Identifying the most important set of weights when modelling bad outputs with the weak disposability approach

Environmental efficiency and the shadow prices of undesirable outputs have received increasing attention in recent years. In data envelopment analysis the shadow prices can be calculated using the dual variables, or weights, from the solved linear programming program. However, the weights, and therefore the shadow prices, for corner points are not necessarily unique and most linear program solvers simply return one of the possible optimal values. To overcome this uncertainty we build on a model suggested by Cooper et al. (2007) which for corner points selects a set of weights that is most supported by other points on the frontier. In the context of incorporating undesirable outputs, we adjust their idea for the weak disposability model and suggest an alternative approach to break ties in case two sets of weights are supported by the same number of points. We illustrate the models using a constructed example dataset and also apply it on a sample of UK dairy farms.