A conic relaxation model for searching for the global optimum of network data envelopment analysis

Abstract

Network data envelopment analysis (DEA) models the internal structures of decision-making units (DMUs). Unlike the standard DEA model, multiplier-based network DEA models are often highly non-linear and cannot be converted into linear programs. As such, obtaining a non-linear network DEA's global optimal solution is a challenge because it corresponds to a nonconvex optimization problem. In this paper, we introduce a conic relaxation model that searches for the global optimum to the general multiplier-based network DEA model. We reformulate the general network DEA models and relax the new models into second order cone programming (SOCP) problems. In comparison with linear relaxation models, which is potentially applicable to general network DEA structures, the conic relaxation model guarantees applicability in general network DEA, since McCormick envelopes involved are ensured to be finite. Furthermore, the conic relaxation model avoids unnecessary linear relaxations of some nonlinear constraints. It generates, in a more convenient manner, feasible approximations and tighter upper bounds on the global optimal overall efficiency. Compared with a line-parameter search method that has been applied to solve non-linear network DEA models, the conic relaxation model keeps track of the distances between the optimal overall efficiency and its approximations. As a result, it is able to determine whether a qualified approximation has been achieved or not, with the help of a branch and bound algorithm. Hence, our proposed approach can substantially reduce the computations involved.