New Formulations for Choice Network Revenue Management

Models incorporating more realistic models of customer behavior, as customers choosing from an offer set, have recently become popular in assortment optimization and revenue management. The dynamic program for these models is intractable and approximated by a deterministic linear program called the choice deterministic linear program (CDLP), which has an exponential number of columns. Column generation has been proposed but finding an entering column is NP-hard when segment consideration sets overlap. In this paper we propose a new approach called segment-based deterministic concave program (SDCP) based on segments and their consideration sets. SDCP is a relaxation of CDLP and hence forms a looser upper bound on the dynamic program, but coincides with CDLP for the case of nonoverlapping segments. If the number of elements in a consideration set for a segment is not very large, SDCP can be applied to any discrete-choice model of consumer behavior. We tighten the SDCP bound by (i) simulations, called the randomized concave programming method, and (ii) by adding cuts to a recent compact formulation (SBLP) of the problem for a latent multinomial-choice model (MNL) of demand. This latter approach turns out to be very effective, essentially obtaining CDLP value, even for overlapping segments. By formulating the problem as a separation problem, we give insight into why CDLP is easy for the MNL with nonoverlapping consideration sets and why generalizations of MNL pose difficulties. Numerical conclusions that we derive from the present paper are the following: (a) The randomized linear programming approach that obtains significant tightening of the linear program upper bound under an older independent-class model seems to have relatively little effect for the choice case; (b) for the MNL choice model, the SBLP+ formulation we give here for overlapping segments is very fast and is potentially scalable to industrial-size problems.