An Enhanced Concave Program Relaxation for Choice Network Revenue Management

Abstract

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 paper, starting with a concave program formulation based on segment-level consideration sets called SDCP , we add a class of constraints called product constraints, that project onto subsets of intersections. In addition we propose a natural direct tightening of the SDCP called κSDCP , and compare the performance of both methods on the benchmark data sets in the literature. Both the product constraints and the κSDCP method are very simple and easy to implement and are applicable to the case of overlapping segment consideration sets. In our computational testing on the benchmark data sets in the literature, SDCP with product constraints achieves the CDLP value at a fraction of the CPU time taken by column generation and we believe is a very promising approach for quickly approximating CDLP when segment consideration sets overlap and the consideration sets themselves are relatively small.

Adaptive dynamic programming algorithms for sequential appointment scheduling with patient preferences

“…In order to avoid the disadvantage of ADP, another group of approximate algorithm is developed recently. This method transforms the MDP model into a liner programming model, and then solves the latter in view of its simplicity [24,25]. However, whether the LP model is resolvable greatly depends on the scale of the model (the number of variables and constraints).…”

New Formulations for Choice Network Revenue Management

“…These are valid constraints for SDCP as shown in Meissner et al (2013). We will just reduce the number of variables by replacing appropriate summations by new variables as done in (21)-so validity of the resulting constraints follows from Meissner et al (2013) and a simple feasibility check.…”

“…We then tighten the bound in two different ways: (i) by a simulation-based randomized concave programming (RCP) method, similar to the randomized linear program (RLP) in the independent-class model (where a one-to-one map between products and segments are assumed, Talluri and van Ryzin 1999); (ii) by adding valid constraints to SDCP. Our cuts are a specialization of the ones developed in Meissner et al (2013) to the compact formulation of Gallego et al (2010) for the multinomial-logit (MNL) choice model. The advantage of these cuts is that the space of the resulting program is exponential only in the number of products in the intersection of two segments' consideration sets, rather than the size of the consideration sets as in Meissner et al (2013).…”

“…Meissner et al (2013) develop a set of valid constraints for SDCP called product cuts (PC-equalities). They are of the following form:…”

Choice-Based Demand Management and Vehicle Routing in E-Fulfillment

“…Vulcano et al [2010]; Suh and Aydin [2011]; Meissner et al [2013]; Vulcano et al [2012]) and although the applications come almost exclusively from the airline sector, the choice behavior follows a similar structure to the choice of delivery slots. The standard problem encountered in the literature considers the optimal pricing of multiple, substitutable flights offered by the same carrier, between the same origin and destination airports.…”