexicographic efficiency is proposed. Numerical results on two instances demonstrate the vitality of our approach and the efficiency of its computation procedure.