This section discusses extensions to the three cases: (1) time-series data, (2) marginal cost-oriented basic cost models, and (3) single driver-single cost pool data.
Suppose the data Yrt are given over time periods indexed by t for a single business unit. Then the MPE model with index j replaced by t can be applied. First, it would be necessary to adjust all the xt cost-pool figures and resulting Yrt data to reflect current dollars using a cost index. This assumes that the estimated ar* cost rates are in terms of current dollars. Next, these rates would be interpretable as follows. The estimated ar* in the current dollar time series case may be interpreted to be the cost rate vector achieved by the unit during its most efficient observation period or periods. The resulting vt suggest periods of more or less efficiency, and would be a useful source for self-study aimed at productivity and process improvements.
The comparability issue for the units under comparison should be easier to accept in this case. However, process or technology changes during the data time span could be problematical. A more complete discussion of limitations for this case is left for specific future applications.
In addition to the NLOB effectiveness test, additional considerations can be brought to bear with respect to an improvement over time dimension. Effectiveness in this respect would be supported by establishing a significant fit of the vt data to a monotone increasing function of time, for example, the reciprocal of a learning curve. Over longer periods of times, learning curve patterns for the estimated gamma parameters could serve similarly. That is, decreasing α indicates improving target effectiveness, while decreasing β would indicate improving precision.
Marginal Cost-Oriented Basic Cost Models
Both the time-series and cross-sectional versions of the MPE model can be adapted to nonlinear basic cost models with marginal cost features. For example, consider the original cross-sectional case, but using the basic cost model
Again s is interpreted as a possible inefficiency due to experiencing higher than benchmark ar and/or br values. The cost of service provided by activity r is ar* yr + br* yr2 for efficient units. By differentiation, the marginal cost by this model for activity r becomes ar* + 2br*yr at the observed driver level. Defining new data elements Yrj(2) =y2rj/xj, the modified MPE model becomes
The constraints (6.4) ensure that cost contributions can only be non-negative, even if some br is negative. In that case, the marginal rate is decreasing; if these coefficients are positive, the corresponding marginal rates are increasing. Here a quadratic basic cost model was used. More generally, other models with different marginal cost structures could be employed (e.g., Cobb-Douglas as in Noreen and Soderstrom, 1994).
Implications for the Single Driver Single Cost Pool Case
The MPE model for this case simplifies to max ∑ a Yj, s.t. a Yj ≤ 1, for all j, and a ≥ 0. The solution of this model is clearly a* = min Yj−1 = min xj/yj. The NLOB criterion requires α* ≤ 1/2 in this case. If this condition fails to hold, then this minimum value may be unreasonably low, perhaps due to an outlier. Deletion of one or a few such tentative outliers would be well supported if the remaining data do, in fact, pass the NLOB test. Otherwise no credible ar estimate is forthcoming from the present method. It should be noted that the simulation method could also be employed for this case, provided the NLOB criterion is met.