Error bound comparisons for aggregation/disaggregation techniques applied to the transportation problem

Abstract

A priori and a posteriori error bounds for a transportation problem model at different levels of aggregation were statistically compared. An experimental design was used to (1) examine the size and significance of correlation between all pairs of the a priori error bound, a posteriori error bound and actual error and (2) quantify the size and significance of the difference of a posteriori error bound from actual error. Two different methods for calculating a posteriori error bounds were utilized. Results are for aggregating customers in a transportation model using one aggregation strategy and varying the level of aggregation on a set of randomly generated problems. Results show significant correlation between a posteriori error and actual error. A priori error is not significantly correlated with actual error. These preliminary results indicate that calculating the a posteriori error bound to select the appropriate aggregation level is a helpful strategy since the a posteriori bound varies in the same way that the actual error varies. In addition, one method of calculating the a posteriori bound is determined to be significantly tighter than the other method of calculating the a posteriori error bound. Scope and purpose Aggregation/disaggregation techniques are used to reduce a large model to a smaller model. Error bounds for the objective function of an aggregated linear programming model quantify the loss of information due to aggregation. There are two types of error bounds, a priori and a posteriori, and several methods to calculate the a posteriori bound. This is an exploratory study to determine if these error bounds vary consistently when different levels of aggregation are applied to the customers of the same transportation problem model and to statistically compare the tightness of these bounds. Results show that under the conditions of this study there is significant correlation between the actual aggregation error and a posteriori error. This implies that aggregation levels that produce large a posteriori error bounds are associated with large actual error and aggregation levels that produce small a posteriori error bounds are associated with small actual error. Results also show that of the two methods used to calculate the a posteriori bound, one consistently determines a tighter bound.