Exact and Approximate Cost Functions for Product Aggregates

We show that certain large, multiproduct problems can be well approximated by smaller models of the same form, representing only one aggregate product (or a few such products). This reduction follows from a new technique to approximate the minimal cost of a special allocation or “disaggregation” problem, by a simple, closed-form aggregate cost function. This function, moreover, has the same form as the cost functions for the original, individual products. For special cases the “approximation” is exact. (This latter result is related to economic-theoretic research on aggregation with homothetic preferences.) The results are applied to a one-stage, multiperiod, multi-item production smoothing model, and then to a similar model of a two-stage system, yielding a reduction in the number of items to one or to relatively few.