An Enhanced Concave Program Relaxation for Choice Network Revenue Management

The network choice revenue management problem models customers as choosing from an offer set, and the firm decides the best subset to offer at any given moment to maximize expected revenue. The resulting dynamic program for the firm is intractable and approximated by a deterministic linear program called the CDLP which has an exponential number of columns. However, under the choice-set paradigm when the segment consideration sets overlap, the CDLP is difficult to solve. Column generation has been proposed but finding an entering column has been shown to be NP-hard. In this study, starting with a concave program formulation called SDCP that is based on segment-level consideration sets, we add a class of constraints called product constraints (sigma PC), that project onto subsets of intersections. In addition, we propose a natural direct tightening of the SDCP called ESDC kappa, and compare the performance of both methods on the benchmark data sets in the literature. In our computational testing on the data sets, 2PC achieves the CDLP value at a fraction of the CPU time taken by column generation. For a large network our 2PC procedure runs under 70 seconds to come within 0.02% of the CDLP value, while column generation takes around 1 hour; for an even larger network with 68 legs, column generation does not converge even in 10 hours for most of the scenarios while 2PC runs under 9 minutes. Thus we believe our approach is very promising for quickly approximating CDLP when segment consideration sets overlap and the consideration sets themselves are relatively small.