Piecewise linear output measures in DEA (third revision)

It is assumed in the standard DEA model that the aggregate output (input) is a pure linear function of each output (input). This means, for example, that if DMU j1 generates twice as much of an output as does another DMU j2, then the former is credited with having created twice as much value. In many situations, however, linear pricing ([mu]ryrj) may not adequately reflect differences in value created from one DMU to another. In this paper, a generalization of the DEA methodology is presented that incorporates piecewise linear functions of factors. We deal specifically with those situations where for certain outputs in an input-oriented model, the weight function f(yrj) is described by either a non-increasing or non-decreasing set of multipliers for larger amounts of the factor. We refer to such a variable r as exhibiting diminishing marginal value (DMV) or increasing marginal value (IMV). The DMV/IMV phenomenon is common in many for-profit applications. For example, in the case that yrj is the amount of a consumer product r generated by DMU j, and [mu]r is the price of that product, it may well be that the market will force lower prices if greater amounts of that product are generated; discounts automatically lead to this DMV situation. Such a phenomenon can arise as well in not-for-profit settings, and we examine such a situation based on earlier work by Cook et al. [Cook, W.D., Roll, Y., Kazakov, A., 1990. A DEA model for measuring the relative efficiency of highway maintenance patrols. INFOR 28 (2), 113-124].