A randomized concave programming method 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 CDLP which has an exponential number of columns. However, 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 we propose a new approach called SDCP to solving CDLP 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 non-overlapping 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 (RCP) method, and (ii) by adding cuts to a recent compact formulation of the problem for a latent multinomial-choice model of demand (SBLP+). This latter approach turns out to be very effective, essentially obtaining CDLP value, and excellent revenue performance in simulations, even for overlapping segments. By formulating the problem as a separation problem, we give insight into why CDLP is easy for the MNL with non-overlapping considerations sets and why generalizations of MNL pose difficulties. We perform numerical simulations to determine the revenue performance of all the methods on reference data sets in the literature.assortment optimization, randomized algorithms, network revenue management.

Dynamic pricing under customer choice behavior for revenue management in passenger railway networks

Revenue management (RM) for passenger railway is a small but active research field with an increasing attention during the past years. However, a detailed look into existing research shows that most of the current models in theory rely on traditional RM techniques and that advanced models are rare. This thesis aims to close the gap by proposing a state-of-the-art passenger railway pricing model that covers the most important properties from practice, with a special focus on the German railway network and long-distance rail company Deutsche Bahn Fernverkehr (DB). The new model has multiple advantages over DB’s current RM system. Particularly, it uses a choice-based demand function rather than a traditional independent demand model, is formulated as a network model instead of the current leg-based approach and finally optimizes prices on a continuous level instead of controlling booking classes. Since each itinerary in the network is considered by multiple heterogeneous customer segments (e.g., differentiated by travel purpose, desired departure time) a discrete mixed multinomial logit model (MMNL) is applied to represent demand. Compared to alternative choice models such as the multinomial logit model (MNL) or the nested logit model (NL), the MMNL is significantly less considered in pricing research. Furthermore, since the resulting deterministic multi-product multi-resource dynamic pricing model under the MMNL turns out to be non- linear non-convex, an open question is still how to obtain a globally optimal solution. To narrow this gap, this thesis provides multiple approaches that make it able to derive a solution close to the global optimum. For medium-sized networks, a mixed-integer programming approach is proposed that determines an upper bound close to the global optimum of the original model (gap < 1.5%). For large-scale networks, a heuristic approach is presented that significantly decreases the solution time (by factor up to 56) and derives a good solution for an application in practice. Based on these findings, the model and heuristic are extended to fit further price constraints from railway practice and are tested in an extensive simulation study. The results show that the new pricing approach outperforms both benchmark RM policies (i.e., DB’s existing model and EMSR-b) with a revenue improvement of approx. +13-15% over DB’s existing approach under a realistic demand scenario. Finally, to prepare data for large-scale railway networks, an algorithm is presented that automatically derives a large proportion of necessary data to solve choice-based network RM models. This includes, e.g., the set of all meaningful itineraries (incl. transfers) and resources in a network, the corresponding resource consumption and product attribute values such as travel time or number of transfers. All taken together, the goal of this thesis is to give a broad picture about choice-based dynamic pricing for passenger railway networks

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 paper, 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 (σPC), that project onto subsets of intersections. In addition we propose a natural direct tightening of the SDCP called ESDCPκ, and compare the performance of both methods on the benchmark data sets in the literature. In our computational testing on the benchmark data sets in the literature, 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

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 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 ?SDCP, and compare the performance of both methods on the benchmark data sets in the literature. Both the product constraints and the ?SDCP 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.discrete-choice models, network revenue management, optimization

On bounds for network revenue management

The Network Revenue Management problem can be formulated as a stochastic dynamic programming problem (DP or the\optimal" solution V *) whose exact solution is computationally intractable. Consequently, a number of heuristics have been proposed in the literature, the most popular of which are the deterministic linear programming (DLP) model, and a simulation based method, the randomized linear programming (RLP) model. Both methods give upper bounds on the optimal solution value (DLP and PHLP respectively). These bounds are used to provide control values that can be used in practice to make accept/deny decisions for booking requests. Recently Adelman [1] and Topaloglu [18] have proposed alternate upper bounds, the affine relaxation (AR) bound and the Lagrangian relaxation (LR) bound respectively, and showed that their bounds are tighter than the DLP bound. Tight bounds are of great interest as it appears from empirical studies and practical experience that models that give tighter bounds also lead to better controls (better in the sense that they lead to more revenue). In this paper we give tightened versions of three bounds, calling themsAR (strong Affine Relaxation), sLR (strong Lagrangian Relaxation) and sPHLP (strong Perfect Hindsight LP), and show relations between them. Speciffically, we show that the sPHLP bound is tighter than sLR bound and sAR bound is tighter than the LR bound. The techniques for deriving the sLR and sPHLP bounds can potentially be applied to other instances of weakly-coupled dynamic programming.revenue management, bid-prices, relaxations, bounds

Tolling, Capacity Selection and Equilibrium Problems with Equilibrium Constraints

An Equilibrium problem with an equilibrium constraint is a mathematical construct that can be applied to private competition in highway networks. In this paper we consider the problem of finding a Nash Equilibrium regarding competition in toll pricing on a network utilising 2 alternative algorithms. In the first algorithm, we utilise a Gauss Siedel fixed point approach based on the cutting constraint algorithm for toll pricing. In the second algorithm, we extend an existing sequential linear complementarity approach for finding Nash equilibrium subject to Wardrop Equilibrium constraints. Finally we consider how the equilibrium may change between the Nash competitive equilibrium and a collusive equilibrium where the two players co-operate to form the equivalent of a monopoly operation